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8,9 [
In triangle $ABC$, angle $B$ is equal to $\arccos \frac{15}{17}$. On side $AC$, a point $K$ is taken such that $AK=12$, $KC=4$. Find the radius of the circle passing through vertex $B$, touching side $AC$ at point $K$, and touching the circumcircle of triangle $ABC$.
#
|
12
|
4/8
|
3. Determine all prime numbers $p$ for which $\frac{p^{2}-p-2}{2}$ is a cube of a natural number.
|
2
|
0/8
|
Consider all triangular pyramids whose base edges are $1, 1, c$, and the side edges opposite these, respectively, are $1, c, c$. Determine the range of possible values for $c$.
|
\left( \dfrac{\sqrt{5} - 1}{2}, \dfrac{\sqrt{5} + 1}{2} \right)
|
1/8
|
3. The range of the function $f(x)=[2 \sin x \cdot \cos x]+[\sin x+\cos x]$ is $\qquad$ (where $[x]$ denotes the greatest integer less than or equal to $x$). Answer: $\{-1,0,1,2\}$.
|
\{-1, 0, 1, 2\}
|
1/8
|
Let there be two different media I and II on either side of the $x$-axis (Figure 2), with the speed of light in media I and II being $c_{1}$ and $c_{2}$, respectively. If a light ray starts from point A in medium I and reaches point B in medium II, what path should the light take to minimize the time of travel?
|
\frac{\sin \theta_1}{\sin \theta_2} = \frac{c_1}{c_2}
|
4/8
|
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$.
|
6
|
4/8
|
Two circles meet at A and B and touch a common tangent at C and D. Show that triangles ABC and ABD have the same area.
|
\text{Triangles } ABC \text{ and } ABD \text{ have the same area.}
|
0/8
|
4. As shown in Figure 3, $A B$ and $C D$ are two chords of $\odot O$, $\angle A O B$ and $\angle O C D$ are supplementary, $\angle O A B$ and $\angle C O D$ are equal. Then the value of $\frac{A B}{C D}$ is $\qquad$
|
\dfrac{3 + \sqrt{5}}{2}
|
5/8
|
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan Chen[/i]
|
336
|
4/8
|
N1. Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$.
|
(2, 4)
|
3/8
|
69. The area of $\triangle A B C$ is $1, D, E$ are points on sides $A B, A C$ respectively, $B E, C D$ intersect at point $P$, and the area of quadrilateral $B C E D$ is twice the area of $\triangle P B C$. Find the maximum value of the area of $\triangle P D E$.
---
The area of $\triangle A B C$ is $1, D, E$ are points on sides $A B, A C$ respectively, $B E, C D$ intersect at point $P$, and the area of quadrilateral $B C E D$ is twice the area of $\triangle P B C$. Find the maximum value of the area of $\triangle P D E$.
|
5\sqrt{2} -7
|
2/8
|
Problem 4. Given an isosceles triangle $ABC$, where $\overline{AC}=\overline{BC}$. On the leg $AC$ are chosen points $M$ and $N$ such that $\measuredangle M B A=\measuredangle C B N$ and $\overline{M N}=\overline{B M}$, with point $M$ between points $A$ and $N$. Determine $\measuredangle N B A$.
|
60^\circ
|
1/8
|
1. In the room, there are knights who always tell the truth, and liars who always lie. 10 of them said: "In this room, there are more knights than liars." 12 said: "In this room, there are more liars than knights." The remaining 22 said: "In this room, there are an equal number of liars and knights." How many liars could there be in the room? If there are multiple possible answers, write them in any order separated by a semicolon.
|
22
|
0/8
|
On the sides \( AB, AC \), and \( BC \) of the equilateral triangle \( ABC \), points \( C_{1}, B_{1} \), and \( A_{1} \) are located respectively, such that the triangle \( A_{1} B_{1} C_{1} \) is equilateral. The segment \( BB_{1} \) intersects the side \( C_{1} A_{1} \) at point \( O \), with \( \frac{BO}{OB_{1}}=k \). Find the ratio of the area of triangle \( ABC \) to the area of triangle \( A_{1} B_{1} C_{1} \).
|
1 + 3k
|
0/8
|
4. The numbers $x$ and $y$ are such that the equations $\sin y + \sin x + \cos 3x = 0$ and $\sin 2y - \sin 2x = \cos 4x + \cos 2x$ are satisfied. What is the greatest value that the sum $\cos y + \cos x$ can take?
|
1 + \dfrac{\sqrt{2 + \sqrt{2}}}{2}
|
0/8
|
134. Someone said to his friend: "Give me 100 rupees, and I will be twice as rich as you," to which the latter replied: "If you give me just 10 rupees, I will be six times richer than you." The question is: how much did each have?
|
40
|
0/8
|
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).
|
(1/20)^{2012}
|
0/8
|
9. (Adapted from the 1st "Hope Cup" Senior High School Competition) Let the function $f(n)=k$, where $n$ is a natural number, and $k$ is the digit at the $n$-th position after the decimal point of the irrational number $\pi=3.1415926535 \cdots$, with the rule that $f(0)=3$. Let $F_{n}=$ $\underbrace{f\{f\{f\{f\{f}(n)\} \cdots\}\}$, then $F[f(1990)+f(5)+f(13)]=$ $\qquad$.
|
1
|
4/8
|
3. Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of 3 , the number of checkers in each column is a multiple of 5 .
Assuming the top left corner of the board is shown below, how many checkers are used in total?
|
30
|
3/8
|
1. A seagull is being fed from a moving boat. A piece of bread is thrown down, the seagull takes 3 seconds to pick up the piece from the sea surface, and then it takes 12 seconds to catch up with the boat. Upon entering the bay, the boat reduces its speed by half. How much time will it now take the seagull to catch up with the boat after picking up the piece of bread?
Om vem: 2 seconds.
|
2
|
5/8
|
For integers $n$ (where $n \geq 2$), let real numbers $x_1, x_2, \ldots, x_n$ satisfy the conditions:
$$
\begin{array}{l}
x_1 + x_2 + \cdots + x_n = 0, \\
x_1^2 + x_2^2 + \cdots + x_n^2 = 1.
\end{array}
$$
For any set $A \subseteq \{1, 2, \ldots, n\}$, define $S_A = \sum_{i \in A} x_i$ (if $A$ is an empty set, then $S_A = 0$).
Prove: For any positive real number $\lambda$, the number of sets $A$ satisfying $S_A \geq \lambda$ is at most $\frac{2^{n-3}}{\lambda^2}$, and determine all $x_1, x_2, \ldots, x_n$ for which equality holds.
|
\dfrac{2^{n-3}}{\lambda^2}
|
4/8
|
8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$
|
4102
|
1/8
|
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.
|
74
|
4/8
|
The equation $\sqrt[3]{\sqrt[3]{x - \frac38} - \frac38} = x^3+ \frac38$ has exactly two real positive solutions $r$ and $s$ . Compute $r + s$ .
|
\frac{1+\sqrt{13}}{4}
|
5/8
|
Complex numbers $a$ , $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$ , and $|a|^2+|b|^2+|c|^2=250$ . The points corresponding to $a$ , $b$ , and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$ . Find $h^2$ .
|
125
|
0/8
|
<u>Round 1</u>**1.1.** A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$ . What is the minimum number of groups needed?**1.2.** On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has
forgotten which is which, so he labels them in random order. The probability that he labels all three countries
correctly can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .**1.3.** In a class of $40$ sixth graders, the class average for their final test comes out to be $90$ (out of a $100$ ). However, a student brings up an issue with problem $5$ , and $10$ students receive credit for this question, bringing the class average to a $90.75$ . How many points was problem $5$ worth?
<u>Round 2</u>**2.1.** Compute $1 - 2 + 3 - 4 + ... - 2022 + 2023$ .**2.2.** In triangle $ABC$ , $\angle ABC = 75^o$ . Point $D$ lies on side $AC$ such that $BD = CD$ and $\angle BDC$ is a right angle. Compute the measure of $\angle A$ .**2.3.** Joe is rolling three four-sided dice each labeled with positive integers from $1$ to $4$ . The probability the sum of the numbers on the top faces of the dice is $6$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p + q$ .
<u>Round 3</u>**3.1.** For positive integers $a, b, c, d$ that satisfy $a + b + c + d = 23$ , what is the maximum value of $abcd$ ?**3.2.** A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games?**3.3.** For a triangle $\vartriangle ABC$ , let $M, N, O$ be the midpoints of $AB$ , $BC$ , $AC$ , respectively. Let $P, Q, R$ be points on $AB$ , $BC$ , $AC$ such that $AP =\frac13 AB$ , $BQ =\frac13 BC$ , and $CR =\frac13 AC$ . The ratio of the areas of $\vartriangle MNO$ and $\vartriangle P QR$ can be expressed as $\frac{m}{n}$ , where $ m$ and $n$ are relatively prime positive integers. Find $m + n$ .
<u>Round 4</u>**4.1.** $2023$ has the special property that leaves a remainder of $1$ when divided by $2$ , $21$ when divided by $22$ , and $22$ when divided by $23$ . Let $n$ equal the lowest integer greater than $2023$ with the above properties. What is $n$ ?**4.2.** Ants $A, B$ are on points $(0, 0)$ and $(3, 3)$ respectively, and ant A is trying to get to $(3, 3)$ while ant $B$ is trying to get to $(0, 0)$ . Every second, ant $A$ will either move up or right one with equal probability, and ant $B$ will move down or left one with equal probability. The probability that the ants will meet each other be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$ .**4.3.** Find the number of trailing zeros of $100!$ in base $ 49$ .
PS. You should use hide for answers. Rounds 5-9 have been posted [here](https://artofproblemsolving.com/community/c3h3129723p28347714). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).
|
8
|
0/8
|
In $\triangle ABC$ , $AB= 425$ , $BC=450$ , and $AC=510$ . An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$ , find $d$ .
|
306
|
3/8
|
Find all positive integers $ n $ with the following property: It is possible to fill a $ n \times n $ chessboard with one of arrows $ \uparrow, \downarrow, \leftarrow, \rightarrow $ such that
1. Start from any grid, if we follows the arrows, then we will eventually go back to the start point.
2. For every row, except the first and the last, the number of $ \uparrow $ and the number of $ \downarrow $ are the same.
3. For every column, except the first and the last, the number of $ \leftarrow $ and the number of $ \rightarrow $ are the same.
|
2
|
1/8
|
<u>Set 4</u>**4.1** Triangle $T$ has side lengths $1$ , $2$ , and $\sqrt7$ . It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.
**4.2** Let $T$ be the answer from the previous problem. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$ , find $k$ .
**4.3** Let $T$ be the answer from the previous problem. $T^2$ congruent squares are arranged in the configuration below (shown for $T = 3$ ), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for $T$ , consider small cases and try to find a general formula in terms of $T$ , such a formula does exist.)
PS. You should use hide for answers.
|
7
|
0/8
|
Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$ , and $k>\frac{|A|}{2}$ ;
(2) for any $a,b\in A$ , there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that $a,b\in A_r\cap A_s\cap A_t$ ;
(3) for any integer $i,j\, (1\leq i<j\leq n)$ , $|A_i\cap A_j|\leq 3$ .
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$ .
|
11
|
0/8
|
3. Given a point on the plane that does not coincide with the origin. How many different points can be obtained from it by sequentially applying symmetries relative to the $O y$ axis and the line $y=-x$ (in any order and any number of times)? If the point itself can also be obtained, it should be counted. If different points can have different answers, list them in any order separated by a semicolon.
|
4;8
|
0/8
|
1. Each of the three equal containers can hold 600 liters of liquid and each is filled exactly to half. From the first container to the second, we pour $18 \%$ of the liquid. Then from the second container to the third, we pour $2 / 3$ of the liquid. After that, from the third container to the first, we pour $3 / 8$ of the liquid and an additional 5 liters of liquid. How many liters of liquid need to be poured from the container with the most liquid to the container with the least liquid so that the amount of liquid in these containers is equal?
|
167
|
0/8
|
4、A mover needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen), and he can carry 40 buns each time. However, since he is very greedy, he will eat 1 bun whether he is walking from the kitchen to the construction site or from the construction site back to the kitchen. How many buns can this mover transport to the construction site at most?
|
191
|
0/8
|
The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is $\tfrac{a+\sqrt{b}}{c}$ where $a$ , $b$ , and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$ .
[asy]
size(200);
real f(real x) {return 1.2*exp(2/3*log(16-x^2));}
path Q=graph(f,-3.99999,3.99999);
path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle};
for(int k=0;k<3;++k)
{
fill(P[k],grey); draw(P[k]);
}
draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]
|
20
|
0/8
|
Let $\triangle ABC$ be a triangle with $AB = 7$ , $AC = 8$ , and $BC = 3$ . Let $P_1$ and $P_2$ be two distinct points on line $AC$ ( $A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ( $A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$ .
|
3
|
0/8
|
p1. Suppose the number $\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3}$ is integer. Calculate it.
p2. A house A is located $300$ m from the bank of a $200$ m wide river. $600$ m above and $500$ m from the opposite bank is another house $B$ . A bridge has been built over the river, that allows you to go from one house to the other covering the minimum distance. What distance is this and in what place of the riverbank is at the bridge?
p3. Show that it is not possible to find $1992$ positive integers $x_1, x_2, ..., x_{1992}$ that satisfy the equation: $$ \sum_{j=1}^{1992}2^{j-1}x_j^{1992}=2^{1992}\prod_{j=2}^{1992}x_j $$ p4. The following division of positive integers with remainder has been carried out. Every cross represents one digit, and each question mark indicates that there are digits, but without specifying how many. Fill in each cross and question mark with the missing digits.
p5. Three externally tangent circles have, respectively, radii $1$ , $2$ , $3$ . Determine the radius of the circumference that passes through the three points of tangency.
p6. $\bullet$ It is possible to draw a continuous curve that cuts each of the segments of the figure exactly one time? It is understood that cuts should not occur at any of the twelve vertices. $\bullet$ What happens if the figure is on a sphere?
p7. Given any $\vartriangle ABC$ , and any point $P$ , let $X_1$ , $X_2$ , $X_3$ be the midpoints of the sides $AB$ , $BC$ , $AC$ respectively, and $Y_1$ , $Y_2$ , $Y_3$ midpoints of $PC$ , $PA$ , $PB$ respectively. Prove that the segments $X_1Y_1$ , $X_2Y_2$ , $X_3Y_3$ concur.
|
2
|
4/8
|
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called *good* if one can put on each of these segments a positive integer not larger than $m$ , so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a *good* number $m$ .
|
11
|
0/8
|
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB=CD=3$ , $BC=DE=10$ , and $AE=14$ . The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? $\textbf{(A) }129\qquad
\textbf{(B) }247\qquad
\textbf{(C) }353\qquad
\textbf{(D) }391\qquad
\textbf{(E) }421\qquad$
|
391
|
0/8
|
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
|
85
|
1/8
|
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$ , let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute $0$ . On the $i$ -th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$ , the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end.
Note: If $r_i=s_i$ , the couple on $s_i$ stays there and does not drop out.
|
0
|
3/8
|
In cyclic quadrilateral $ABCD$ , $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$ . The diagonals of $ABCD$ meet at $E$ . If $\frac{BE}{ED} = 2$ and $CD = 60$ , compute $AD$ . (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)
|
\frac{30}{\sqrt{7}}
|
0/8
|
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$ , respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$ , respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$ . If $O_1O_2=25$ , find the area of quadrilateral $O_1AO_2B$ .
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(12cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179; /* image dimensions */
/* draw figures */
draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2));
draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2));
draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2));
/* dots and labels */
dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle);
label(" $O_1$ ", (-4.252707018231291,-1.545940604327141), N * labelscalefactor);
dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle);
label(" $O_2$ ", (21.704189347819636,-1.250863978037686), NE * labelscalefactor);
dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle);
label(" $A$ ", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor);
dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle);
label(" $B$ ", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
*Proposed by Deyuan Li and Andrew Milas*
|
84
|
4/8
|
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$ . Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$ .
|
\sqrt{3}
|
4/8
|
p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$ .
Find the number of subsets of $H$ .
p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$ , then the magician will know that the hidden number is $3$ .
a. Give an example of your own from the above process.
b. Explain mathematically the general form of the process.
p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn?
p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made.
p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$ ?
|
64
|
1/8
|
**p1.** $17.5\%$ of what number is $4.5\%$ of $28000$ ?**p2.** Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$ . The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p3.** In the $xy$ -plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$ . Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at?**p4.** What are the last two digits of the sum of the first $2021$ positive integers?**p5.** A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$ th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p6.** How many terms are in the arithmetic sequence $3$ , $11$ , $...$ , $779$ ?**p7.** Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$ ?**p8.** What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$ ?**p9.** Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$ , $AB = 10$ , $BC = 9$ , and the area of $\vartriangle ABC$ is $36$ . Compute the length of $AC$ .
**p10.** If $x + y - xy = 4$ , and $x$ and $y$ are integers, compute the sum of all possible values of $ x + y$ .**p11.** What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap?**p12.** $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$ . Compute the smallest possible value of $N$ .**p13.** Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years?**p14.** Say there is $1$ rabbit on day $1$ . After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$ ?**15.** Ajit draws a picture of a regular $63$ -sided polygon, a regular $91$ -sided polygon, and a regular $105$ -sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have?**p16.** Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m + n$ .**p17.** Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$ 's adjacent.**p18.** From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$ , where $a$ and $b$ are positive integers. Compute $a + b$ .**p19.** Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$ . He starts by putting the first marble in bucket $1$ , the second marble in bucket $2$ , the third marble in bucket $3$ , etc. After placing a marble in bucket $9$ , he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag?
**p20.** What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$ ?
PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).
|
1
|
0/8
|
4. 12 baskets of apples and 14 baskets of pears weigh 6 c 92 kg. Moreover, the weight of one basket of pears is 10 kg less than the weight of one basket of apples. How much does one basket of pears and one basket of apples weigh separately?
|
32
|
2/8
|
Every time my brother tells the truth, our grandmother sneezes. One day, my brother said that he got a "5" in math, but grandmother did not sneeze. Then, with a slight doubt in his initial words, he said he got a "4", and grandmother sneezed. Encouraged by grandmother's sneezing, he confirmed that he definitely got at least a 3, but grandmother did not sneeze anymore. So, what grade did my brother actually get in math?
|
2
|
0/8
|
a) Each vertex at the base of a triangle is connected by straight lines to \( n \) points located on the opposite side. Into how many parts do these lines divide the triangle?
b) Each of the three vertices of a triangle is connected by straight lines to \( n \) points located on the opposite side of the triangle. Into how many parts do these lines divide the triangle, assuming that no three of them intersect at the same point?
|
(n + 1)^2
|
1/8
|
Question 13: Given that $A$ and $B$ are two subsets of $\{1,2, \ldots, 100\}$, satisfying: $|A|=|B|, A \cap B=\emptyset$, and for any $x \in A, 2 x+2 \in B$. Try to find the maximum value of $|A \cup B|$.
---
The translation maintains the original format and line breaks as requested.
|
66
|
0/8
|
The combination for a certain lock is set by assigning either the number 0 or 1 to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$ and simultaneously coloring each vertex either red or blue, such that any two adjacent vertices must share at least one same number or color. How many different possible combinations are there for this lock?
|
3^n + (-1)^n + 2
|
0/8
|
471. A motorcyclist left A for B and at the same time a pedestrian set off from B to A. Upon meeting the pedestrian, the motorcyclist gave him a ride, brought him to A, and immediately set off again for B. As a result, the pedestrian reached A 4 times faster than he would have if he had walked the entire way. How many times faster would the motorcyclist have arrived in B if he had not had to return?
|
\dfrac{11}{4}
|
4/8
|
5. Let $n$ be a given positive integer. If a set $K$ of integer points in the plane satisfies the following condition, it is called "connected": for any pair of points $R, S \in K$, there exists a positive integer $l$ and a sequence of points in $K$
$$
R=T_{0}, T_{1}, \cdots, T_{l}=S,
$$
where the distance between each $T_{i}$ and $T_{i+1}$ is 1.
For such a set $K$, define
$$
\Delta(K)=\{\overrightarrow{R S} \mid R, S \in K\} \text {. }
$$
For all connected sets $K$ of $2 n+1$ integer points in the plane, find the maximum possible value of $|\Delta(K)|$.
|
2n^2 + 4n + 1
|
0/8
|
[ Signs of Perpendicularity]
The height of the triangular pyramid $A B C D$, dropped from vertex $D$, passes through the point of intersection of the heights of triangle $A B C$. In addition, it is known that $D B=3, D C=2, \angle B D C=90^{\circ}$. Find the ratio of the area of face $A D B$ to the area of face $A D C$.
#
|
\dfrac{3}{2}
|
2/8
|
## Task 5 - 010735
Rolf claims to know a math problem that uses only the number 7 and whose result is the year 1962.
a) Try to set up such a math problem!
b) Can a math problem also be set up that uses only the number 1962 and whose result is 7? If yes, provide this math problem!
|
[(7 + 7) \times (7 + 7)] \times [7 + \frac{7 + 7 + 7}{7}] + \frac{7 + 7}{7} = 1962
|
2/8
|
6. A square is inscribed in a circle, and a rectangle is inscribed in the square. Another circle is circumscribed about the rectangle, and a smaller circle is tangent to three sides of the rectangle, as shown below. The shaded area between the two larger circles is eight times the area of the smallest circle, which is also shaded. What fraction of the largest circle is shaded?
|
\dfrac{9}{25}
|
0/8
|
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.
|
n
|
2/8
|
13. There are five cards below, each with a number: 00123. Using them, many different five-digit numbers can be formed. Find the average of all these five-digit numbers.
|
21111
|
2/8
|
97.1. Let A be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of A satisfying $x<y$ and $x+y=z$.
|
9
|
5/8
|
13. As shown in Figure 5, in the circle $\odot O$ with radius $r$, $AB$ is the diameter, $C$ is the midpoint of $\overparen{AB}$, and $D$ is the one-third point of $\overparen{CB}$. Moreover, the length of $\overparen{DB}$ is twice the length of $\overparen{CD}$; connect $AD$ and extend it to intersect the tangent line $CE$ of $\odot O$ at point $E$ ($C$ is the point of tangency). Find the length of $AE$.
|
2r
|
5/8
|
8.1. Usually, we write the date in the format of day, month, and year (for example, 17.12.2021). In the USA, however, it is customary to write the month number, day number, and year in sequence (for example, 12.17.2021). How many days in a year cannot be determined unequivocally by its writing?
|
132
|
4/8
|
Let $f$ be a function defined on and taking values in the set of non-negative real numbers. Find all functions $f$ that satisfy the following conditions:
(i) $f[x \cdot f(y)] \cdot f(y)=f(x+y)$;
(ii) $f(2)=0$;
(iii) $f(x) \neq 0$, when $0 \leq x < 2$.
|
f(x) = \begin{cases} \dfrac{2}{2 - x} & \text{if } 0 \leq x < 2, \\ 0 & \text{if } x \geq 2. \end{cases}
|
3/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Carol`, `Bob`, `Alice`, `Peter`
- Each person prefers a unique type of vacation: `mountain`, `camping`, `beach`, `cultural`, `city`, `cruise`
- Everyone has something unique for lunch: `spaghetti`, `soup`, `stir fry`, `stew`, `grilled cheese`, `pizza`
- Each person has a unique favorite drink: `boba tea`, `root beer`, `water`, `milk`, `coffee`, `tea`
- People have unique heights: `tall`, `average`, `very tall`, `short`, `very short`, `super tall`
## Clues:
1. The person who enjoys mountain retreats and the person who prefers city breaks are next to each other.
2. Alice is in the fourth house.
3. The coffee drinker is the person who is a pizza lover.
4. The person who is very tall is directly left of the person who enjoys mountain retreats.
5. Peter is not in the first house.
6. The boba tea drinker is in the fourth house.
7. The person who loves stir fry is somewhere to the left of Eric.
8. The person who likes milk is Bob.
9. The person who loves the soup is not in the second house.
10. The person who loves the spaghetti eater is not in the second house.
11. The person who loves the stew is Eric.
12. The root beer lover is the person who is short.
13. The person who loves beach vacations is in the first house.
14. The person who loves stir fry is the person who enjoys mountain retreats.
15. The person who loves the soup is somewhere to the left of the person who loves the spaghetti eater.
16. The boba tea drinker is somewhere to the left of the person who is a pizza lover.
17. Arnold is the person who likes going on cruises.
18. The person who is very short is somewhere to the right of the person who has an average height.
19. The person who is tall is not in the fourth house.
20. The person who prefers city breaks is the coffee drinker.
21. Bob and the one who only drinks water are next to each other.
22. The person who loves the stew is the person who enjoys camping trips.
23. Arnold is the person who is very short.
24. Carol is in the fifth house.
What is the value of attribute Name for the person whose attribute Vacation is cultural? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Bob`, `Peter`, `Arnold`
- People have unique favorite book genres: `fantasy`, `romance`, `mystery`, `science fiction`, `biography`
- Each person has a unique level of education: `bachelor`, `master`, `high school`, `doctorate`, `associate`
## Clues:
1. The person who loves fantasy books is Alice.
2. Bob is directly left of the person with a bachelor's degree.
3. The person with a high school diploma is the person who loves science fiction books.
4. The person with an associate's degree is Bob.
5. The person who loves fantasy books is somewhere to the right of the person with a master's degree.
6. Arnold is the person with a high school diploma.
7. Eric is not in the second house.
8. The person who loves biography books is the person with a doctorate.
9. The person who loves science fiction books is in the third house.
10. The person who loves mystery books is not in the first house.
11. The person with a master's degree is somewhere to the left of the person with a doctorate.
What is the value of attribute Name for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
1/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Carol`, `Arnold`
- Everyone has a unique favorite cigar: `dunhill`, `blends`, `blue master`, `yellow monster`, `pall mall`, `prince`
- People have unique favorite book genres: `fantasy`, `romance`, `science fiction`, `mystery`, `historical fiction`, `biography`
## Clues:
1. The person partial to Pall Mall is Alice.
2. The Prince smoker is not in the sixth house.
3. Carol is directly left of the person who loves fantasy books.
4. The Prince smoker is not in the fifth house.
5. Peter is in the first house.
6. The person partial to Pall Mall is directly left of the Dunhill smoker.
7. The person partial to Pall Mall is in the second house.
8. Carol is the person who loves mystery books.
9. Bob is not in the fourth house.
10. The person who smokes Yellow Monster is not in the sixth house.
11. The person who loves biography books is not in the third house.
12. The person who loves historical fiction books is in the first house.
13. The person who smokes many unique blends is not in the fifth house.
14. The person who loves romance books is somewhere to the right of Eric.
15. There is one house between Arnold and the person who smokes Yellow Monster.
16. There is one house between the person who smokes many unique blends and Bob.
What is the value of attribute Name for the person whose attribute Cigar is prince? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
- Each person has a unique level of education: `bachelor`, `high school`, `associate`, `master`
- The mothers' names in different houses are unique: `Holly`, `Janelle`, `Aniya`, `Kailyn`
- The people keep unique animals: `fish`, `horse`, `cat`, `bird`
- Each person has a unique birthday month: `jan`, `feb`, `sept`, `april`
- People have unique favorite music genres: `rock`, `pop`, `jazz`, `classical`
## Clues:
1. The person with an associate's degree is The person whose mother's name is Aniya.
2. Peter is the person whose birthday is in September.
3. The person with a bachelor's degree is directly left of the person who loves pop music.
4. Arnold is The person whose mother's name is Janelle.
5. The person with a master's degree is Arnold.
6. The person whose mother's name is Janelle is the person who loves classical music.
7. The person whose birthday is in April is somewhere to the right of the person who loves rock music.
8. The cat lover is in the second house.
9. The fish enthusiast is in the third house.
10. Eric is the person whose birthday is in April.
11. The person who loves pop music is the person whose birthday is in January.
12. The person whose mother's name is Holly is the person with a bachelor's degree.
13. The person who keeps horses is the person with a master's degree.
14. The person whose mother's name is Aniya is the fish enthusiast.
15. Arnold is somewhere to the left of the fish enthusiast.
What is the value of attribute Mother for the person whose attribute Animal is cat? Please reason step by step, and put your final answer within \boxed{}
|
Holly
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
- Everyone has something unique for lunch: `grilled cheese`, `stew`, `pizza`, `spaghetti`
- People have unique hair colors: `blonde`, `black`, `red`, `brown`
- Each person has a favorite color: `red`, `white`, `yellow`, `green`
- The mothers' names in different houses are unique: `Holly`, `Aniya`, `Janelle`, `Kailyn`
## Clues:
1. Alice is the person whose favorite color is red.
2. The person who has black hair is the person who loves the stew.
3. The person who loves white is Arnold.
4. The person whose favorite color is red and The person whose mother's name is Kailyn are next to each other.
5. The person who loves yellow and the person who has black hair are next to each other.
6. The person who is a pizza lover is somewhere to the left of the person whose favorite color is green.
7. Peter is The person whose mother's name is Janelle.
8. The person who has blonde hair is the person who loves white.
9. The person who has red hair is Alice.
10. The person who loves white is somewhere to the right of the person who loves eating grilled cheese.
11. The person who loves white is not in the second house.
12. The person whose mother's name is Kailyn is the person who is a pizza lover.
13. The person whose mother's name is Holly is the person who has red hair.
What is the value of attribute Food for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}
|
grilled cheese
|
0/8
|
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Eric`, `Arnold`
- Each person has a unique birthday month: `jan`, `april`, `sept`
- Each person has a unique type of pet: `cat`, `dog`, `fish`
- Everyone has something unique for lunch: `pizza`, `spaghetti`, `grilled cheese`
- Each person prefers a unique type of vacation: `beach`, `mountain`, `city`
- Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`
## Clues:
1. The person whose birthday is in January is somewhere to the left of the person who is a pizza lover.
2. The person who owns a dog is in the second house.
3. The person whose birthday is in September is the person who loves the spaghetti eater.
4. The person whose birthday is in January is the person who enjoys mountain retreats.
5. The Watermelon smoothie lover is the person with an aquarium of fish.
6. The person whose birthday is in September is not in the third house.
7. Eric is somewhere to the left of the person who prefers city breaks.
8. The person who has a cat is Peter.
9. The person who loves beach vacations is in the third house.
10. The person who likes Cherry smoothies is the person whose birthday is in April.
What is the value of attribute Food for the person whose attribute Birthday is april? Please reason step by step, and put your final answer within \boxed{}
|
pizza
|
1/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Carol`, `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
- Each person prefers a unique type of vacation: `camping`, `cruise`, `beach`, `city`, `cultural`, `mountain`
- People have unique favorite sports: `volleyball`, `baseball`, `basketball`, `swimming`, `soccer`, `tennis`
## Clues:
1. The person who loves tennis and Peter are next to each other.
2. Bob is in the second house.
3. There is one house between the person who prefers city breaks and the person who loves tennis.
4. Peter is somewhere to the left of Alice.
5. Carol is the person who loves basketball.
6. The person who loves swimming is the person who likes going on cruises.
7. The person who loves tennis and the person who loves beach vacations are next to each other.
8. The person who loves swimming is Bob.
9. The person who loves volleyball is the person who enjoys camping trips.
10. Eric is in the first house.
11. Arnold is somewhere to the left of the person who enjoys mountain retreats.
12. Eric is the person who loves soccer.
13. The person who prefers city breaks is in the first house.
14. The person who enjoys camping trips is not in the sixth house.
What is the value of attribute House for the person whose attribute FavoriteSport is volleyball? Please reason step by step, and put your final answer within \boxed{}
|
5
|
2/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Bob`, `Carol`, `Arnold`, `Eric`
- Each person has an occupation: `engineer`, `doctor`, `lawyer`, `artist`, `nurse`, `teacher`
- Everyone has something unique for lunch: `soup`, `stew`, `grilled cheese`, `spaghetti`, `pizza`, `stir fry`
- Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `watermelon`, `lime`, `blueberry`, `desert`
- Each person prefers a unique type of vacation: `camping`, `cultural`, `city`, `cruise`, `beach`, `mountain`
- People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `samsung galaxy s21`, `xiaomi mi 11`, `huawei p50`
## Clues:
1. The person who likes Cherry smoothies is directly left of the person who enjoys camping trips.
2. The person who is a lawyer is the person who uses a Xiaomi Mi 11.
3. The person who is a doctor is Eric.
4. The person who is a doctor and the Desert smoothie lover are next to each other.
5. The person who is an artist is the person who prefers city breaks.
6. The person who loves beach vacations is Alice.
7. There are two houses between the person who is a pizza lover and the person who uses a Huawei P50.
8. The person who uses a Xiaomi Mi 11 is the person who enjoys camping trips.
9. The person who is an artist is directly left of the person who loves eating grilled cheese.
10. The person who uses a Google Pixel 6 is in the first house.
11. There is one house between the person who is a teacher and the person who loves stir fry.
12. There are two houses between the person who is a pizza lover and the person who is an engineer.
13. The person who is a pizza lover is somewhere to the left of the person who loves eating grilled cheese.
14. The Watermelon smoothie lover is the person who uses a Xiaomi Mi 11.
15. The person who loves eating grilled cheese is Carol.
16. Alice is the person who drinks Blueberry smoothies.
17. The person who likes Cherry smoothies is the person who loves the stew.
18. The person who loves the soup is the person who uses a Samsung Galaxy S21.
19. The person who uses an iPhone 13 is directly left of the Dragonfruit smoothie lover.
20. The person who loves the stew is somewhere to the left of Peter.
21. Arnold is somewhere to the right of the Desert smoothie lover.
22. The person who likes going on cruises is the person who uses a Google Pixel 6.
23. The person who goes on cultural tours is not in the fifth house.
24. The person who is a teacher is Alice.
What is the value of attribute Occupation for the person whose attribute Vacation is camping? Please reason step by step, and put your final answer within \boxed{}
|
lawyer
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`
- People own unique car models: `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`
- Each person has a unique birthday month: `feb`, `april`, `sept`, `jan`
- Everyone has something unique for lunch: `pizza`, `stew`, `spaghetti`, `grilled cheese`
- People have unique heights: `average`, `tall`, `very short`, `short`
- The mothers' names in different houses are unique: `Janelle`, `Kailyn`, `Holly`, `Aniya`
## Clues:
1. The person who is very short is the person who loves the stew.
2. The person who is a pizza lover is Alice.
3. The person who loves the spaghetti eater is the person who owns a Honda Civic.
4. The person who owns a Toyota Camry is somewhere to the left of the person who is a pizza lover.
5. The person whose birthday is in February is not in the fourth house.
6. Eric is the person who owns a Tesla Model 3.
7. The person who owns a Toyota Camry is somewhere to the right of The person whose mother's name is Janelle.
8. The person who is tall is in the fourth house.
9. The person whose birthday is in September is Arnold.
10. The person who has an average height is somewhere to the right of Arnold.
11. The person whose mother's name is Aniya and Alice are next to each other.
12. The person whose birthday is in April and The person whose mother's name is Holly are next to each other.
13. The person who is short is in the second house.
14. The person who loves the stew and The person whose mother's name is Janelle are next to each other.
What is the value of attribute Height for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{}
|
short
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
- Everyone has a favorite smoothie: `lime`, `dragonfruit`, `desert`, `watermelon`, `cherry`
- The people keep unique animals: `horse`, `dog`, `bird`, `fish`, `cat`
- The people are of nationalities: `german`, `swede`, `norwegian`, `brit`, `dane`
## Clues:
1. The Swedish person is directly left of the dog owner.
2. There are two houses between the dog owner and the British person.
3. The Dane is the person who keeps horses.
4. The bird keeper is somewhere to the right of the cat lover.
5. The dog owner is directly left of the person who drinks Lime smoothies.
6. Eric is the cat lover.
7. Bob is the bird keeper.
8. The person who likes Cherry smoothies is directly left of Peter.
9. The bird keeper is the Watermelon smoothie lover.
10. The Desert smoothie lover is the dog owner.
11. The person who keeps horses is in the third house.
12. The Norwegian is Alice.
What is the value of attribute Animal for the person whose attribute Smoothie is lime? Please reason step by step, and put your final answer within \boxed{}
|
horse
|
1/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Bob`, `Eric`, `Arnold`
- The people are of nationalities: `norwegian`, `german`, `dane`, `brit`, `swede`
- People have unique favorite book genres: `fantasy`, `biography`, `romance`, `mystery`, `science fiction`
- Everyone has something unique for lunch: `stir fry`, `grilled cheese`, `pizza`, `spaghetti`, `stew`
- Each person has a favorite color: `red`, `green`, `blue`, `yellow`, `white`
- The people keep unique animals: `bird`, `dog`, `cat`, `horse`, `fish`
## Clues:
1. The person who loves fantasy books is the Norwegian.
2. The cat lover and the person who loves biography books are next to each other.
3. The German is Bob.
4. The person who loves yellow is Bob.
5. The person whose favorite color is green is Peter.
6. There is one house between the Dane and the person who is a pizza lover.
7. The person who loves blue is somewhere to the left of the Dane.
8. The person who loves eating grilled cheese is somewhere to the left of the Norwegian.
9. The person who loves the spaghetti eater is Peter.
10. The person who keeps horses is Alice.
11. The fish enthusiast is directly left of the person who loves science fiction books.
12. There is one house between the Norwegian and Arnold.
13. The person who loves romance books is the British person.
14. There are two houses between the Norwegian and Alice.
15. The bird keeper is the person whose favorite color is red.
16. The dog owner is directly left of the fish enthusiast.
17. The person who loves the stew is the Norwegian.
What is the value of attribute Nationality for the person whose attribute BookGenre is biography? Please reason step by step, and put your final answer within \boxed{}
|
swede
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
- People use unique phone models: `iphone 13`, `samsung galaxy s21`, `google pixel 6`, `oneplus 9`
- The people are of nationalities: `norwegian`, `swede`, `dane`, `brit`
- People have unique favorite sports: `tennis`, `swimming`, `soccer`, `basketball`
- Each person has a favorite color: `red`, `white`, `yellow`, `green`
- Everyone has a favorite smoothie: `desert`, `watermelon`, `cherry`, `dragonfruit`
## Clues:
1. The person who loves basketball is the British person.
2. The person who likes Cherry smoothies is directly left of Arnold.
3. The person who uses an iPhone 13 is somewhere to the left of the person who loves tennis.
4. The person who loves soccer is the person whose favorite color is red.
5. Eric is the person who loves basketball.
6. Alice is the Watermelon smoothie lover.
7. The Norwegian is not in the fourth house.
8. Eric is somewhere to the left of the person who loves yellow.
9. The Desert smoothie lover is not in the second house.
10. The person who loves tennis is the person whose favorite color is green.
11. The person who uses a OnePlus 9 is the Norwegian.
12. Alice is the person who uses a Samsung Galaxy S21.
13. Eric is in the third house.
14. Peter is somewhere to the left of the Dane.
What is the value of attribute FavoriteSport for the person whose attribute PhoneModel is iphone 13? Please reason step by step, and put your final answer within \boxed{}
|
soccer
|
0/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Carol`, `Eric`, `Peter`, `Bob`, `Arnold`
- People have unique favorite music genres: `classical`, `hip hop`, `jazz`, `pop`, `rock`, `country`
- The mothers' names in different houses are unique: `Sarah`, `Penny`, `Aniya`, `Janelle`, `Kailyn`, `Holly`
- Each mother is accompanied by their child: `Alice`, `Fred`, `Timothy`, `Bella`, `Samantha`, `Meredith`
- People have unique heights: `very short`, `tall`, `short`, `very tall`, `super tall`, `average`
- The people keep unique animals: `bird`, `dog`, `horse`, `rabbit`, `cat`, `fish`
## Clues:
1. The person who loves pop music is the cat lover.
2. The rabbit owner is directly left of The person whose mother's name is Aniya.
3. The person whose mother's name is Holly is directly left of Carol.
4. The person whose mother's name is Holly is the person's child is named Alice.
5. The person whose mother's name is Holly is the person who loves classical music.
6. The person who loves jazz music is The person whose mother's name is Sarah.
7. The person's child is named Meredith is somewhere to the right of The person whose mother's name is Aniya.
8. The person who is super tall is The person whose mother's name is Holly.
9. The person who is the mother of Timothy is Bob.
10. The person who is very short is somewhere to the left of The person whose mother's name is Aniya.
11. Eric is the fish enthusiast.
12. The person's child is named Samantha is somewhere to the right of the person who is very tall.
13. The person who loves rock music is The person whose mother's name is Janelle.
14. There is one house between the person who keeps horses and the person's child is named Meredith.
15. The person's child is named Bella is somewhere to the right of Peter.
16. The fish enthusiast is somewhere to the left of the bird keeper.
17. The fish enthusiast is somewhere to the right of the person's child is named Alice.
18. There is one house between the person's child is named Bella and the person who loves rock music.
19. The person who is short is the cat lover.
20. Alice is directly left of the person who loves classical music.
21. The person's child is named Bella is The person whose mother's name is Aniya.
22. There are two houses between The person whose mother's name is Penny and the person who is short.
23. The person who loves hip-hop music is in the first house.
24. Carol is the person who is tall.
What is the value of attribute MusicGenre for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}
|
rock
|
0/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Bob`, `Carol`, `Eric`, `Alice`, `Arnold`
- Each person has a unique type of pet: `bird`, `dog`, `cat`, `rabbit`, `fish`, `hamster`
- Each person lives in a unique style of house: `victorian`, `ranch`, `modern`, `mediterranean`, `colonial`, `craftsman`
- Each person has a unique birthday month: `mar`, `sept`, `may`, `feb`, `jan`, `april`
## Clues:
1. The person with a pet hamster is somewhere to the right of the person whose birthday is in March.
2. The person whose birthday is in January is somewhere to the left of the person whose birthday is in September.
3. The person whose birthday is in May is in the second house.
4. The person living in a colonial-style house is in the second house.
5. Carol is in the third house.
6. The person in a Mediterranean-style villa is not in the sixth house.
7. The person with an aquarium of fish is somewhere to the right of Bob.
8. Eric is in the sixth house.
9. There is one house between the person who has a cat and the person residing in a Victorian house.
10. There are two houses between the person residing in a Victorian house and the person with a pet hamster.
11. The person in a Craftsman-style house is Arnold.
12. The person living in a colonial-style house is somewhere to the left of the person in a modern-style house.
13. The person with an aquarium of fish is not in the second house.
14. Peter is the person living in a colonial-style house.
15. The person whose birthday is in January is directly left of the person whose birthday is in April.
16. There is one house between the person who keeps a pet bird and the person in a modern-style house.
17. Carol is the person whose birthday is in March.
18. The person in a Craftsman-style house is in the fourth house.
19. The person who owns a dog is in the fourth house.
What is the value of attribute House for the person whose attribute Birthday is mar? Please reason step by step, and put your final answer within \boxed{}
|
3
|
0/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Carol`, `Peter`, `Bob`, `Alice`, `Arnold`
- People have unique favorite sports: `swimming`, `basketball`, `volleyball`, `tennis`, `soccer`, `baseball`
- Each person has a unique birthday month: `mar`, `feb`, `may`, `sept`, `april`, `jan`
- Everyone has a favorite smoothie: `cherry`, `lime`, `desert`, `watermelon`, `blueberry`, `dragonfruit`
- People have unique favorite music genres: `country`, `hip hop`, `jazz`, `rock`, `pop`, `classical`
## Clues:
1. The person whose birthday is in March is not in the fifth house.
2. Peter is not in the sixth house.
3. The person whose birthday is in September is not in the sixth house.
4. The person whose birthday is in March is somewhere to the left of the Desert smoothie lover.
5. The person who loves jazz music is Eric.
6. The person who loves swimming is somewhere to the right of the person who loves basketball.
7. The person who loves swimming is the person who loves jazz music.
8. There is one house between the person whose birthday is in September and the Desert smoothie lover.
9. The Watermelon smoothie lover is the person who loves jazz music.
10. There are two houses between the person whose birthday is in January and the Desert smoothie lover.
11. The person whose birthday is in February and the person who drinks Lime smoothies are next to each other.
12. There is one house between Bob and the person who loves jazz music.
13. The Watermelon smoothie lover is somewhere to the right of the person whose birthday is in April.
14. The person who loves rock music is Carol.
15. The person who loves volleyball is directly left of the Dragonfruit smoothie lover.
16. The person who loves rock music is directly left of the person whose birthday is in March.
17. The Desert smoothie lover is the person who loves soccer.
18. The person whose birthday is in January is the person who loves pop music.
19. There are two houses between the person who drinks Blueberry smoothies and the Desert smoothie lover.
20. The person whose birthday is in January is somewhere to the left of the person whose birthday is in March.
21. The Dragonfruit smoothie lover is directly left of the person who loves baseball.
22. Alice is the person who loves hip-hop music.
23. The person who loves classical music is in the sixth house.
What is the value of attribute Smoothie for the person whose attribute Birthday is feb? Please reason step by step, and put your final answer within \boxed{}
|
watermelon
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
- Each mother is accompanied by their child: `Meredith`, `Bella`, `Timothy`, `Fred`, `Samantha`
- They all have a unique favorite flower: `tulips`, `roses`, `lilies`, `daffodils`, `carnations`
## Clues:
1. The person who loves a bouquet of daffodils is not in the first house.
2. The person's child is named Fred is Bob.
3. The person's child is named Fred is not in the second house.
4. The person's child is named Fred is somewhere to the left of the person who loves a carnations arrangement.
5. Peter is not in the first house.
6. The person's child is named Fred and the person who loves the boquet of lilies are next to each other.
7. Arnold is the person who is the mother of Timothy.
8. The person's child is named Bella is somewhere to the right of Eric.
9. The person who is the mother of Timothy is not in the first house.
10. Eric is the person who loves the boquet of lilies.
11. There is one house between the person's child is named Meredith and the person's child is named Fred.
12. The person who loves the vase of tulips is in the second house.
What is the value of attribute House for the person whose attribute Flower is tulips? Please reason step by step, and put your final answer within \boxed{}
|
2
|
2/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Carol`, `Bob`, `Eric`
- Each person has an occupation: `artist`, `nurse`, `engineer`, `lawyer`, `teacher`, `doctor`
- The mothers' names in different houses are unique: `Holly`, `Penny`, `Aniya`, `Janelle`, `Sarah`, `Kailyn`
- People have unique favorite music genres: `pop`, `rock`, `classical`, `jazz`, `hip hop`, `country`
## Clues:
1. The person who is an engineer is directly left of the person who is an artist.
2. The person whose mother's name is Janelle is the person who is an artist.
3. The person who loves classical music is directly left of Peter.
4. The person whose mother's name is Janelle and the person who loves pop music are next to each other.
5. The person whose mother's name is Sarah is somewhere to the left of Arnold.
6. The person who loves country music is The person whose mother's name is Aniya.
7. The person whose mother's name is Penny is not in the second house.
8. Bob is the person who loves jazz music.
9. Carol is not in the sixth house.
10. The person who is an engineer is the person who loves classical music.
11. The person who is a nurse is somewhere to the right of the person who loves rock music.
12. There are two houses between Bob and The person whose mother's name is Penny.
13. The person who is a lawyer is directly left of the person who is a doctor.
14. The person whose mother's name is Penny is not in the third house.
15. The person who is an engineer is not in the fourth house.
16. The person who is a doctor is The person whose mother's name is Kailyn.
17. Carol is somewhere to the right of the person who loves country music.
18. The person who loves classical music is Alice.
19. The person who is a nurse is The person whose mother's name is Holly.
What is the value of attribute MusicGenre for the person whose attribute Name is Bob? Please reason step by step, and put your final answer within \boxed{}
|
jazz
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
- People own unique car models: `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
- Each person has a unique favorite drink: `tea`, `milk`, `water`, `root beer`, `coffee`
- The people are of nationalities: `dane`, `brit`, `german`, `norwegian`, `swede`
- People use unique phone models: `oneplus 9`, `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `huawei p50`
## Clues:
1. The person who uses a Google Pixel 6 is somewhere to the right of the Dane.
2. The person who uses a Google Pixel 6 is somewhere to the right of the root beer lover.
3. The person who owns a Tesla Model 3 is the British person.
4. The person who owns a Ford F-150 is the Dane.
5. Eric is the tea drinker.
6. The British person is directly left of the person who owns a Honda Civic.
7. Peter is somewhere to the right of the person who likes milk.
8. Bob is directly left of the Swedish person.
9. The person who uses an iPhone 13 is somewhere to the right of the person who uses a Huawei P50.
10. Arnold is the person who uses a Google Pixel 6.
11. Eric is not in the third house.
12. The person who owns a BMW 3 Series is somewhere to the right of the person who uses a Samsung Galaxy S21.
13. The person who owns a Tesla Model 3 is not in the first house.
14. Alice is somewhere to the left of the person who owns a Ford F-150.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Samsung Galaxy S21.
16. The German is somewhere to the right of the person who owns a Tesla Model 3.
17. Eric is not in the fourth house.
18. Peter is directly left of the person who uses a Samsung Galaxy S21.
19. The coffee drinker is Bob.
What is the value of attribute Name for the person whose attribute CarModel is ford f150? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Bob`, `Eric`, `Peter`, `Arnold`
- Each person has a unique type of pet: `hamster`, `bird`, `fish`, `dog`, `cat`
- Each person has a unique favorite drink: `coffee`, `milk`, `root beer`, `water`, `tea`
- People have unique favorite sports: `soccer`, `swimming`, `tennis`, `baseball`, `basketball`
- People use unique phone models: `oneplus 9`, `google pixel 6`, `iphone 13`, `huawei p50`, `samsung galaxy s21`
## Clues:
1. The root beer lover is the person who owns a dog.
2. The person who uses an iPhone 13 is the root beer lover.
3. There is one house between the person who owns a dog and the person who uses a Google Pixel 6.
4. Bob is somewhere to the right of the person who uses a OnePlus 9.
5. Alice is not in the fourth house.
6. Eric is in the fifth house.
7. The person who likes milk is the person who uses a Samsung Galaxy S21.
8. The person who loves baseball is the person who uses a Google Pixel 6.
9. The coffee drinker is in the third house.
10. Peter is the tea drinker.
11. The person who loves basketball is the person who owns a dog.
12. The person with a pet hamster is the person who loves tennis.
13. There are two houses between Alice and the tea drinker.
14. The person who loves soccer is the tea drinker.
15. The person who keeps a pet bird is directly left of the person with an aquarium of fish.
16. The person with an aquarium of fish is the person who likes milk.
What is the value of attribute Drink for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}
|
tea
|
0/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Carol`, `Alice`, `Eric`
- Everyone has something unique for lunch: `stir fry`, `stew`, `soup`, `pizza`, `grilled cheese`, `spaghetti`
- Each person has an occupation: `doctor`, `artist`, `engineer`, `teacher`, `lawyer`, `nurse`
- Each person has a unique type of pet: `dog`, `bird`, `hamster`, `cat`, `fish`, `rabbit`
- People have unique favorite music genres: `pop`, `country`, `jazz`, `classical`, `rock`, `hip hop`
- Everyone has a favorite smoothie: `blueberry`, `desert`, `watermelon`, `cherry`, `lime`, `dragonfruit`
## Clues:
1. The person who loves the spaghetti eater is the Dragonfruit smoothie lover.
2. The person who drinks Lime smoothies is in the second house.
3. Carol is somewhere to the left of the Desert smoothie lover.
4. The person who is an engineer is not in the second house.
5. The person who loves the soup is in the second house.
6. The person who owns a dog is not in the second house.
7. The person who is a pizza lover is Alice.
8. Arnold is somewhere to the left of Eric.
9. The person who loves classical music is the Dragonfruit smoothie lover.
10. The person who owns a rabbit is somewhere to the left of the Desert smoothie lover.
11. The person who is a nurse is not in the sixth house.
12. There are two houses between the person who is a nurse and the person who loves the spaghetti eater.
13. The person who likes Cherry smoothies is the person with an aquarium of fish.
14. The person who has a cat is Alice.
15. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
16. There is one house between the person who drinks Blueberry smoothies and the person who loves eating grilled cheese.
17. The person who loves classical music is somewhere to the left of the person who loves the stew.
18. The person who is a lawyer is the person who loves classical music.
19. The person who loves jazz music is Bob.
20. Bob is the person who drinks Lime smoothies.
21. The person who loves pop music is the person who is a teacher.
22. The person who is an engineer is directly left of the person who loves rock music.
23. The person who loves hip-hop music is not in the first house.
24. The person with an aquarium of fish is Peter.
25. The person who drinks Blueberry smoothies is the person who keeps a pet bird.
26. The person who has a cat is in the first house.
27. The person who is a doctor is not in the sixth house.
28. The person who owns a rabbit is not in the second house.
29. Carol is not in the fourth house.
What is the value of attribute Food for the person whose attribute Pet is cat? Please reason step by step, and put your final answer within \boxed{}
|
pizza
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
- Everyone has something unique for lunch: `grilled cheese`, `spaghetti`, `pizza`, `stew`
- People use unique phone models: `iphone 13`, `google pixel 6`, `samsung galaxy s21`, `oneplus 9`
- Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`
## Clues:
1. Peter is the person who is a pizza lover.
2. The person who is a pizza lover and the person who loves the spaghetti eater are next to each other.
3. The person who loves eating grilled cheese is directly left of Eric.
4. Eric is the person who loves cooking.
5. The person who uses a Google Pixel 6 is in the fourth house.
6. The photography enthusiast is in the second house.
7. The person who is a pizza lover is the person who uses a OnePlus 9.
8. The person who uses a Samsung Galaxy S21 is in the third house.
9. Arnold is directly left of the person who enjoys gardening.
What is the value of attribute Food for the person whose attribute House is 1? Please reason step by step, and put your final answer within \boxed{}
|
pizza
|
4/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
- The people keep unique animals: `fish`, `dog`, `horse`, `bird`, `cat`
- People own unique car models: `honda civic`, `bmw 3 series`, `ford f150`, `toyota camry`, `tesla model 3`
- Each person has a unique favorite drink: `coffee`, `tea`, `root beer`, `milk`, `water`
## Clues:
1. Arnold is not in the fifth house.
2. Alice and Bob are next to each other.
3. The person who owns a Tesla Model 3 is the person who keeps horses.
4. There are two houses between Arnold and the root beer lover.
5. Eric is in the second house.
6. The person who likes milk is the dog owner.
7. Arnold is the person who owns a Tesla Model 3.
8. The bird keeper is the person who owns a Honda Civic.
9. The one who only drinks water and the person who owns a Toyota Camry are next to each other.
10. The one who only drinks water is directly left of the cat lover.
11. There is one house between the person who owns a Ford F-150 and the fish enthusiast.
12. The dog owner is not in the fifth house.
13. The tea drinker is Bob.
What is the value of attribute Name for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
0/8
|
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Peter`, `Eric`
- Each person has a unique hobby: `gardening`, `photography`, `cooking`
- Each person prefers a unique type of vacation: `mountain`, `city`, `beach`
- Everyone has a unique favorite cigar: `pall mall`, `prince`, `blue master`
- The mothers' names in different houses are unique: `Janelle`, `Holly`, `Aniya`
- People have unique heights: `short`, `average`, `very short`
## Clues:
1. The person who loves beach vacations is somewhere to the right of the person who is very short.
2. The person whose mother's name is Aniya is somewhere to the right of the person who loves cooking.
3. Eric is not in the second house.
4. The person who is short is in the third house.
5. The person who prefers city breaks is the photography enthusiast.
6. The person who enjoys gardening is the person who enjoys mountain retreats.
7. The person partial to Pall Mall is the photography enthusiast.
8. The person who enjoys mountain retreats is The person whose mother's name is Holly.
9. The person who enjoys gardening is the person who smokes Blue Master.
10. Peter is The person whose mother's name is Aniya.
What is the value of attribute Height for the person whose attribute Cigar is blue master? Please reason step by step, and put your final answer within \boxed{}
|
very short
|
1/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
- Each person has a unique level of education: `high school`, `trade school`, `bachelor`, `doctorate`, `master`, `associate`
- Each person has a favorite color: `purple`, `blue`, `white`, `red`, `green`, `yellow`
- Each person prefers a unique type of vacation: `cultural`, `cruise`, `beach`, `mountain`, `camping`, `city`
- People have unique favorite book genres: `science fiction`, `biography`, `historical fiction`, `fantasy`, `romance`, `mystery`
## Clues:
1. Alice is the person with a master's degree.
2. There is one house between the person with an associate's degree and Peter.
3. The person who loves historical fiction books is somewhere to the left of the person who loves biography books.
4. The person who loves historical fiction books is the person with a high school diploma.
5. Bob is the person with a bachelor's degree.
6. Carol is somewhere to the right of Eric.
7. The person with a master's degree is somewhere to the right of the person who enjoys camping trips.
8. The person who loves science fiction books is somewhere to the left of the person who loves yellow.
9. The person who loves historical fiction books is somewhere to the left of the person whose favorite color is green.
10. The person with a high school diploma is not in the second house.
11. The person who loves beach vacations is the person with a doctorate.
12. The person who loves mystery books is somewhere to the left of the person with a bachelor's degree.
13. The person with a doctorate is in the first house.
14. The person with a high school diploma is directly left of the person who enjoys camping trips.
15. The person who loves romance books is not in the third house.
16. There is one house between the person who loves purple and the person with a bachelor's degree.
17. The person who loves biography books is the person whose favorite color is red.
18. There is one house between the person who loves fantasy books and the person who loves blue.
19. The person who goes on cultural tours is the person who loves blue.
20. Carol is not in the fifth house.
21. The person who loves purple is the person with a doctorate.
22. The person who enjoys mountain retreats is somewhere to the right of the person who loves science fiction books.
23. The person who prefers city breaks is the person whose favorite color is green.
What is the value of attribute House for the person whose attribute Name is Carol? Please reason step by step, and put your final answer within \boxed{}
|
2
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Peter`, `Bob`, `Alice`, `Eric`
- People own unique car models: `toyota camry`, `tesla model 3`, `bmw 3 series`, `honda civic`, `ford f150`
- Each person has a favorite color: `blue`, `red`, `green`, `white`, `yellow`
- The mothers' names in different houses are unique: `Janelle`, `Penny`, `Holly`, `Kailyn`, `Aniya`
## Clues:
1. The person who owns a Honda Civic is the person who loves yellow.
2. The person whose favorite color is red is the person who owns a Tesla Model 3.
3. The person who owns a BMW 3 Series is not in the fourth house.
4. The person whose mother's name is Aniya is the person who loves blue.
5. The person whose favorite color is green is Eric.
6. The person whose favorite color is red is somewhere to the left of the person who owns a Ford F-150.
7. Alice is directly left of Eric.
8. The person whose mother's name is Holly is in the first house.
9. Arnold is the person who loves white.
10. The person whose mother's name is Janelle is the person who loves white.
11. The person whose mother's name is Kailyn is Alice.
12. Arnold is somewhere to the left of Peter.
13. The person who owns a BMW 3 Series is Eric.
14. Bob and the person who owns a Ford F-150 are next to each other.
What is the value of attribute CarModel for the person whose attribute Mother is Kailyn? Please reason step by step, and put your final answer within \boxed{}
|
honda civic
|
0/8
|
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`, `Bob`
- Each person has a unique level of education: `associate`, `doctorate`, `high school`, `bachelor`, `master`
- Each mother is accompanied by their child: `Fred`, `Meredith`, `Bella`, `Timothy`, `Samantha`
- Everyone has a unique favorite cigar: `prince`, `pall mall`, `blue master`, `dunhill`, `blends`
- The people are of nationalities: `swede`, `brit`, `dane`, `german`, `norwegian`
- People have unique favorite book genres: `romance`, `mystery`, `biography`, `science fiction`, `fantasy`
## Clues:
1. The person with a master's degree is the person partial to Pall Mall.
2. The person with a high school diploma is in the fourth house.
3. The person's child is named Samantha is in the first house.
4. The Norwegian and the person with a master's degree are next to each other.
5. The German is the Dunhill smoker.
6. The person who is the mother of Timothy is directly left of the person with a master's degree.
7. Arnold and the Dane are next to each other.
8. The Swedish person is the person who loves mystery books.
9. The Swedish person is Alice.
10. Arnold is the person with an associate's degree.
11. The person with a doctorate is in the second house.
12. Peter is somewhere to the right of the person who loves biography books.
13. The person who smokes many unique blends is Alice.
14. The person who loves fantasy books is in the first house.
15. Peter is the Norwegian.
16. The person with a doctorate is the person's child is named Meredith.
17. The Prince smoker is somewhere to the left of the person who smokes many unique blends.
18. The person partial to Pall Mall is the person's child is named Bella.
19. The person who loves romance books is Bob.
What is the value of attribute Nationality for the person whose attribute Children is Samantha? Please reason step by step, and put your final answer within \boxed{}
|
german
|
0/8
|
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`, `Carol`
- People have unique heights: `very tall`, `tall`, `average`, `short`, `super tall`, `very short`
- Each person has a unique birthday month: `sept`, `may`, `jan`, `april`, `feb`, `mar`
- People have unique hair colors: `blonde`, `red`, `black`, `auburn`, `gray`, `brown`
- People have unique favorite book genres: `science fiction`, `romance`, `biography`, `mystery`, `fantasy`, `historical fiction`
- Each person has a unique type of pet: `bird`, `hamster`, `fish`, `dog`, `cat`, `rabbit`
## Clues:
1. The person with an aquarium of fish is the person who loves mystery books.
2. The person who has black hair is the person who owns a dog.
3. The person who has gray hair is the person who loves science fiction books.
4. The person who loves historical fiction books is somewhere to the left of the person who has black hair.
5. The person who loves historical fiction books is directly left of the person who owns a rabbit.
6. The person whose birthday is in February is the person who owns a dog.
7. Bob is the person who keeps a pet bird.
8. The person who keeps a pet bird is the person who loves romance books.
9. Alice is directly left of the person who loves fantasy books.
10. The person who owns a rabbit is directly left of the person who is very short.
11. The person who is short is somewhere to the left of the person who is super tall.
12. The person who has an average height is in the sixth house.
13. The person who has red hair is not in the first house.
14. The person who is very tall is somewhere to the right of the person who is super tall.
15. There are two houses between the person who loves romance books and Eric.
16. The person with an aquarium of fish is in the sixth house.
17. The person who has blonde hair is the person who loves historical fiction books.
18. The person who is tall is somewhere to the right of the person whose birthday is in February.
19. The person who has auburn hair is Carol.
20. The person with an aquarium of fish is the person whose birthday is in March.
21. The person who loves science fiction books is directly left of the person with an aquarium of fish.
22. The person whose birthday is in September is not in the first house.
23. The person who loves biography books is directly left of the person who has a cat.
24. The person whose birthday is in January is the person who loves science fiction books.
25. Arnold is the person whose birthday is in May.
What is the value of attribute Pet for the person whose attribute Name is Arnold? Please reason step by step, and put your final answer within \boxed{}
|
rabbit
|
0/8
|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
- People have unique favorite book genres: `romance`, `mystery`, `fantasy`, `science fiction`
- People have unique heights: `tall`, `very short`, `short`, `average`
- Everyone has a favorite smoothie: `dragonfruit`, `cherry`, `desert`, `watermelon`
- Each person has a favorite color: `red`, `green`, `yellow`, `white`
## Clues:
1. Arnold is not in the fourth house.
2. Peter is the person who loves mystery books.
3. The person who likes Cherry smoothies is somewhere to the right of the Desert smoothie lover.
4. The person who loves science fiction books is directly left of Eric.
5. The Desert smoothie lover is the person who is very short.
6. The person who loves white is the person who loves mystery books.
7. The person who loves fantasy books is the person who loves yellow.
8. The Desert smoothie lover is somewhere to the right of the person who is short.
9. The Dragonfruit smoothie lover and the person whose favorite color is green are next to each other.
10. The Dragonfruit smoothie lover is the person who loves mystery books.
11. The person who loves yellow is in the second house.
12. The person who likes Cherry smoothies is not in the third house.
13. The person who has an average height is the person who likes Cherry smoothies.
What is the value of attribute Smoothie for the person whose attribute Height is short? Please reason step by step, and put your final answer within \boxed{}
|
watermelon
|
0/8
|
8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 16 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is considered constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
|
\dfrac{1}{2}
|
0/8
|
## Task 21/72
Given the equation $x^{4}+x^{3}+x^{2}+(62-k) x+k=0$ with the real solutions $x_{i}(i=1 ; 2 ; 3 ; 4)$. What real values can $k$ take if
$$
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}>5
$$
is to be satisfied?
|
(-\frac{31}{2}, 0)
|
3/8
|
Example 3: On a plane, there are $n$ lines, no two of which are parallel, and no three of which intersect at the same point. How many intersection points do these $n$ lines have in total?
Example 4: Following the previous example, now we want to know how many non-overlapping regions will be formed by such $n$ lines dividing the plane?
|
\dfrac{n(n + 1)}{2} + 1
|
0/8
|
6. (10 points) A convoy of trucks is delivering supplies to a disaster victim resettlement point. Each truck has a carrying capacity of 10 tons. If each tent is allocated 1.5 tons of supplies, there will be less than one truck's worth of supplies left over. If each tent is allocated 1.6 tons of supplies, there will be a shortage of more than 2 trucks' worth of supplies. How many tents are there at the resettlement point at a minimum?
|
213
|
0/8
|
Folkpor
In a convex quadrilateral $A B C D: A C \perp B D, \angle B C A=10^{\circ}, \angle B D A=20^{\circ}, \angle B A C=40^{\circ}$. Find $\angle B D C$.
|
60^\circ
|
3/8
|
8. Let the rational number $r=\frac{p}{q} \in(0,1)$, where $p, q$ are coprime positive integers, and $pq$ divides 3600. The number of such rational numbers $r$ is $\qquad$ .
|
112
|
4/8
|
12.313. The height of the cone is $H$, and the angle between the generatrix and the base plane is $\alpha$. The total surface area of this cone is divided in half by a plane perpendicular to its height. Find the distance from this plane to the base of the cone.
|
H \left(1 - \cos \frac{\alpha}{2}\right)
|
0/8
|
15. The teachers and students of a township primary school went to the county town for a visit. It was stipulated that the bus would depart from the county town and arrive at the school at 7:00 AM to pick up the visiting teachers and students and immediately head to the county town. However, the bus broke down on its way to the school and had to stop for repairs. The teachers and students at the school waited until 7:10 AM but still did not see the bus, so they started walking towards the county town. On their way, they met the repaired bus, immediately got on, and headed to the county town, arriving 30 minutes later than the originally scheduled time. If the speed of the bus is 6 times the walking speed, how long did the bus spend on the road for repairs?
|
38
|
0/8
|
7. Given that the modulus of the complex number $z$ is 1, if when $z=z_{1}$ and $z=z_{2}$, $|z+1+\mathrm{i}|$ attains its maximum and minimum values respectively, then $z_{1}-z_{2}=$ $\qquad$ .
|
\sqrt{2}(1+i)
|
1/8
|
Example 3 In the tetrahedron $A-B C D$, $\angle D A B+$ $\angle B A C+\angle D A C=90^{\circ}$, and $\angle A D B=\angle B D C$ $=\angle A D C=90^{\circ}$. If $D B=a, D C=b$, try to calculate the volume of the tetrahedron $A-B C D$.
|
\dfrac{ab(a + b)}{6}
|
3/8
|
Problem 6. Calculate the maximum number of distinct real roots that a polynomial $P$ can have, which satisfies the following property: the product of two distinct roots of $P$ is still a root of $P$.
|
4
|
0/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.