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Let \( A_{1} A_{2} \ldots A_{100} \) be the vertices of a regular 100-gon. Let \( \pi \) be a randomly chosen permutation of the numbers from 1 through 100. The segments \( A_{\pi(1)} A_{\pi(2)}, A_{\pi(2)} A_{\pi(3)}, \ldots, A_{\pi(99)} A_{\pi(100)}, A_{\pi(100)} A_{\pi(1)} \) are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the 100-gon.
|
\frac{4850}{3}
|
1/8
|
In Figure 1, two rectangles with widths 4 and 5 units cross each other at \( 30^{\circ} \). Find the area of the overlapped region.
|
40
|
2/8
|
In a convex quadrilateral $ABCD$, side $BC$ is half the length of side $AD$. Diagonal $AC$ is perpendicular to side $CD$, and diagonal $BD$ is perpendicular to side $AB$. Find the larger acute angle of this quadrilateral if the smaller one is $36^\circ$.
|
84
|
1/8
|
For positive integers \( n \), let \( c_{n} \) be the smallest positive integer for which \( n^{c_{n}}-1 \) is divisible by 210, if such a positive integer exists, and \( c_{n} = 0 \) otherwise. What is \( c_{1} + c_{2} + \cdots + c_{210} \)?
|
329
|
2/8
|
Petya wants to write out all possible sequences of 100 natural numbers, in each of which the number three appears at least once, and any two neighboring elements differ by no more than 1. How many sequences will he have to write out?
|
3^{100}-2^{100}
|
0/8
|
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares \(\square_{1}, \square_{2}, \square_{3}\) on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set \(\{1,2,3,4,5\}\) such that no integer is used more than once. Ana's goal is to minimize the minimum value \(M\) that the polynomial \(a_{1} x^{2} + a_{2} x + a_{3}\) attains over all real \(x\), where \(a_{1}, a_{2}, a_{3}\) are the integers written in \(\square_{1}, \square_{2}, \square_{3}\) respectively. Banana aims to maximize \(M\). Assuming both play optimally, compute the final value of \(100 a_{1} + 10 a_{2} + a_{3}\).
|
451
|
0/8
|
A robotic grasshopper jumps \(1 \text{ cm}\) to the east, then \(2 \text{ cm}\) to the north, then \(3 \text{ cm}\) to the west, then \(4 \text{ cm}\) to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: \(1 \text{ cm}\) to the east, then \(2 \text{ cm}\) to the north, then \(3 \text{ cm}\) to the west, then \(4 \text{ cm}\) to the south. After a total of \(n\) jumps, the position of the grasshopper is \(162 \text{ cm}\) to the west and \(158 \text{ cm}\) to the south of its original position. The sum of the squares of the digits of \(n\) is
(A) 22
(B) 29
(C) 17
(D) 14
(E) 13
|
22
|
0/8
|
The country is shaped like a square and is divided into 25 identical square counties. In each county, there is either a knight count, who always tells the truth, or a liar count, who always lies. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that could possibly be? (Counts are considered neighbors if their counties share a common side.)
|
8
|
1/8
|
A cross composed of two identical large squares and two identical small squares is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is \( 810 \mathrm{~cm}^{2} \).
|
36
|
0/8
|
In three out of six circles in a diagram, the numbers 4, 14, and 6 are written. How many ways can natural numbers be placed in the remaining three circles so that the products of the triplets of numbers along each of the three sides of the triangular diagram are the same?
|
6
|
3/8
|
A plane passing through the vertex of a cone intersects the base along a chord whose length is equal to the radius of the base. Determine the ratio of the volumes of the resulting parts of the cone.
|
\frac{2\pi-3\sqrt{3}}{10\pi+3\sqrt{3}}
|
0/8
|
Znayka cut a semicircle out of paper. Neznaika marked a point \( D \) on the diameter \( AB \) of this semicircle and cut out two semicircles from Znayka's semicircle with diameters \( AD \) and \( DB \). Find the area of the remaining figure if the length of the chord passing through point \( D \) and perpendicular to \( AB \) is 6. If necessary, round your answer to two decimal places.
|
9\pi
|
0/8
|
The 277th National Junior High School Mathematics Competition consists of 14 questions (5 multiple-choice questions, 5 fill-in-the-blank questions, 4 problem-solving questions), with a full score of 150 points. Among them, each correct answer for multiple-choice and fill-in-the-blank questions earns 7 points, and a wrong answer earns 0 points, with no other point values; each problem-solving question is worth 20 points, and the step scores can only be $0, 5, 10, 15, 20$ points, with no other point values. How many different possible scores are there?
|
127
|
3/8
|
2. Vasya and Petya, participating in a school sports and entertainment game, need to cover a distance of 3 km as quickly as possible with only one pair of roller skates between them. They start simultaneously, one just running, the other running on roller skates. At any time, the one running on roller skates can leave them to their friend and continue running without them. Such an exchange can be made as many times as needed. Find the minimum time required to complete the distance (determined by the last one to arrive), if Vasya's speeds for regular running and running on roller skates are 4 km/h and 8 km/h, and Petya's speeds are 5 km/h and 10 km/h. Assume that no time is lost when switching between roller skates and regular running.
|
\dfrac{1}{2}
|
3/8
|
Amy has divided a square up into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the smallest possible value of \( x \)?
|
2
|
1/8
|
71. Find all triples $(p, q, r)$ that satisfy the following conditions:
1) $p, q, r \in \mathbf{N}, p \geqslant q \geqslant r$;
2) At least two of $p, q, r$ are prime numbers;
3) $\frac{(p+q+r)^{2}}{p q r}$ is a positive integer.
|
(3, 2, 1)
|
0/8
|
4. $A B C D$ is a cyclic quadrilateral such that $|D A|=|B C|=2$, and $|A B|=4$. If $|C D|>|A B|$ and the lines $D A$ and $B C$ intersect at an angle of $60^{\circ}$, find the radius of the circumscribing circle.
|
\dfrac{2\sqrt{21}}{3}
|
4/8
|
7. Consider a positive integer,
$$
\mathrm{N}=9+99+999+\ldots \ldots+\underset{2018}{999 \ldots 9}
$$
How many times does the digit 1 occur in its decimal representation?
|
2014
|
4/8
|
13. In a math competition, there were three problems: A, B, and C. Among the 25 participating students, each student solved at least one problem. Among the students who did not solve problem A, the number of students who solved problem B is twice the number of students who solved problem C; the number of students who only solved problem A is one more than the number of students who solved problem A among the remaining students; among the students who only solved one problem, half did not solve problem A. How many students only solved problem B?
|
6
|
3/8
|
4. To build a batch of identical houses with a total area of $2500 \mathrm{~m}^{2}$, the cost of a $a \mathrm{~m}^{2}$ house is the sum of material cost $100 p_{1} a^{\frac{3}{2}}$ yuan, construction cost $100 p_{2} a$ yuan, and other various expenses $100 p_{3} a^{\frac{1}{2}}$ yuan, where the numbers $p_{1} 、 p_{2} 、 p_{3}$ are three consecutive terms of a geometric sequence, their sum is 21, and their product is 64. If 63 such houses are built, the material cost will be less than the sum of the construction cost and other various expenses. To minimize the total cost, what is the maximum number of houses that can be built?
|
156
|
4/8
|
3. (8 points) Xiaoshu, Xiaoxue, Xiaohua, Xiaoyuan, and Tanmi won the top 5 places (no ties) in the long jump competition. They said:
Xiaoshu: “My ranking is better than Xiaoxue”;
Xiaohua: “My ranking is not as good as Xiaoyuan”;
Tanmi: “My ranking is not as good as Xiaoxue”.
Xiaoxue: “My ranking is better than Xiaohua”;
Xiaoyuan: “My ranking is not as good as Tanmi”;
It is known that Xiaoshu, Xiaoxue, Xiaohua, Xiaoyuan, and Tanmi respectively obtained the $A$, $B$, $C$, $D$, $E$ places, and they are all honest students who never lie. Therefore, the five-digit number $\overline{\mathrm{ABCDE}}$ is $\qquad$
|
12543
|
2/8
|
8.3. On the table, 28 coins of the same size but possibly different masses are arranged in a triangular shape (see figure). It is known that the total mass of any triplet of coins that touch each other pairwise is 10 g. Find the total mass of all 18 coins on the boundary of the triangle.
|
60
|
2/8
|
[ Product of chord segments or secant segments ]
In triangle $A B C$, the bisector $A P$ is drawn. It is known that $B P=16, P C=20$ and that the center of the circumcircle of triangle $A B P$ lies on the segment $A C$. Find the side $A B$.
#
|
\dfrac{144\sqrt{5}}{5}
|
2/8
|
6.1. Find the area of the figure defined on the coordinate plane by the inequality $\sqrt{\arcsin \frac{x}{3}} \leq \sqrt{\arccos \frac{y}{3}} \cdot$ In your answer, specify the integer closest to the found value of the area.
|
16
|
2/8
|
$4 \cdot 39$ Given 9 points in space, where no 4 points are coplanar. Find the smallest natural number $n$, such that when any $n$ line segments are drawn between the given points and each line segment is colored either red or blue, there will always exist a triangle with all sides of the same color.
|
33
|
4/8
|
[ Properties and characteristics of a parallelogram ] [ Isosceles, inscribed, and circumscribed trapezoids ]
A circle passing through the vertices $A, B$, and $C$ of parallelogram $A B C D$ intersects the lines $A D$ and $C D$ at points $M$ and $N$ respectively. Point $M$ is at distances 4, 3, and 2 from vertices $B, C$, and $D$ respectively. Find $M N$.
|
\dfrac{8}{3}
|
1/8
|
99.2. Consider 7-gons inscribed in a circle such that all sides of the 7-gon are of different length. Determine the maximal number of $120^{\circ}$ angles in this kind of a 7-gon.
|
2
|
0/8
|
7. (1976 Polish Mathematical Olympiad) A fishing boat is fishing in the territorial waters of a foreign country without permission, and each time it casts a net, it causes a loss of equal value to the country's fishing yield. The probability that the boat is detained by the foreign coast guard during each net casting is $1 / k$, where $k$ is a natural number of the country. Assume that the events of the boat being detained or not during each net casting are independent of the previous fishing process. If the fishing boat is detained by the foreign coast guard, all the fish caught are confiscated, and it cannot fish in the future. The captain plans to leave the foreign territorial waters after casting the net for the $n$th time. Because the possibility of the fishing boat being detained by the foreign coast guard cannot be ruled out, the fishing income is a random variable. Find the number $n$ that maximizes the expected value of the fishing income.
|
k - 1
|
2/8
|
1. Calculation of EMF
## TASK 4
## SOLUTION
The amount of potassium hydroxide and nitric acid in the solutions:
$$
\begin{gathered}
v(\mathrm{KOH})=\omega \rho V / M=0.062 \cdot 1.055 \text { g/ml } \cdot 22.7 \text { ml / } 56 \text { g/ mol }=0.0265 \text { mol. } \\
v\left(\mathrm{HNO}_{3}\right)=\text { C }_{\mathrm{M}} \cdot \mathrm{V}=2.00 \text { mol/l } \cdot 0.0463 \text { l }=0.0926 \text { mol. }
\end{gathered}
$$
Since $v(\mathrm{KOH})<v(\mathrm{HNO} 3)$, and the coefficients of these substances in the reaction equation are the same and equal to one, potassium hydroxide is in deficiency, and its amount will determine the heat effect of the reaction, which will be:
$$
\text { Q = 55.6 kJ/mol } \cdot \text { 0.0265 mol = 1.47 kJ. }
$$
|
1.47\,\text{kJ}
|
0/8
|
7. Plot the figure $\Phi$ on the plane, consisting of points $(x ; y)$ of the coordinate plane such that the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
\sqrt{y^{2}-8 x^{2}-6 y+9} \leqslant 3 y-1 \\
x^{2}+y^{2} \leqslant 9
\end{array}\right.
$$
Determine how many parts the figure $\Phi$ consists of.
|
1
|
1/8
|
5. Given that $M$ is the midpoint of edge $C_{1} D_{1}$ of the cube $A_{1} B_{1} C_{1} D_{1}-A B C D$, $O$ is the midpoint of $B D_{1}$, and $O M / /$ plane $\beta$, where plane $\beta$ passes through point $B$ and is different from plane $B_{1} B C C_{1}$. If point $P \in \beta$, and $P$ is within or on the boundary of the square $B_{1} B C C_{1}$, let $\theta$ be the angle between $A_{1} P$ and plane $B_{1} B C C_{1}$, then the maximum value of $\tan \theta$ is $\qquad$
|
\sqrt{2}
|
5/8
|
The cinema has two doors, one wide and one narrow. After a screening, all the viewers exit the hall through both doors within 3 minutes and 45 seconds. If the viewers exit through only the wide door, it will take 4 minutes less than if they exit only through the narrow door. How much time is required for the viewers to exit the cinema hall through each door individually?
|
6
|
2/8
|
11. From May 1 to May 3, the school plans to arrange six leaders for duty, requiring each person to be on duty for 1 day, with two people scheduled each day. If among the six leaders, A cannot be on duty on the 2nd, and B cannot be on duty on the 3rd, then the number of different ways to arrange the duty schedule is $\qquad$ kinds.
|
42
|
5/8
|
8-5. In a tournament, 6 teams $P, Q, R, S, T$ and $U$ participate, and each team must play against every other team exactly once. Each day, they are divided into 3 pairs, and all three matches are played simultaneously. The "Sports" channel has chosen which match it will broadcast each day:
$$
\begin{array}{c|c|c|c|c}
1 \text { day } & 2 \text { day } & 3 \text { day } & 4 \text { day } & 5 \text { day } \\
\hline P-Q & R-S & P-T & T-U & P-R
\end{array}
$$
On which day can teams $S$ and $U$ play against each other? Mark all possible options.
|
1
|
1/8
|
## Problem 3
On the set of real numbers $\mathbb{R}$, the composition law "*" is defined with the following properties:
1) $\left(\frac{a+1}{3}\right) *\left(\frac{a}{2}\right)=1, \forall a \in \mathbb{R}$
2) $(a * b) \cdot c=(a \cdot c) *(b \cdot c), \forall a, b, c \in \mathbb{R}$.
Calculate $10 * 14$.
|
2
|
2/8
|
U ohně seděli náčelníci tří indiánských kmenů se třemi stejnými dýmkami. Měli válečnou poradu a kouřili. První z nich vykouří celou dýmku za deset minut, druhý za půl hodiny a třetí za hodinu. Jak si mají náčelníci mezi sebou měnit dýmky, aby se mohli radit co nejdéle.
(Bednářová)
#
By the fire sat the chiefs of three Indian tribes with three identical pipes. They were having a war council and smoking. The first of them can smoke a whole pipe in ten minutes, the second in half an hour, and the third in an hour. How should the chiefs exchange pipes among themselves so that they can deliberate for as long as possible.
(Bednářová)
#
|
20
|
1/8
|
5. A diagonal of a convex polygon (i.e., a polygon where all interior angles are less than $180^{\circ}$) is called "bisecting" if and only if the diagonal simultaneously bisects the area and the perimeter of the polygon. How many diagonals of a convex pentagon can be bisecting at most?
---
Please note that the mathematical notation and symbols have been preserved in the translation.
|
2
|
5/8
|
Let \( G \) be an infinite complete graph. Show that if the edges of \( G \) are colored with a finite number of colors, then \( G \) contains an infinite monochromatic subgraph.
Hint: The idea is to construct a sequence of distinct vertices \(\left(u_{n}\right)_{n \in \mathbb{N}}\) and a sequence of colors (where a color may appear multiple times) \(\left(c_{n}\right)_{n \in \mathbb{N}}\) with the following property: if \( p > q \), then the edge between \( u_{p} \) and \( u_{q} \) is colored \( c_{q} \).
|
G \text{ contains an infinite monochromatic subgraph.}
|
3/8
|
8. Find the smallest odd number $a$ greater than 5 that satisfies the following conditions: there exist positive integers $m_{1}, n_{1}, m_{2}, n_{2}$, such that
$$
a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2} \text {, }
$$
and $m_{1}-n_{1}=m_{2}-n_{2}$.
|
261
|
1/8
|
4. In the plane, there are 16 black points, as shown in the figure. What is the minimum number of these points that we need to color red so that no square with vertices in the remaining black points and sides parallel to the coordinate axes exists? Justify your answer.
Solve the problems independently. You have 210 minutes for solving. The use of notes, literature, or a pocket calculator is not allowed.
## 51st Mathematical Competition for High School Students in Slovenia Maribor, April 21, 2007
## Problems for 3rd Year
|
4
|
4/8
|
4. Points $A, B, C, D$ lie on the circumference of a circle, and $B C=D C=4, A E=6$, the lengths of segments $B E$ and $D E$ are both integers, find the length of $B D$.
|
7
|
1/8
|
In the acute isosceles triangle \(ABC\) with \(AB = BC\), altitudes \(AD\) and \(CE\) intersect at point \(H\). The circumcircle of triangle \(ACH\) intersects segment \(BC\) at point \(F\). Prove that \(\angle ABH = \angle BFH\).
|
\angle ABH = \angle BFH
|
4/8
|
Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.
|
\beta^{-1/\alpha}
|
2/8
|
The diagram shows a 16 meter by 16 meter wall. Three grey squares are painted on the wall as shown. The two smaller grey squares are equal in size and each makes an angle of \(45^\circ\) with the edge of the wall. The grey squares cover a total area of \(B\) square meters. What is the value of \(B\)?
|
128
|
2/8
|
In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B$; $B$ takes lessons from $C$; $C$ takes lessons from $A$. What is the largest number of players there can be?
|
4
|
4/8
|
There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The viewer can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and answers all of them honestly without announcing the questions out loud. What is the minimum number of notes that need to be sent to definitely find out where the prize is?
|
99
|
4/8
|
A man travels 10 km at a speed of 4 km/h and another 10 km at a speed of 6 km/h. If the average speed of the whole journey is \( x \) km/h, find \( x \).
|
\frac{24}{5}
|
0/8
|
The hare is running a 2024-meter race. At the start, it jumps off with its left leg and throughout the race alternates regularly among the left leg, right leg, and both legs. When the hare jumps with its left leg, it jumps $35 \mathrm{dm}$; with its right leg, it jumps $15 \mathrm{dm}$; and with both legs, it jumps $61 \mathrm{dm}$.
How many jumps will the hare make before reaching the finish line? And which leg will it use for the final jump?
|
548\,
|
0/8
|
If \( 0^{\circ} \leq \theta < 360^{\circ} \), the equation in \( \theta \): \( 3 \cos \theta + \frac{1}{\cos \theta} = 4 \) has \( p \) roots. Find \( p \).
If \( x - \frac{1}{x} = p \) and \( x^{3} - \frac{1}{x^{3}} = q \), find \( q \).
A circle is inscribed in an equilateral triangle of perimeter \( q \) cm. If the area of the circle is \( k \pi \) cm\(^{2} \), find \( k \).
Each interior angle of a regular polygon of \( k \) sides is \( m^{\circ} \). Find \( m \).
|
150
|
0/8
|
In a computer program written in Turbo Pascal, a function $\operatorname{Random}(x)$ is used, generating random integers from 1 to $x$. What is the probability that a number divisible by 5 will appear when this function is executed if $x=100$?
|
0.2
|
0/8
|
It is known that the distance from the center of the circumscribed circle to side \(AB\) of triangle \(ABC\) is equal to half the radius of this circle. Find the height of triangle \(ABC\) dropped to side \(AB\), if it is less than \(\sqrt{\frac{3}{2}}\), and the other two sides of the triangle are 2 and 3.
|
3\sqrt{\frac{3}{19}}
|
0/8
|
12. Simplify: $\sum_{k=1}^{n} \frac{\cos 3^{k} x+3 \cos 3^{k-1} x}{3^{k-1} \sin 3^{k} x}$.
|
\dfrac{3}{2} \left( \cot x - \dfrac{1}{3^{n}} \cot 3^{n} x \right)
|
0/8
|
In the angle with vertex $A$, equal to $60^{\circ}$, a circle with center $O$ is inscribed. A tangent to this circle intersects the sides of the angle at points $B$ and $C$. Segment $B C$ intersects segment $A O$ at point $M$. Find the radius of the circle inscribed in triangle $A B C$, if $A M: M O=2: 3$ and $B C=7$.
#
|
\dfrac{7\sqrt{3}}{9}
|
1/8
|
1. Sei $A B C$ ein spitzwinkliges Dreieck mit $A B \neq B C$ und Umkreis $k$. Seien $P$ und $Q$ die Schnittpunkte von $k$ mit der Winkelhalbierenden beziehungsweise der Aussenwinkelhalbierenden von $\angle C B A$. Sei $D$ der Schnittpunkt von $A C$ und $P Q$. Bestimme das Verhältnis $A D: D C$.
## 1st solution:
|
1
|
5/8
|
Example 6 In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (when A is a friend of B, B is also a friend of A. No one is a friend of themselves). How many friends does the person with the most friends have in this carriage?
(5th National Training Team Selection Test)
|
m
|
4/8
|
3. (4 points) Positive numbers $b_{1}, b_{2}, b_{3}, b_{4}, b_{5}$ form a geometric progression. The sum of the logarithms base 2 of these numbers is 15. Find these numbers if $\log _{2} b_{1} \cdot \log _{2} b_{5}=8$.
|
4
|
3/8
|
On an island, there are knights who always tell the truth and liars who always lie. There are 11 players on the island's football team. Player number 1 said, "There are as many knights as liars on our team." Player number 2 said, "The number of knights and liars on our team differs by one," and so on. Player number 11 said, "The number of knights and liars on our team differs by ten." How many knights are on the team, and if there are any, what are their numbers? Find all possible answers to this question.
|
0
|
0/8
|
4. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.
|
88327
|
3/8
|
3. As shown in Figure 4, in $\triangle A B C$, $\angle C=90^{\circ}, D$ is a point on side $B C$, $\angle A D C=3 \angle B A D, B D=9$, $D C=5$. Then $A B=$
|
21
|
4/8
|
9.48 The total weight of a pile of stones is 100 kilograms, where the weight of each stone does not exceed 2 kilograms. By taking out some of the stones in various ways and calculating the difference between the sum of the weights of these stones and 10 kilograms. Among all these differences, the minimum value of their absolute values is denoted as $d$. Among all piles of stones that meet the above conditions, find the maximum value of $d$.
|
\dfrac{10}{11}
|
5/8
|
For many years, every day at noon, a mail steamship departs from Le Havre to New York and at the same time a steamship from New York departs for Le Havre of the same company. Each of these steamships is at sea for exactly seven days, and they follow the same route.
How many steamships of its company will a steamship traveling from Le Havre to New York meet on its way?
|
15
|
0/8
|
ЕЕооокимов M.A. Anya calls a date beautiful if all 6 digits in its notation are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in 2023?
|
30
|
4/8
|
A unit-edged parallelepiped has a base $A_{1} A_{2} A_{3} A_{4}$ and a top face $B_{1} B_{2} B_{3} B_{4}$ such that the vertices $A_{i}$ and $B_{i}$ are connected by an edge. Within what bounds does the sum of squares $A_{1} B_{2}^{2}+A_{2} B_{3}^{2}+A_{3} B_{4}^{2}+A_{4} B_{1}^{2}$ vary?
|
8
|
0/8
|
Given a hexagon \(ABCDEF\) inscribed in a circle where the sides \(AB = CD = EF = r\), with \(r\) being the radius of the circle, let \(G\), \(H\), and \(K\) be the midpoints of \(BC\), \(DE\), and \(FA\). Prove that triangle \(\triangle GHK\) is an equilateral triangle.
|
\triangle GHK \text{ is an equilateral triangle}
|
2/8
|
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$. Similarly, let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?
|
120
|
3/8
|
We take $100$ consecutive natural numbers $a_1, a_2, \ldots, a_{100}$. Determine the last two digits of the number $a_1^8 + a_2^8 + \ldots + a_{100}^8$.
|
30
|
4/8
|
The leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum number of integer values for $x$ such that $|P(x)| < 50$.
|
2
|
1/8
|
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has a length of $5$, and after each hop, the grasshopper is at a point whose coordinates are both integers. Thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
|
578
|
2/8
|
Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
|
9
|
0/8
|
Squares $ABCD$ and $AEFG$, each with side length $12$, overlap such that $\triangle AED$ is an equilateral triangle, as shown in the diagram. The area of the region that lies within both squares, which is shaded in the diagram, is given by $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Determine the value of $m + n$.
|
51
|
1/8
|
Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. Initially, $B$ moves more slowly than $A$ until they meet. After meeting, $B$ doubles his speed and next meets $A$ at the starting point. Let $d$ m be the distance traveled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.
|
41
|
2/8
|
For a positive integer $n$, let $f(n)$ be the integer formed by reversing the digits of $n$ (and removing any leading zeroes). For example, $f(14172) = 27141$. Define a sequence of numbers $\{a_n\}_{n \ge 0}$ by $a_0 = 1$ and for all $i \ge 0$, $a_{i+1} = 11a_i$ or $a_{i+1} = f(a_i)$. How many possible values are there for $a_8$?
|
13
|
0/8
|
In a classroom of 32 students, a majority bought the same number of pens, with each student purchasing more than 1 pen, and the cost of each pen in cents exceeded the number of pens bought by each student. If the total sum spent on the pens was $21.16, determine the cost of one pen in cents.
|
23
|
4/8
|
In a tournament, there are eight teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. Furthermore, the top four teams earned the same number of total points. Calculate the greatest possible number of total points for each of the top four teams.
|
33
|
4/8
|
Let Martha the moose take 50 equal steps between consecutive lamp posts on a city street. Let Percy the pronghorn cover the same distance in 15 equal bounds. The lamp posts are evenly spaced, and the 51st post is exactly 2 miles or 10560 feet from the first post. Calculate the difference in length between Percy's bound and Martha's step.
|
9.856
|
0/8
|
The basic subscription price of the online streaming service is $15 per month. Calculate the maximum percentage decrease in the number of subscribers that the service can tolerate in order to keep their total income at least the same despite a 20% price increase.
|
16.67\%
|
1/8
|
A $4 \times 4$ square is partitioned into $16$ unit squares. Each unit square is painted either white or black with each color being equally likely, independently and at random. The square is then rotated $180\,^{\circ}$ about its center, and every white square that lands in a position formerly occupied by a black square is painted black, while black squares moved to a position formerly occupied by white squares are turned white. Determine the probability that the entire grid is now black.
|
\frac{1}{65536}
|
5/8
|
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`, `Alice`, `Arnold`, `Carol`, `Eric`
- The people are of nationalities: `swede`, `chinese`, `norwegian`, `german`, `brit`, `dane`
- People own unique car models: `tesla model 3`, `toyota camry`, `ford f150`, `chevrolet silverado`, `honda civic`, `bmw 3 series`
- Each person has a unique favorite drink: `root beer`, `milk`, `tea`, `water`, `coffee`, `boba tea`
## Clues:
1. The boba tea drinker is the person who owns a Ford F-150.
2. The British person is directly left of Eric.
3. Alice is the person who owns a Honda Civic.
4. The tea drinker is not in the first house.
5. Eric is the Chinese.
6. The tea drinker is the person who owns a BMW 3 Series.
7. The German is in the second house.
8. The person who owns a Chevrolet Silverado is somewhere to the left of the person who likes milk.
9. Bob is in the fifth house.
10. The person who owns a Toyota Camry is directly left of the one who only drinks water.
11. The root beer lover is directly left of the Norwegian.
12. The Dane is somewhere to the right of the person who owns a BMW 3 Series.
13. Arnold is in the second house.
14. The person who owns a Honda Civic is directly left of the person who owns a Ford F-150.
15. Peter is not in the third house.
16. The person who owns a Toyota Camry is in the third house.
What is the value of attribute House for the person whose attribute Drink is coffee? Please reason step by step, and put your final answer within \boxed{}
|
1
|
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`, `Alice`, `Arnold`, `Eric`
- Each person lives in a unique style of house: `victorian`, `ranch`, `craftsman`, `colonial`
- People have unique favorite music genres: `pop`, `classical`, `rock`, `jazz`
- They all have a unique favorite flower: `roses`, `lilies`, `carnations`, `daffodils`
- Each person prefers a unique type of vacation: `city`, `beach`, `mountain`, `cruise`
- Each person has a favorite color: `red`, `white`, `yellow`, `green`
## Clues:
1. The person who loves classical music is Peter.
2. The person in a ranch-style home is not in the fourth house.
3. The person who loves a bouquet of daffodils is the person living in a colonial-style house.
4. The person living in a colonial-style house is the person who loves white.
5. The person who prefers city breaks and the person whose favorite color is red are next to each other.
6. The person who loves the rose bouquet is not in the fourth house.
7. The person who likes going on cruises is in the second house.
8. The person who prefers city breaks is the person who loves rock music.
9. The person who loves pop music is Alice.
10. The person who loves the rose bouquet is the person who prefers city breaks.
11. Eric is the person who enjoys mountain retreats.
12. The person who enjoys mountain retreats is directly left of the person who loves a bouquet of daffodils.
13. The person who loves yellow is somewhere to the left of the person who likes going on cruises.
14. The person who enjoys mountain retreats is the person residing in a Victorian house.
15. The person who loves classical music is the person who likes going on cruises.
16. Peter is somewhere to the right of the person who loves the boquet of lilies.
What is the value of attribute House for the person whose attribute MusicGenre is rock? Please reason step by step, and put your final answer within \boxed{}
|
3
|
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`, `Eric`, `Bob`, `Arnold`, `Carol`, `Peter`
- Each person has a unique hobby: `gardening`, `painting`, `knitting`, `photography`, `cooking`, `woodworking`
- People own unique car models: `chevrolet silverado`, `toyota camry`, `bmw 3 series`, `ford f150`, `honda civic`, `tesla model 3`
- Each person lives in a unique style of house: `ranch`, `colonial`, `victorian`, `mediterranean`, `craftsman`, `modern`
## Clues:
1. Arnold is somewhere to the right of the person who owns a BMW 3 Series.
2. The person in a Mediterranean-style villa is the person who loves cooking.
3. The person in a ranch-style home is Alice.
4. The person residing in a Victorian house is somewhere to the right of Eric.
5. Arnold is not in the fourth house.
6. There is one house between the person who owns a Tesla Model 3 and the person who enjoys gardening.
7. The person who enjoys knitting is the person who owns a Toyota Camry.
8. Bob is somewhere to the right of the person residing in a Victorian house.
9. There are two houses between the person who enjoys gardening and Arnold.
10. The person who owns a Toyota Camry is not in the fifth house.
11. Alice is not in the third house.
12. Peter is not in the fourth house.
13. The person living in a colonial-style house is the woodworking hobbyist.
14. The person who owns a Ford F-150 is in the second house.
15. The person who loves cooking and the person who enjoys gardening are next to each other.
16. The person in a ranch-style home is the person who owns a Chevrolet Silverado.
17. Carol is somewhere to the left of the person in a ranch-style home.
18. The person who owns a Toyota Camry is Bob.
19. The person in a modern-style house is the person who enjoys knitting.
20. Peter is not in the third house.
21. The photography enthusiast and Peter are next to each other.
What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \boxed{}
|
1
|
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`
- Each person has an occupation: `doctor`, `engineer`, `artist`, `teacher`
- Each person has a unique birthday month: `sept`, `april`, `jan`, `feb`
- Each person has a favorite color: `red`, `white`, `green`, `yellow`
## Clues:
1. Arnold is in the first house.
2. The person whose birthday is in February and the person who is a teacher are next to each other.
3. The person who is a teacher is the person whose favorite color is red.
4. The person whose favorite color is red is not in the third house.
5. The person who loves yellow is directly left of Peter.
6. The person whose favorite color is green and the person whose favorite color is red are next to each other.
7. The person who is an engineer is the person whose favorite color is green.
8. Eric is the person who is a teacher.
9. The person who loves yellow is not in the second house.
10. The person whose birthday is in September is somewhere to the left of the person whose favorite color is green.
11. The person who is a doctor is somewhere to the right of the person whose birthday is in April.
What is the value of attribute Color for the person whose attribute Occupation is engineer? Please reason step by step, and put your final answer within \boxed{}
|
green
|
5/8
|
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: `dane`, `swede`, `norwegian`, `brit`, `german`
- Everyone has a favorite smoothie: `desert`, `cherry`, `watermelon`, `dragonfruit`, `lime`
- They all have a unique favorite flower: `tulips`, `roses`, `daffodils`, `carnations`, `lilies`
- Each person lives in a unique style of house: `ranch`, `modern`, `victorian`, `colonial`, `craftsman`
- The people keep unique animals: `dog`, `horse`, `bird`, `fish`, `cat`
## Clues:
1. The person living in a colonial-style house is Peter.
2. The person who keeps horses is the person who drinks Lime smoothies.
3. Alice is the British person.
4. The Swedish person is in the first house.
5. The person who keeps horses and the dog owner are next to each other.
6. There is one house between the person in a Craftsman-style house and the Dane.
7. The person who loves the boquet of lilies is directly left of the fish enthusiast.
8. The person who loves the boquet of lilies is the person in a modern-style house.
9. The Desert smoothie lover is in the first house.
10. The person residing in a Victorian house is somewhere to the right of the Watermelon smoothie lover.
11. The person who loves a bouquet of daffodils is Alice.
12. The Dragonfruit smoothie lover is not in the third house.
13. The person who loves a carnations arrangement is the German.
14. The Dragonfruit smoothie lover is Eric.
15. Bob is somewhere to the right of the person who loves the rose bouquet.
16. The Watermelon smoothie lover is the Norwegian.
17. Bob is the person in a modern-style house.
18. The person residing in a Victorian house is not in the fifth house.
19. The person who loves a carnations arrangement is somewhere to the left of the person in a modern-style house.
20. The Desert smoothie lover is the bird keeper.
What is the value of attribute HouseStyle for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{}
|
modern
|
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`, `Alice`, `Eric`, `Arnold`, `Bob`, `Carol`
- The people are of nationalities: `dane`, `brit`, `norwegian`, `swede`, `german`, `chinese`
- People have unique favorite book genres: `historical fiction`, `mystery`, `romance`, `fantasy`, `science fiction`, `biography`
- The mothers' names in different houses are unique: `Holly`, `Aniya`, `Penny`, `Kailyn`, `Sarah`, `Janelle`
## Clues:
1. The person whose mother's name is Janelle is the Swedish person.
2. Alice is the person who loves science fiction books.
3. The person who loves romance books is not in the fifth house.
4. Carol and Peter are next to each other.
5. The person whose mother's name is Sarah is the Norwegian.
6. There is one house between Arnold and the British person.
7. The German and the person who loves historical fiction books are next to each other.
8. There is one house between the German and Carol.
9. The Swedish person is the person who loves fantasy books.
10. Arnold is in the second house.
11. The person whose mother's name is Sarah is in the third house.
12. The person who loves historical fiction books is somewhere to the right of The person whose mother's name is Penny.
13. The Chinese is not in the sixth house.
14. There are two houses between Alice and the person who loves mystery books.
15. Carol is somewhere to the right of Alice.
16. Arnold is the Dane.
17. The person whose mother's name is Aniya is in the fourth house.
18. The person whose mother's name is Penny is somewhere to the left of the Swedish person.
19. The person whose mother's name is Holly and Eric are next to each other.
What is the value of attribute Mother for the person whose attribute BookGenre is biography? Please reason step by step, and put your final answer within \boxed{}
|
Holly
|
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`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`
- Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`
- People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`
- Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`
- The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`
- Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`
## Clues:
1. Carol is directly left of the person who loves eating grilled cheese.
2. Eric is not in the second house.
3. The person whose mother's name is Holly is somewhere to the right of Carol.
4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.
5. Eric is directly left of Carol.
6. The person who loves pop music is not in the third house.
7. Eric is the person who loves country music.
8. The person who loves classical music is in the sixth house.
9. The coffee drinker is Bob.
10. The person who smokes many unique blends is Peter.
11. The person who loves the stew is not in the fifth house.
12. The root beer lover is directly left of The person whose mother's name is Janelle.
13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.
14. Eric is the tea drinker.
15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.
16. The person who loves the soup is Bob.
17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.
18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.
19. The one who only drinks water is directly left of the person who smokes Blue Master.
20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.
21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.
22. The person who loves hip-hop music is directly left of the root beer lover.
23. The one who only drinks water is the person who loves the stew.
24. The Dunhill smoker is not in the second house.
25. The person who likes milk is The person whose mother's name is Janelle.
26. Eric is The person whose mother's name is Aniya.
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 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`, `Arnold`, `Bob`, `Eric`, `Alice`
- People use unique phone models: `iphone 13`, `google pixel 6`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`, `xiaomi mi 11`
- People have unique hair colors: `gray`, `auburn`, `red`, `brown`, `black`, `blonde`
- Each person has a unique favorite drink: `coffee`, `water`, `root beer`, `tea`, `milk`, `boba tea`
- People have unique favorite music genres: `classical`, `jazz`, `rock`, `country`, `hip hop`, `pop`
## Clues:
1. Alice is the person who uses a Huawei P50.
2. The person who uses a Xiaomi Mi 11 is somewhere to the left of the one who only drinks water.
3. The one who only drinks water is somewhere to the left of Bob.
4. Carol is the person who has gray hair.
5. Eric is the tea drinker.
6. The person who has black hair is not in the sixth house.
7. The person who uses a Samsung Galaxy S21 is directly left of the person who loves classical music.
8. The person who uses a OnePlus 9 is not in the third house.
9. Alice is somewhere to the right of Carol.
10. The person who loves jazz music is Alice.
11. The person who loves jazz music is the boba tea drinker.
12. There is one house between the person who uses a OnePlus 9 and Eric.
13. The person who has brown hair is the person who loves hip-hop music.
14. The person who loves pop music is not in the first house.
15. The root beer lover is in the second house.
16. Eric is the person who uses an iPhone 13.
17. The person who has red hair is the person who uses an iPhone 13.
18. The person who has red hair and the coffee drinker are next to each other.
19. Peter is in the first house.
20. Arnold is somewhere to the right of the person who has auburn hair.
21. The person who has gray hair is not in the second house.
22. The person who loves country music is in the fifth house.
What is the value of attribute PhoneModel for the person whose attribute HairColor is red? Please reason step by step, and put your final answer within \boxed{}
|
iphone 13
|
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`, `Arnold`, `Peter`
- The people keep unique animals: `cat`, `fish`, `horse`, `bird`
- Each person has a favorite color: `yellow`, `green`, `white`, `red`
- Each person prefers a unique type of vacation: `cruise`, `city`, `beach`, `mountain`
- Everyone has something unique for lunch: `pizza`, `spaghetti`, `stew`, `grilled cheese`
- People have unique favorite sports: `soccer`, `swimming`, `basketball`, `tennis`
## Clues:
1. The person whose favorite color is red is the person who loves the spaghetti eater.
2. The person who loves the stew is the cat lover.
3. The person who loves yellow is not in the fourth house.
4. Alice is somewhere to the right of Arnold.
5. The person who loves the spaghetti eater is somewhere to the right of the person who loves soccer.
6. The person who loves white is the person who loves tennis.
7. The fish enthusiast is in the second house.
8. Arnold is the person who loves the stew.
9. The person who prefers city breaks is the person who is a pizza lover.
10. The person who prefers city breaks is directly left of the person who likes going on cruises.
11. The person who loves swimming is the person who loves beach vacations.
12. The person who keeps horses and the person whose favorite color is red are next to each other.
13. The person whose favorite color is green is directly left of Eric.
14. The person who prefers city breaks is somewhere to the left of the cat lover.
What is the value of attribute Animal for the person whose attribute Food is pizza? Please reason step by step, and put your final answer within \boxed{}
|
horse
|
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: `Arnold`, `Bob`, `Eric`, `Carol`, `Alice`, `Peter`
- Each person has a unique level of education: `doctorate`, `trade school`, `associate`, `high school`, `bachelor`, `master`
- People own unique car models: `chevrolet silverado`, `ford f150`, `honda civic`, `tesla model 3`, `bmw 3 series`, `toyota camry`
- Each person has a unique favorite drink: `tea`, `water`, `boba tea`, `coffee`, `root beer`, `milk`
- They all have a unique favorite flower: `carnations`, `iris`, `lilies`, `tulips`, `daffodils`, `roses`
## Clues:
1. Eric is the one who only drinks water.
2. The person with an associate's degree is the person who owns a Toyota Camry.
3. The boba tea drinker is in the fourth house.
4. The person who loves the vase of tulips is directly left of the person who owns a Tesla Model 3.
5. The person with a master's degree is somewhere to the left of the person who owns a BMW 3 Series.
6. The person who owns a Toyota Camry is directly left of the person who loves a bouquet of daffodils.
7. The person who loves the boquet of iris is somewhere to the left of the root beer lover.
8. The person who likes milk is somewhere to the right of the person who owns a Honda Civic.
9. Carol is the person who owns a Honda Civic.
10. The person with a high school diploma is in the second house.
11. The person with a master's degree is the boba tea drinker.
12. The person who owns a Tesla Model 3 is Arnold.
13. The person who loves the boquet of lilies is the person who owns a Tesla Model 3.
14. The person who loves a carnations arrangement is somewhere to the left of Alice.
15. The person with a bachelor's degree is somewhere to the right of the person who likes milk.
16. The one who only drinks water is the person who owns a Ford F-150.
17. Carol is somewhere to the left of the person with a doctorate.
18. There are two houses between the person who owns a Toyota Camry and the person who loves the vase of tulips.
19. The person who owns a Tesla Model 3 is somewhere to the left of the coffee drinker.
20. Bob is in the third house.
What is the value of attribute House for the person whose attribute Education is master? Please reason step by step, and put your final answer within \boxed{}
|
4
|
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`, `Arnold`, `Eric`, `Carol`, `Alice`, `Bob`
- People have unique favorite music genres: `country`, `classical`, `pop`, `hip hop`, `jazz`, `rock`
- Each person lives in a unique style of house: `modern`, `mediterranean`, `victorian`, `craftsman`, `ranch`, `colonial`
- People use unique phone models: `huawei p50`, `iphone 13`, `google pixel 6`, `oneplus 9`, `xiaomi mi 11`, `samsung galaxy s21`
- Each person has a unique type of pet: `hamster`, `dog`, `cat`, `rabbit`, `fish`, `bird`
## Clues:
1. The person who has a cat is the person in a Mediterranean-style villa.
2. Peter is in the second house.
3. The person in a modern-style house is the person who uses a Samsung Galaxy S21.
4. The person who keeps a pet bird is the person who uses a Google Pixel 6.
5. The person who keeps a pet bird is directly left of the person who uses a Xiaomi Mi 11.
6. Arnold is the person residing in a Victorian house.
7. The person who loves rock music is the person living in a colonial-style house.
8. The person in a Craftsman-style house is Carol.
9. The person who owns a rabbit is in the sixth house.
10. The person who loves pop music is somewhere to the left of the person living in a colonial-style house.
11. The person who uses a Huawei P50 and the person who loves country music are next to each other.
12. The person who loves hip-hop music is the person who uses a Samsung Galaxy S21.
13. Carol and Peter are next to each other.
14. The person who uses a OnePlus 9 is Peter.
15. The person with a pet hamster is not in the first house.
16. The person in a ranch-style home is somewhere to the left of the person who loves classical music.
17. There are two houses between Carol and Alice.
18. The person who owns a dog is in the fifth house.
19. The person in a modern-style house is Eric.
20. Peter is the person living in a colonial-style house.
21. The person who has a cat is not in the fourth house.
What is the value of attribute House for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}
|
4
|
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: `Carol`, `Peter`, `Eric`, `Bob`, `Alice`, `Arnold`
- Each person prefers a unique type of vacation: `cruise`, `mountain`, `city`, `camping`, `cultural`, `beach`
- Each person has a unique hobby: `knitting`, `photography`, `cooking`, `painting`, `gardening`, `woodworking`
- Everyone has something unique for lunch: `stew`, `pizza`, `spaghetti`, `grilled cheese`, `stir fry`, `soup`
- People own unique car models: `chevrolet silverado`, `honda civic`, `toyota camry`, `ford f150`, `tesla model 3`, `bmw 3 series`
- They all have a unique favorite flower: `tulips`, `daffodils`, `carnations`, `lilies`, `iris`, `roses`
## Clues:
1. Arnold is the person who enjoys camping trips.
2. The photography enthusiast is the person who loves the soup.
3. Carol is the person who loves eating grilled cheese.
4. The person who owns a Honda Civic is the person who loves a carnations arrangement.
5. The person who loves beach vacations is not in the first house.
6. The person who enjoys gardening is the person who owns a Toyota Camry.
7. There is one house between the person who loves the boquet of iris and the person who loves the vase of tulips.
8. The person who prefers city breaks is the person who loves the stew.
9. The person who loves a bouquet of daffodils is Carol.
10. The person who prefers city breaks is Eric.
11. Carol is the person who loves cooking.
12. Eric is directly left of the photography enthusiast.
13. The person who loves the rose bouquet is Bob.
14. There is one house between Arnold and the person who likes going on cruises.
15. The person who owns a Tesla Model 3 and the person who enjoys mountain retreats are next to each other.
16. The person who loves the rose bouquet is the person who owns a Toyota Camry.
17. The person who owns a BMW 3 Series is somewhere to the right of the person who loves cooking.
18. The woodworking hobbyist is not in the third house.
19. The person who owns a Ford F-150 is the person who enjoys mountain retreats.
20. The person who is a pizza lover is not in the second house.
21. The person who loves the vase of tulips is not in the fourth house.
22. The person who loves the spaghetti eater is somewhere to the left of the person who enjoys knitting.
23. Peter is somewhere to the right of the photography enthusiast.
24. The person who owns a Honda Civic is the person who is a pizza lover.
25. The person who loves cooking is in the fifth house.
26. The person who likes going on cruises is the person who enjoys knitting.
What is the value of attribute Name for the person whose attribute Hobby is knitting? Please reason step by step, and put your final answer within \boxed{}
|
Peter
|
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 House for the person whose attribute Pet is bird? Please reason step by step, and put your final answer within \boxed{}
|
1
|
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`, `Eric`, `Alice`, `Bob`, `Peter`
- Each person prefers a unique type of vacation: `mountain`, `city`, `cruise`, `beach`, `camping`
- Each person has a unique level of education: `doctorate`, `high school`, `bachelor`, `associate`, `master`
- Each person has a favorite color: `blue`, `red`, `white`, `yellow`, `green`
- People use unique phone models: `google pixel 6`, `iphone 13`, `oneplus 9`, `huawei p50`, `samsung galaxy s21`
- Everyone has something unique for lunch: `grilled cheese`, `stir fry`, `pizza`, `spaghetti`, `stew`
## Clues:
1. The person who loves the stew is not in the first house.
2. There are two houses between the person who loves stir fry and the person with an associate's degree.
3. The person who enjoys mountain retreats is the person with a bachelor's degree.
4. The person with a doctorate is somewhere to the right of Bob.
5. The person who uses a Samsung Galaxy S21 is in the third house.
6. Eric is the person with a doctorate.
7. The person with a doctorate is in the third house.
8. The person who loves stir fry is the person with a bachelor's degree.
9. The person with a doctorate is the person who is a pizza lover.
10. The person whose favorite color is green is somewhere to the right of Peter.
11. The person who enjoys camping trips is the person who uses an iPhone 13.
12. The person who likes going on cruises is Alice.
13. There is one house between the person with a high school diploma and the person who uses a Samsung Galaxy S21.
14. The person who uses a Google Pixel 6 is Arnold.
15. The person who uses a OnePlus 9 is somewhere to the right of the person who uses a Huawei P50.
16. Arnold is the person who loves eating grilled cheese.
17. The person who loves eating grilled cheese is not in the fourth house.
18. There are two houses between the person with a bachelor's degree and the person whose favorite color is red.
19. The person who loves beach vacations is somewhere to the right of the person who prefers city breaks.
20. The person whose favorite color is green is not in the second house.
21. The person who loves blue is somewhere to the right of Peter.
22. There is one house between the person who enjoys camping trips and the person who loves yellow.
What is the value of attribute Color for the person whose attribute Education is doctorate? Please reason step by step, and put your final answer within \boxed{}
|
yellow
|
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: `Eric`, `Peter`, `Alice`, `Arnold`, `Bob`
- People own unique car models: `toyota camry`, `honda civic`, `tesla model 3`, `bmw 3 series`, `ford f150`
- Everyone has a unique favorite cigar: `dunhill`, `blends`, `prince`, `blue master`, `pall mall`
## Clues:
1. Alice is the person who owns a Honda Civic.
2. The person who owns a Ford F-150 is not in the third house.
3. Alice is the person who smokes many unique blends.
4. The person who smokes many unique blends is not in the third house.
5. Alice is not in the second house.
6. The person who owns a Tesla Model 3 is somewhere to the left of Eric.
7. Arnold is somewhere to the right of the person who smokes many unique blends.
8. Arnold is the Dunhill smoker.
9. Bob is the person who owns a BMW 3 Series.
10. The person partial to Pall Mall is directly left of the person who smokes many unique blends.
11. There is one house between the Prince smoker and the person partial to Pall Mall.
12. The person who owns a Toyota Camry is Arnold.
What is the value of attribute CarModel for the person whose attribute House is 4? Please reason step by step, and put your final answer within \boxed{}
|
honda civic
|
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`, `Carol`, `Alice`, `Eric`, `Arnold`, `Peter`
- Everyone has something unique for lunch: `pizza`, `spaghetti`, `soup`, `grilled cheese`, `stew`, `stir fry`
- Each person has a unique level of education: `associate`, `doctorate`, `trade school`, `high school`, `master`, `bachelor`
- The people keep unique animals: `horse`, `fish`, `rabbit`, `cat`, `bird`, `dog`
- Each person lives in a unique style of house: `ranch`, `victorian`, `modern`, `craftsman`, `mediterranean`, `colonial`
## Clues:
1. The person residing in a Victorian house is in the fourth house.
2. The fish enthusiast is in the first house.
3. The person in a ranch-style home is the bird keeper.
4. The person residing in a Victorian house is somewhere to the left of the person who loves the soup.
5. Carol is directly left of the person who keeps horses.
6. The person who loves stir fry is the person with a doctorate.
7. The person with a master's degree is somewhere to the left of the person with a doctorate.
8. There is one house between the person in a Mediterranean-style villa and the person with a bachelor's degree.
9. The person who attended trade school is the cat lover.
10. The person with a master's degree is the person who is a pizza lover.
11. The person who is a pizza lover is somewhere to the right of Alice.
12. The person living in a colonial-style house is the person who loves the stew.
13. Arnold is the person with a master's degree.
14. There is one house between Arnold and the person who loves eating grilled cheese.
15. The person in a modern-style house and the person who attended trade school are next to each other.
16. Eric is directly left of the person with a doctorate.
17. Alice and the person in a Mediterranean-style villa are next to each other.
18. The person with an associate's degree is directly left of Eric.
19. The dog owner is Bob.
What is the value of attribute House for the person whose attribute Animal is bird? Please reason step by step, and put your final answer within \boxed{}
|
3
|
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`, `Eric`, `Alice`, `Bob`, `Arnold`
- People have unique favorite book genres: `mystery`, `fantasy`, `biography`, `science fiction`, `romance`
- Each person prefers a unique type of vacation: `mountain`, `camping`, `city`, `beach`, `cruise`
- The mothers' names in different houses are unique: `Holly`, `Janelle`, `Kailyn`, `Penny`, `Aniya`
## Clues:
1. Bob is the person who loves fantasy books.
2. The person whose mother's name is Janelle and Eric are next to each other.
3. The person who enjoys camping trips is the person who loves biography books.
4. Peter is The person whose mother's name is Kailyn.
5. Peter and The person whose mother's name is Janelle are next to each other.
6. The person who loves beach vacations is Bob.
7. Peter is somewhere to the right of the person who loves romance books.
8. The person whose mother's name is Penny is the person who loves biography books.
9. Bob is directly left of the person who prefers city breaks.
10. The person who loves science fiction books is in the fourth house.
11. Arnold and Bob are next to each other.
12. The person whose mother's name is Aniya is the person who enjoys mountain retreats.
What is the value of attribute House for the person whose attribute Vacation is mountain? Please reason step by step, and put your final answer within \boxed{}
|
1
|
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`, `Alice`, `Arnold`, `Carol`, `Peter`, `Bob`
- Each person lives in a unique style of house: `mediterranean`, `modern`, `craftsman`, `ranch`, `colonial`, `victorian`
- People have unique favorite music genres: `country`, `hip hop`, `pop`, `jazz`, `classical`, `rock`
- Each person has a unique hobby: `cooking`, `painting`, `photography`, `woodworking`, `gardening`, `knitting`
## Clues:
1. The person who loves rock music is in the fifth house.
2. The person who loves classical music and the woodworking hobbyist are next to each other.
3. The person in a Mediterranean-style villa is the person who loves hip-hop music.
4. There are two houses between Arnold and the person residing in a Victorian house.
5. The person who loves jazz music is directly left of Eric.
6. The person who loves hip-hop music is somewhere to the left of the person who enjoys knitting.
7. Carol is the person who loves hip-hop music.
8. The person in a Craftsman-style house is Arnold.
9. The person in a ranch-style home is Eric.
10. The woodworking hobbyist is the person residing in a Victorian house.
11. The person who loves country music is in the first house.
12. There is one house between the person who paints as a hobby and the person living in a colonial-style house.
13. Alice is the photography enthusiast.
14. The person who enjoys gardening is Eric.
15. Bob is in the third house.
What is the value of attribute House for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}
|
6
|
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`, `Arnold`, `Peter`, `Eric`
- Each person has a unique level of education: `doctorate`, `bachelor`, `associate`, `master`, `high school`
- People use unique phone models: `oneplus 9`, `samsung galaxy s21`, `google pixel 6`, `iphone 13`, `huawei p50`
- People have unique favorite music genres: `rock`, `classical`, `jazz`, `pop`, `hip hop`
- The people keep unique animals: `bird`, `dog`, `horse`, `cat`, `fish`
- Each person has a unique hobby: `painting`, `cooking`, `knitting`, `gardening`, `photography`
## Clues:
1. The person who loves hip-hop music is in the second house.
2. The person who loves classical music is not in the fifth house.
3. The person who loves classical music is the fish enthusiast.
4. The person who uses an iPhone 13 is the person with a doctorate.
5. The person with a master's degree is not in the second house.
6. The person who enjoys knitting is not in the first house.
7. The person with a high school diploma is somewhere to the left of the photography enthusiast.
8. The person who paints as a hobby is somewhere to the right of the person who loves rock music.
9. Arnold is the bird keeper.
10. The person with a bachelor's degree is the person who uses a Google Pixel 6.
11. The person with a bachelor's degree is somewhere to the right of the person who paints as a hobby.
12. Bob is not in the fourth house.
13. The person who loves cooking is the person who loves classical music.
14. The person who loves cooking is the person who uses a Huawei P50.
15. The person who keeps horses is somewhere to the right of Arnold.
16. The person who paints as a hobby is the person who uses a OnePlus 9.
17. The person who uses a OnePlus 9 is somewhere to the left of the person with an associate's degree.
18. The person who uses an iPhone 13 is the dog owner.
19. The person with a high school diploma is Alice.
20. Peter is the person who enjoys gardening.
21. The person who paints as a hobby is the person who loves jazz music.
What is the value of attribute MusicGenre for the person whose attribute PhoneModel is oneplus 9? Please reason step by step, and put your final answer within \boxed{}
|
jazz
|
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`, `Bob`, `Alice`, `Peter`, `Arnold`, `Carol`
- They all have a unique favorite flower: `daffodils`, `carnations`, `roses`, `tulips`, `iris`, `lilies`
- People have unique heights: `very tall`, `super tall`, `average`, `short`, `very short`, `tall`
- Each person has a favorite color: `blue`, `yellow`, `purple`, `red`, `green`, `white`
- Everyone has a unique favorite cigar: `yellow monster`, `pall mall`, `prince`, `blue master`, `blends`, `dunhill`
- People use unique phone models: `google pixel 6`, `iphone 13`, `xiaomi mi 11`, `samsung galaxy s21`, `huawei p50`, `oneplus 9`
## Clues:
1. The person who uses an iPhone 13 is the person who loves white.
2. Peter is the person whose favorite color is red.
3. The Prince smoker is directly left of the person partial to Pall Mall.
4. The person who smokes Blue Master and the Dunhill smoker are next to each other.
5. The person who loves blue is the person who uses a Xiaomi Mi 11.
6. The person who loves white is Alice.
7. The person who uses a OnePlus 9 is somewhere to the left of Arnold.
8. The person who is tall is the person who smokes Blue Master.
9. The person who uses a Samsung Galaxy S21 is the Dunhill smoker.
10. Bob is directly left of the person who is very short.
11. The person who loves a bouquet of daffodils is Eric.
12. The person who loves the boquet of iris is the person who uses a OnePlus 9.
13. The person who uses a Huawei P50 is the person who is very short.
14. The person who smokes many unique blends is directly left of Peter.
15. The Prince smoker is not in the first house.
16. The person who is very tall is the person who loves a bouquet of daffodils.
17. The person who uses a Google Pixel 6 is not in the fifth house.
18. The person who is tall is not in the third house.
19. The person who uses a Huawei P50 is somewhere to the left of the person who uses a Samsung Galaxy S21.
20. Arnold is somewhere to the left of the person who is short.
21. The person who loves the vase of tulips is not in the first house.
22. The person who loves the boquet of lilies is directly left of the person who loves yellow.
23. Arnold is the person who loves purple.
24. The person who uses a OnePlus 9 and the person who uses a Samsung Galaxy S21 are next to each other.
25. The person who has an average height is not in the first house.
26. The person who loves the rose bouquet is Alice.
What is the value of attribute Cigar for the person whose attribute Color is yellow? Please reason step by step, and put your final answer within \boxed{}
|
pall mall
|
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`, `Eric`, `Peter`
- Everyone has something unique for lunch: `spaghetti`, `stew`, `pizza`, `grilled cheese`
- The people keep unique animals: `fish`, `horse`, `bird`, `cat`
## Clues:
1. The fish enthusiast is Alice.
2. Eric is in the second house.
3. The person who is a pizza lover is directly left of the fish enthusiast.
4. Arnold is the bird keeper.
5. The person who loves eating grilled cheese is Peter.
6. The cat lover and the person who loves the stew are next to each other.
7. The person who is a pizza lover is somewhere to the right of Eric.
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{}
|
grilled cheese
|
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: `Eric`, `Carol`, `Bob`, `Peter`, `Arnold`, `Alice`
- Everyone has something unique for lunch: `stew`, `stir fry`, `soup`, `grilled cheese`, `pizza`, `spaghetti`
- Each person has a unique favorite drink: `root beer`, `milk`, `boba tea`, `water`, `coffee`, `tea`
- People have unique heights: `super tall`, `short`, `tall`, `very short`, `very tall`, `average`
- Each person has a unique birthday month: `sept`, `feb`, `may`, `mar`, `jan`, `april`
## Clues:
1. The person whose birthday is in January is Peter.
2. The person who loves eating grilled cheese is the person whose birthday is in May.
3. The person whose birthday is in April is the person who loves stir fry.
4. The person who loves the stew is the one who only drinks water.
5. The person whose birthday is in September is not in the fourth house.
6. The person who likes milk is somewhere to the left of the person who loves stir fry.
7. The coffee drinker is in the fifth house.
8. The person who loves eating grilled cheese is directly left of the one who only drinks water.
9. The person whose birthday is in May is directly left of Arnold.
10. The root beer lover is somewhere to the left of Alice.
11. There are two houses between Eric and the person who is a pizza lover.
12. The person who loves the spaghetti eater is the person who is very short.
13. The person who is super tall is the tea drinker.
14. The person who is tall is not in the third house.
15. Peter is directly left of the person who is a pizza lover.
16. The person who likes milk is somewhere to the right of the one who only drinks water.
17. The person who is very short is the coffee drinker.
18. The person who is super tall is Bob.
19. The person who is super tall and the person whose birthday is in March are next to each other.
20. Carol is somewhere to the left of the person who has an average height.
21. The person who is very tall is the person who loves the stew.
22. There is one house between the person who loves eating grilled cheese and Peter.
What is the value of attribute Birthday for the person whose attribute Food is stew? Please reason step by step, and put your final answer within \boxed{}
|
sept
|
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`, `Peter`, `Alice`, `Eric`
- Each person has a unique favorite drink: `milk`, `root beer`, `coffee`, `tea`, `water`
- Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
- They all have a unique favorite flower: `daffodils`, `roses`, `lilies`, `tulips`, `carnations`
- Each person has a unique hobby: `painting`, `cooking`, `photography`, `gardening`, `knitting`
## Clues:
1. Alice is not in the fourth house.
2. The root beer lover is the person who enjoys gardening.
3. The person whose favorite color is green is the coffee drinker.
4. The person whose favorite color is green is the person who loves the boquet of lilies.
5. The person who loves blue is somewhere to the right of the person who loves a bouquet of daffodils.
6. The person who loves cooking is the person who loves blue.
7. Eric is directly left of the tea drinker.
8. The one who only drinks water is Peter.
9. Arnold is the photography enthusiast.
10. The person who loves white is the person who loves the rose bouquet.
11. There is one house between the person who loves a carnations arrangement and the person whose favorite color is red.
12. The person who loves cooking is somewhere to the left of the person who paints as a hobby.
13. The one who only drinks water is in the third house.
14. The person who loves a carnations arrangement is the root beer lover.
15. The person who loves white is in the second house.
What is the value of attribute Flower for the person whose attribute Name is Alice? Please reason step by step, and put your final answer within \boxed{}
|
lilies
|
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.