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Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients \textit{equivalent} if $p(\alpha)\equiv q(\alpha)\pmod{3}$. It is known that every such polynomial is equivalent to exactly one of $0, 1, x, x^2, \ldots, x^{727}$. Find the largest integer $n < 728$ for which there exists a polynomial $p$ such that $p^3 - p - x^n$ is equivalent to $0$.
|
727
|
1/8
|
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
|
495
|
3/8
|
A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained within the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
|
323
|
4/8
|
The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.
|
16184704896
|
2/8
|
You and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses $5$ "lucky" buckets by marking an "X" on the bottom that you cannot see. However, he tells you that they either form a vertical or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:
- Either $\{(a, b), (a, b + 1), (a, b + 2), (a, b + 3), (a, b + 4)\}$,
- Or $\{(b, a), (b + 1, a), (b + 2, a), (b + 3, a), (b + 4, a)\}$,
where $1 \le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a "lucky" bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is "lucky"?
|
9
|
0/8
|
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
|
118
|
5/8
|
Let $T$ be a trapezoid with two right angles and side lengths $4$, $4$, $5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.
|
120
|
0/8
|
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2} = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
|
108
|
0/8
|
Given the expression:
\[ A = 3\sum_{m=1}^{n^2}\left(\frac{1}{2} - \{\sqrt{m}\}\right) \]
where $n$ is a positive integer. Find the largest integer $k$ such that $n^k$ divides $\lfloor A \rfloor$ (the greatest integer less than or equal to $A$).
|
1
|
2/8
|
In a local frisbee league, teams have 7 members including at least one woman per team, and each of the 5 teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on the tournament committee, except the host team, which selects three members, ensuring at least one woman from each team is included. How many possible 11-member tournament committees are there?
|
97200
|
3/8
|
In triangle $ABC$, point $E$ is on side $BC$, $\angle BAE = 20^\circ$ and $\angle EAC = 40^\circ$. Determine the measure of angle $BEA$, in degrees.
|
80^\circ
|
1/8
|
Ed and Sue cycle, jog, and swim at constant rates. Ed covers 66 kilometers biking for 3 hours, jogging for 2 hours, and swimming for 4 hours, while Sue covers 96 kilometers after jogging for 3 hours, swimming for 2 hours, and biking for 4 hours. Their cycling, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's cycling, jogging, and swimming rates.
|
612
|
3/8
|
Our school's boys basketball team has 16 players, including a set of twins, Ben and Jerry, and a set of triplets, Tom, Tim, and Ted. In how many ways can we choose 5 players if at most one of the triplets and one of the twins can be in the starting lineup?
|
3102
|
4/8
|
The planet Zircon orbits its star in an elliptical path with the star located at one of the focusses. The closest Zircon gets to its star (perigee) is 3 astronomical units (AU) and its furthest distance (apogee) is 15 AU. Determine the distance from Zircon to its star when Zircon is halfway through its orbital path.
|
9 \text{ AU}
|
3/8
|
Each triangle in a sequence of four connected triangles is a 30-60-90 triangle, and the hypotenuse of each triangle is the longer leg of the subsequent triangle. The hypotenuse of the largest triangle is 16 centimeters. Determine the length of the longer leg of the smallest triangle.
|
9 \text{ cm}
|
4/8
|
Points $P$, $Q$, $R$, $S$, and $T$ are located in 2-dimensional space with $PQ = QR = RS = ST = TP = 3$ units and $\angle PQR = \angle RST = \angle STP = 120^\circ$. The line through $PQ$ is parallel to the segment $ST$. What is the perimeter of triangle $QRS$?
|
9 \text{ units}
|
1/8
|
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.3 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
draw((-8,i)--(8,i),gray(0.7));
}
draw((-8,0)--(8,0),Arrows(6));
draw((0,-8)--(0,8),Arrows(6));
label("$x$", (8,0), E);
label("$y$", (0,8), N);
draw(graph(func,-3,3),red);
label("$y = f(x)$", (4,-3), UnFill);
[/asy]
The graph of $y = h(x)$ is obtained by applying a vertical reflection to $f(x)$, then reflecting it on the $y$-axis, and finally shifting 5 units to the right. What is $h(x)$ in terms of $f(x)$?
|
h(x) = -f(5 - x)
|
0/8
|
A school has eight identical copies of a specific textbook. During the school day, some of these textbooks can be in the classroom, and others in the library. How many different ways are there for some of the books to be in the classroom and the rest in the library if at least one book must be in both locations?
|
7
|
2/8
|
The time right now is exactly 8:30 a.m. What time will it be in 2423 minutes?
|
12:53 \text{ a.m.}
|
3/8
|
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a rectangular field. The rectangle has three times as many posts on its longer side as on the shorter side. What is the perimeter, in feet, of the fence?
|
236 \text{ feet}
|
5/8
|
Chandra has five bowls of different colors: red, blue, yellow, green, and purple. She also has matching glasses for each bowl except for the purple bowl, for which she has no glass. If she chooses a bowl and a glass from the cupboard, excluding the possibility of choosing the purple bowl unless she chooses no glass, how many valid pairings are possible?
|
17
|
1/8
|
In a 6 by 6 grid, each of the 36 small squares measures 1.5 cm by 1.5 cm. The grid is fully shaded in grey. Six unshaded shapes are then added: one medium hexagon, four small circles placed symmetrically, and one larger circle in the center. The diameter of the small circles equals the side of a small square (1.5 cm), and the larger circle has a diameter of 3 cm; the side length of the hexagon equals 1.5 cm. Assume all circles and the hexagon are placed such that no parts of these shapes overlap. The area of the visible shaded region can be written in the form $A-B\pi - C\sqrt{3}$. What is the value $A+B+C$?
|
88.875
|
0/8
|
Define a set of integers as "spacy" if it includes no more than one out of any three consecutive integers. Determine the count of all "spacy" subsets of the set $\{1, 2, \dots, 15\}$, including the empty set.
|
406
|
5/8
|
In the diagram below, lines $m$ and $n$ are parallel. A transversal cuts lines $m$ and $n$ at points A and B respectively, forming a $40^\circ$ angle with line $m$. A second transversal intersects $m$ at C and $n$ at D, not at right angles. Find the measure of angle $y$ which is the angle between the second transversal and line $n$ at D. Assume the angle formed between the first transversal and the second transversal on the same side of the parallel lines is $60^\circ$.
|
80^\circ
|
3/8
|
Berengere and her friend Liam, who is visiting from the US, are at a café in Paris that accepts both euros and dollars. They want to buy a pastry that costs 8 euros. Liam has a ten-dollar bill. How many euros does Berengere need to contribute to the cost of the pastry if 1 euro = 1.10 USD?
|
0 \text{ euros}
|
3/8
|
The circumference of a particular circle is 36 cm. Calculate both the area and the diameter of the circle. Express the area as a common fraction in terms of $\pi$ and the diameter in centimeters.
|
\frac{36}{\pi}
|
0/8
|
Let \( x, y, z \) be natural numbers satisfying the condition \(\frac{1}{x} - \frac{1}{y} = \frac{1}{z}\). Prove that \(\gcd(x, y, z) \cdot xyz\) is the square of a natural number.
|
\gcd(x, y, z) \cdot xyz \text{ is a perfect square}
|
2/8
|
3. Note that $(n+1)^{2}-(n-1)^{2}=4 n$, i.e., numbers $a=4 n$ that are multiples of four can be represented as the difference of squares of two integers, and therefore, with the condition of zero, can be represented as the sum or difference of four squares. If $a=4 n \pm 1$, then the representation is
$$
a=(n+1)^{2}-(n-1)^{2} \pm 1^{2}
$$
If $a=4 n+2$, then it can be represented as the sum or difference of four squares:
$$
a=(n+1)^{2}-(n-1)^{2}+1^{2}+1^{2}
$$
The number 2021 can be written using representation (1), so, for example, $2021=4 \cdot 505+1, n=505,2021=506^{2}-504^{2}+1^{2}$.
|
2021 = 506^2 - 504^2 + 1^2
|
1/8
|
$2 \cdot 6$ In the set $M$ of the first 100 odd numbers $1,3, \cdots, 199$, select a subset such that no number in the subset can divide another. How many elements can this subset have at most?
|
67
|
2/8
|
## Task 6B - 191246B
In a dark room, there are 20 individual gloves of the same size, namely
- 5 white gloves for the right hand
- 5 white gloves for the left hand
- 5 black gloves for the right hand
- 5 black gloves for the left hand
Two gloves are considered a matching pair if and only if they are of the same color and one is for the right hand, the other for the left hand.
A draw is understood to be the removal of a single glove, without the possibility of selecting by color and form. A game of $n$ draws consists of performing $n$ draws in succession, collecting the gloves thus removed, and only after these $n$ draws determining whether there is (at least) one matching pair among the $n$ removed gloves. The game is considered successful if and only if this is the case.
a) Determine the smallest natural number $n$ with the property that a game of $n$ draws is guaranteed to be successful!
b) Determine the smallest natural number $k$ with the property that a game of $k$ draws is successful with a probability greater than 0.99!
|
11
|
0/8
|
Given a tree with $n$ vertices, $n \geq 2$, numbers $x_{1}, x_{2}, \ldots, x_{n}$ are assigned to the vertices. On each edge, the product of the numbers at its endpoints is written. Let $S$ denote the sum of the numbers on all the edges. Prove that $\sqrt{n-1}\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right) \geq 2 S$.
|
\sqrt{n-1}\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right) \geq 2 S
|
4/8
|
1. Determine the number of distinct values of the expression
$$
\frac{n^{2}-2}{n^{2}-n+2}
$$
where $n \in\{1,2, \ldots, 100\}$.
|
98
|
5/8
|
6.30 There is a sports competition consisting of $M$ events, with athletes $A, B, C$ participating. In each event, the first, second, and third places receive $p_{1}, p_{2}, p_{3}$ points, respectively, where $p_{1}, p_{2}, p_{3}$ are positive integers, and $p_{1}>p_{2}>p_{3}$. In the end, $A$ scores 22 points, $B$ and $C$ both score 9 points, and $B$ comes in first in the 100-meter dash.
Find the value of $M$, and determine who came in second in the high jump.
|
5
|
0/8
|
12. Given four points $O, A, B, C$ on a plane, satisfying $O A=4, O B=3, O C=2, \overrightarrow{O B} \cdot \overrightarrow{O C}=3$, then the maximum value of the area of $\triangle A B C$ is $\qquad$.
|
\dfrac{4\sqrt{7} + 3\sqrt{3}}{2}
|
2/8
|
2. What is the minimum number of factors that need to be crossed out from the number 99! (99! is the product of all numbers from 1 to 99) so that the product of the remaining factors ends in 2?
|
20
|
2/8
|
## Task A-4.6.
Let $M$ and $N$ be the feet of the altitudes from vertices $A$ and $B$ of an acute-angled triangle $ABC$. Let $Q$ be the midpoint of segment $\overline{M N}$, and $P$ be the midpoint of side $\overline{A B}$. If $|M N|=10$ and $|A B|=26$, determine the length of $|P Q|$.
|
12
|
5/8
|
In a picture trading, the price of the frames is directly proportional to the value of the paintings inside them. To reduce the price difference between certain paintings, the dealer exchanges the frames of two paintings. In one case, a painting that cost five times as much as the other, after the frames were exchanged, only cost three times as much. How will the price ratio of the paintings titled "Winter Landscape" and "Village Road" change if, before the frames were exchanged, the "Winter Landscape" cost nine times as much as the "Village Road"?
|
4
|
5/8
|
19. (3 points) A male and a female track and field athletes are practicing running on a 110-meter slope (the top of the slope is $A$, the bottom is $B$). Both start from point $A$ at the same time, running back and forth between $A$ and $B$ continuously. If the male athlete's uphill speed is 3 meters per second and downhill speed is 5 meters per second; the female athlete's uphill speed is 2 meters per second and downhill speed is 3 meters per second, then the location of their second face-to-face meeting is $\qquad$ meters away from point $A$.
|
38.5
|
0/8
|
## Task A-1.4.
Determine the smallest natural number whose sum of digits is divisible by 7 and has the property that the sum of the digits of its successor is also divisible by 7.
|
69999
|
4/8
|
For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$.
|
15
|
3/8
|
Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$.
|
35
|
0/8
|
Consider the permutation of $1, 2, \ldots, n$, which we denote as $\{a_1, a_2, \ldots, a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions:
1. $a_1 = 1$
2. $|a_i - a_{i-1}| \le 2$, for $i = 2, 3, \ldots, n$
What is the residue when we divide $f(2015)$ by $4$?
|
3
|
0/8
|
A sequence consists of the digits $122333444455555\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501^{\text{st}}$ and $4052^{\text{nd}}$ digits of this sequence.
|
13
|
4/8
|
Evan has $66000$ omons, particles that can cluster into groups of a perfect square number of omons. An omon in a cluster of $n^2$ omons has a potential energy of $\frac{1}{n}$. Evan accurately computes the sum of the potential energies of all the omons. Compute the smallest possible value of his result.
|
284
|
1/8
|
On a straight line $\ell$, there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn, which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.
|
30
|
3/8
|
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$.
|
30024
|
4/8
|
Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?
|
990
|
0/8
|
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
|
7
|
0/8
|
Define a common chord between two intersecting circles as the line segment connecting their two intersection points. Let $\omega_1, \omega_2, \omega_3$ be three circles with radii $3, 5,$ and $7$, respectively. Suppose they are arranged such that:
1. The common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$.
2. The common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$.
3. The common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$.
Compute the square of the area of the triangle formed by the centers of the three circles.
|
96
|
4/8
|
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruits is the banana!" The last fruit responds "No, one of the \emph{last} forty fruits is the banana!" The fruit in the middle yells "I'm the banana!" In how many positions could the banana be?
|
21
|
2/8
|
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and 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
|
Let $x_1, x_2, \ldots, x_6$ be non-negative reals such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ and $x_1x_3x_5 + x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of the expression:
$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2.$$
Find $p+q$.
|
559
|
1/8
|
A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We define a pair of players $A$ and $B$ as a \textit{dominant pair} if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
|
2015
|
2/8
|
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
|
7
|
2/8
|
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.
Find the total number of pieces of candy given to the children who always tell the truth.
|
7
|
1/8
|
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. Find the area of the region inside both the square and the rectangle. The original answer is in \frac{k}{m} format, please give the value of k + m.
|
133
|
0/8
|
Suppose \(x\) and \(y\) are positive real numbers such that
\(x^y=2^{64}\) and \((\log_2{x})^{\log_2{y}}=2^{7}.\)
The original answer is in the form \(k+\sqrt{m}\). Please find the value of \(k + m\).
|
5
|
2/8
|
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself.
|
359
|
3/8
|
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$? If the answer is $x^\circ$, what is the value of $x$?
|
113
|
3/8
|
Isosceles trapezoid \(ABCD\) has parallel sides \(\overline{AD}\) and \(\overline{BC},\) with \(BC < AD\) and \(AB = CD.\) There is a point \(P\) in the plane such that \(PA=1, PB=2, PC=3,\) and \(PD=4.\) Find the value of \(\frac{BC}{AD}.\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is in simplest form. Please give the value of \(k + m\).
|
4
|
2/8
|
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$?
$\begin{array}{|c|c|c|}\hline-2&9&5\\\hline&&-1\\\\\hline x&&8\\\hline\end{array}$
|
8
|
4/8
|
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. If the sum of the areas of all circles in $S$ is $A\pi$, what is the value of $A$?
|
136
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Movie-Genre, Nationality, Sport.
The attributes have the following possible values:
- Job: accountant, bartender, lawyer, writer
- Movie-Genre: animation, disaster, drama, horror
- Nationality: chinese, nigerian, polish, turkish
- Sport: ice-hockey, rowing, soccer, swimming
Given the following premises about the line of people:
- Job:writer is not to the left of Movie-Genre:drama
- Sport:ice-hockey != Nationality:chinese
- Nationality:nigerian is on the left or right of Movie-Genre:horror
- Sport:rowing == Job:accountant or Sport:rowing == Nationality:turkish, but not both
- Job:lawyer is on the far left or far right
- Movie-Genre:horror is somewhere to the right of Nationality:nigerian
- Job:writer is not to the right of Nationality:nigerian
- Sport:soccer is on the left or right of Nationality:chinese
- Job:writer == Sport:rowing or Job:writer == Movie-Genre:animation or both
- Job:lawyer is on the left or right of Job:bartender
- Job:writer is in an even position
- Sport:soccer is on the far right
Answer the following question:
Question: Question: What is the Nationality of the person who has/is drama??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
polish
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Food, Hobby, Pet, Transport.
The attributes have the following possible values:
- Food: cauliflower, cucumber, kale, lemon
- Hobby: hiking, rock-climbing, skydiving, video-games
- Pet: bird, cat, rabbit, rat
- Transport: boat, car, motorbike, snowmobile
Given the following premises about the line of people:
- Food:lemon != Transport:car
- Transport:car is somewhere between Food:kale and Transport:motorbike
- Pet:rabbit is somewhere to the left of Food:cucumber
- Hobby:rock-climbing and Food:cucumber have different parity positions
- Food:kale is not to the left of Hobby:rock-climbing
- Hobby:video-games is not to the right of Pet:rabbit
- Pet:rat == Food:cucumber or Pet:rat == Transport:snowmobile or both
- Hobby:skydiving is not to the left of Hobby:rock-climbing
- Pet:rat is somewhere to the left of Food:cucumber
- Pet:bird is in an odd position
- Pet:rabbit != Hobby:video-games
- Transport:boat is somewhere to the right of Food:kale
Answer the following question:
Question: Question: What is the Hobby of the person who has/is rat??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
rock-climbing
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Food, Music-Genre, Nationality, Transport.
The attributes have the following possible values:
- Job: fisherman, librarian, paramedic, social-worker, videographer
- Food: broccoli, cherry, cucumber, lime, zucchini
- Music-Genre: classical, electronic, folk, pop, rock
- Nationality: argentine, canadian, italian, malaysian, pakistani
- Transport: airplane, jet-ski, subway, taxi, train
Given the following premises about the line of people:
- Food:cucumber and Nationality:pakistani have different parity positions
- Food:cucumber == Transport:jet-ski or Nationality:canadian == Food:cucumber or both
- Food:lime == Nationality:pakistani or Food:lime == Transport:taxi or both
- Job:paramedic and Nationality:argentine have the same parity positions
- Food:broccoli != Music-Genre:rock
- Music-Genre:classical is on the far right
- Food:cucumber == Music-Genre:classical or Music-Genre:classical == Job:fisherman or both
- Nationality:argentine == Music-Genre:folk or Food:lime == Nationality:argentine, but not both
- Transport:airplane is somewhere between Nationality:italian and Nationality:canadian
- Food:lime is on the left or right of Transport:taxi
- Music-Genre:pop is not to the left of Transport:jet-ski
- Job:videographer is on the far right
- Food:zucchini and Transport:jet-ski have the same parity positions
- Food:cherry == Transport:subway or Transport:subway == Job:paramedic or both
- Music-Genre:rock is in an odd position
- Job:librarian is somewhere to the right of Food:broccoli
- Transport:jet-ski is on the left or right of Job:fisherman
- Transport:airplane is somewhere to the right of Music-Genre:pop
- Nationality:italian and Job:fisherman have different parity positions
- Nationality:argentine is on the left or right of Music-Genre:classical
Answer the following question:
Question: Question: What is the Nationality of the person who is fisherman??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
pakistani
|
0/8
|
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Beverage, Sport, Transport.
The attributes have the following possible values:
- Job: architect, doctor, journalist, project-manager, security-guard, social-worker
- Beverage: almond-milk, coffee, lemonade, milk, mirinda, water
- Sport: climbing, ice-hockey, skateboarding, snowboarding, surfing, water-polo
- Transport: bike, boat, jet-ski, quad-bike, roller, train
Given the following premises about the line of people:
- Sport:skateboarding is on the left of Transport:bike
- Sport:ice-hockey is somewhere to the left of Job:journalist
- Beverage:coffee is somewhere to the left of Job:doctor
- Transport:boat is on the left or right of Beverage:almond-milk
- Transport:jet-ski is on the right of Job:doctor
- Job:social-worker == Beverage:coffee or Transport:bike == Job:social-worker or both
- Beverage:coffee is not to the left of Beverage:mirinda
- Sport:climbing is on the left or right of Beverage:water
- Sport:snowboarding is in an odd position
- Transport:jet-ski is somewhere to the left of Job:architect
- Sport:surfing is on the left or right of Beverage:almond-milk
- Sport:ice-hockey and Job:journalist have different parity positions
- Sport:skateboarding != Job:security-guard
- Job:social-worker is on the left of Transport:quad-bike
- Beverage:coffee is not to the left of Transport:train
- Transport:roller != Job:architect or Transport:roller != Beverage:lemonade or both
- Sport:climbing and Job:doctor have the same parity positions
- Sport:snowboarding is on the left of Beverage:lemonade
- Sport:climbing is somewhere to the left of Sport:ice-hockey
- Transport:boat and Sport:climbing have different parity positions
Answer the following question:
Question: Question: What is the Job of the person who has/is water??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
security-guard
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Beverage, Transport.
The attributes have the following possible values:
- Job: chef, librarian, photographer, pilot
- Hobby: baking, filmmaking, traveling, woodworking
- Beverage: iced-tea, soy-milk, sprite, tea
- Transport: airplane, jet-ski, subway, van
Given the following premises about the line of people:
- Beverage:iced-tea is somewhere between Hobby:filmmaking and Hobby:woodworking
- Transport:van and Hobby:filmmaking have different parity positions
- Transport:van is not to the right of Beverage:sprite
- Beverage:tea is not to the left of Transport:jet-ski
- Transport:subway and Job:chef have different parity positions
- Hobby:woodworking == Job:chef or Hobby:woodworking == Beverage:sprite, but not both
- Transport:van is not to the left of Beverage:tea
- Hobby:filmmaking is somewhere between Hobby:baking and Job:chef
- Hobby:woodworking is not to the right of Beverage:iced-tea
- Beverage:sprite and Hobby:filmmaking have the same parity positions
- Transport:subway and Job:librarian have different parity positions
- Job:pilot is not to the left of Hobby:filmmaking
Answer the following question:
Question: Question: What is the Job of the person who has/is soy-milk??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
pilot
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Hobby, Beverage, Movie-Genre.
The attributes have the following possible values:
- Job: coach, entrepreneur, musician, project-manager, writer
- Hobby: board-games, fishing, magic-tricks, sudoku, woodworking
- Beverage: cola, hot-chocolate, juice, sprite, water
- Movie-Genre: action, crime, mystery, western, zombie
Given the following premises about the line of people:
- Beverage:cola is somewhere to the left of Movie-Genre:mystery
- Job:coach is on the far left or far right
- Job:project-manager is in the middle
- Hobby:sudoku is not to the left of Movie-Genre:action
- Hobby:fishing is in the middle
- Job:musician is somewhere to the right of Hobby:board-games
- Beverage:cola is on the left or right of Beverage:water
- Job:musician is somewhere to the left of Movie-Genre:action
- Beverage:juice is somewhere to the left of Job:entrepreneur
- Hobby:board-games != Movie-Genre:western
- Hobby:fishing is not to the left of Beverage:water
- Beverage:juice is somewhere to the left of Job:coach
- Hobby:woodworking is on the far right
- Hobby:sudoku != Beverage:hot-chocolate
- Movie-Genre:western and Beverage:juice have the same parity positions
- Beverage:sprite is on the left or right of Movie-Genre:crime
- Beverage:hot-chocolate is not to the right of Job:entrepreneur
Answer the following question:
Question: Question: What is the Job of the person who has/is board-games??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
writer
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Food, Hobby, Beverage, Music-Genre, Sport.
The attributes have the following possible values:
- Job: coach, designer, nurse, project-manager, security-guard
- Food: carrot, cucumber, pear, pumpkin, zucchini
- Hobby: camping, gardening, reading, singing, sudoku
- Beverage: 7up, almond-milk, milk, mirinda, tea
- Music-Genre: classical, folk, gospel, house, r&b
- Sport: baseball, golf, skiing, volleyball, water-polo
Given the following premises about the line of people:
- Beverage:mirinda is somewhere to the left of Job:designer
- Beverage:7up is on the right of Food:pumpkin
- Food:cucumber is on the right of Job:project-manager
- Food:cucumber is somewhere to the right of Music-Genre:classical
- Sport:baseball is somewhere to the right of Hobby:camping
- Beverage:milk == Sport:volleyball
- Food:pear is between Music-Genre:folk and Music-Genre:r&b
- Sport:water-polo == Music-Genre:r&b
- Music-Genre:house is somewhere to the left of Food:pumpkin
- Music-Genre:gospel is somewhere to the left of Job:nurse
- Hobby:singing is on the far left
- Beverage:tea is on the left or right of Sport:skiing
- Job:coach is somewhere to the right of Hobby:gardening
- Job:coach is somewhere to the right of Beverage:almond-milk
- Hobby:sudoku == Job:nurse
- Beverage:milk is on the far left
- Beverage:tea != Music-Genre:folk
- Food:carrot is somewhere to the left of Hobby:reading
Answer the following question:
Question: Question: What is the Food of the person who has/is r&b??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
zucchini
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Food, Nationality, Pet, Sport.
The attributes have the following possible values:
- Job: dressmaker, mechanic, photographer, scientist
- Hobby: card-games, chess, hiking, singing
- Food: kale, orange, papaya, raspberry
- Nationality: italian, pakistani, polish, thai
- Pet: cat, chinchilla, mouse, pony
- Sport: badminton, biathlon, volleyball, weightlifting
Given the following premises about the line of people:
- Pet:cat is somewhere between Hobby:hiking and Job:scientist
- Sport:biathlon and Job:scientist have different parity positions
- Nationality:polish is not to the left of Food:kale
- Pet:mouse is not to the right of Pet:chinchilla
- Hobby:chess is somewhere between Hobby:hiking and Food:raspberry
- Nationality:thai and Sport:weightlifting have different parity positions
- Job:dressmaker is somewhere between Job:mechanic and Nationality:polish
- Hobby:card-games and Job:scientist have different parity positions
- Job:scientist is somewhere between Hobby:chess and Sport:volleyball
- Food:raspberry == Nationality:thai or Food:raspberry == Pet:mouse, but not both
- Pet:mouse == Nationality:italian or Nationality:italian == Food:kale or both
- Sport:badminton is not to the left of Job:scientist
- Food:orange and Nationality:pakistani have the same parity positions
- Food:kale is not to the left of Food:raspberry
- Sport:volleyball is not to the left of Sport:badminton
Answer the following question:
Question: Question: What is the Nationality of the person who has/is kale??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
polish
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Movie-Genre, Music-Genre, Beverage, Sport.
The attributes have the following possible values:
- Nationality: chinese, german, pakistani, russian
- Hobby: gardening, puzzles, singing, traveling
- Movie-Genre: documentary, fantasy, satire, sports
- Music-Genre: classical, country, disco, folk
- Beverage: 7up, coffee, hot-chocolate, mirinda
- Sport: climbing, ice-hockey, rugby, tennis
Given the following premises about the line of people:
- Hobby:puzzles is not to the left of Hobby:traveling
- Nationality:russian != Movie-Genre:sports
- Hobby:traveling is on the right of Music-Genre:disco
- Nationality:german == Music-Genre:disco
- Movie-Genre:satire is not to the right of Beverage:mirinda
- Beverage:7up and Nationality:german have different parity positions
- Hobby:puzzles is not to the right of Music-Genre:country
- Movie-Genre:fantasy is on the right of Hobby:puzzles
- Beverage:hot-chocolate is on the far right
- Movie-Genre:sports != Beverage:7up
- Movie-Genre:documentary != Sport:rugby
- Nationality:chinese == Hobby:gardening
- Beverage:coffee is not to the left of Hobby:traveling
- Hobby:gardening is on the right of Sport:climbing
- Music-Genre:country is on the left of Music-Genre:folk
- Sport:rugby is on the left or right of Sport:tennis
Answer the following question:
Question: Question: What is the Hobby of the person who is pakistani??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
puzzles
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Food, Pet, Transport.
The attributes have the following possible values:
- Job: dressmaker, nurse, photographer, social-worker, writer
- Food: broccoli, orange, pomegranate, strawberry, tomato
- Pet: chinchilla, ferret, guinea-pig, lizard, rat
- Transport: car, jet-ski, motorbike, roller, skateboard
Given the following premises about the line of people:
- Food:broccoli is on the left or right of Food:pomegranate
- Transport:motorbike is somewhere between Transport:roller and Transport:jet-ski
- Pet:chinchilla is on the right of Job:dressmaker
- Food:orange is somewhere to the right of Pet:chinchilla
- Food:strawberry is on the left or right of Job:nurse
- Transport:skateboard is somewhere to the left of Pet:lizard
- Pet:rat is somewhere to the right of Job:photographer
- Transport:skateboard is on the left of Food:orange
- Pet:guinea-pig is on the left of Pet:lizard
- Job:writer and Pet:guinea-pig have different parity positions
- Transport:roller is in an even position
- Food:pomegranate is not to the left of Pet:lizard
- Pet:rat is on the left or right of Food:strawberry
Answer the following question:
Question: Question: What is the Food of the person who is nurse??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
orange
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Beverage, Hobby, Pet, Transport.
The attributes have the following possible values:
- Beverage: cola, fanta, iced-tea, juice, mirinda
- Hobby: camping, magic-tricks, reading, singing, traveling
- Pet: ferret, fish, frog, goldfish, pony
- Transport: car, jet-ski, ship, train, trike
Given the following premises about the line of people:
- Beverage:fanta is somewhere to the left of Hobby:singing
- Hobby:camping is somewhere to the left of Hobby:reading
- Hobby:traveling == Pet:fish
- Transport:train is between Hobby:magic-tricks and Transport:ship
- Hobby:traveling is in an odd position
- Transport:jet-ski is somewhere to the right of Pet:ferret
- Transport:jet-ski is somewhere to the left of Hobby:singing
- Hobby:camping == Transport:car
- Pet:ferret == Hobby:reading
- Beverage:fanta == Pet:goldfish
- Hobby:reading is between Beverage:juice and Pet:pony
- Hobby:magic-tricks is somewhere to the left of Beverage:mirinda
- Beverage:cola == Pet:ferret
Answer the following question:
Question: Question: What is the Pet of the person who has/is cola??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
ferret
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Food, Movie-Genre, Music-Genre.
The attributes have the following possible values:
- Job: electrician, engineer, manager, police-officer
- Hobby: camping, chess, drawing, hiking
- Food: apple, blueberry, cranberry, pomegranate
- Movie-Genre: adventure, horror, scientific, thriller
- Music-Genre: classical, country, metal, r&b
Given the following premises about the line of people:
- Hobby:hiking is somewhere to the left of Music-Genre:r&b
- Job:engineer is not to the right of Hobby:drawing
- Job:electrician is on the right of Music-Genre:metal
- Hobby:drawing is on the right of Hobby:hiking
- Movie-Genre:adventure is on the left of Job:electrician
- Food:apple is on the left of Music-Genre:r&b
- Movie-Genre:thriller is in an odd position
- Food:apple == Movie-Genre:scientific
- Music-Genre:classical is not to the left of Movie-Genre:horror
- Job:police-officer is somewhere to the right of Hobby:camping
- Food:cranberry is somewhere between Food:blueberry and Music-Genre:country
- Job:engineer is on the left of Job:police-officer
Answer the following question:
Question: Question: What is the Movie-Genre of the person who has/is apple??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
scientific
|
0/8
|
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Hobby, Beverage, Sport.
The attributes have the following possible values:
- Job: analyst, bartender, coach, designer, mechanic, pilot
- Hobby: chess, cooking, dancing, puzzles, sudoku, traveling
- Beverage: 7up, almond-milk, fanta, milk, sprite, tea
- Sport: cycling, golf, parkour, swimming, volleyball, weightlifting
Given the following premises about the line of people:
- Job:pilot is on the right of Beverage:fanta
- Sport:cycling is not to the right of Job:designer
- Hobby:sudoku is somewhere between Beverage:milk and Sport:parkour
- Beverage:tea is on the left of Job:bartender
- Sport:weightlifting is on the left of Beverage:almond-milk
- Beverage:almond-milk is not to the right of Sport:golf
- Beverage:fanta is on the right of Beverage:tea
- Beverage:fanta is in an even position
- Job:coach is on the far left or far right
- Sport:cycling is not to the left of Hobby:cooking
- Sport:parkour is in an odd position
- Sport:parkour is somewhere to the left of Job:designer
- Hobby:dancing is on the right of Sport:swimming
- Hobby:traveling is not to the left of Sport:swimming
- Job:mechanic is on the right of Job:analyst
- Job:coach is not to the left of Job:mechanic
- Beverage:sprite is somewhere to the right of Hobby:chess
- Sport:parkour is on the left of Hobby:sudoku
- Hobby:puzzles != Beverage:fanta
Answer the following question:
Question: Question: What is the Job of the person who has/is sprite??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
mechanic
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Movie-Genre, Pet, Sport, Transport.
The attributes have the following possible values:
- Nationality: american, canadian, nigerian, spanish
- Movie-Genre: adventure, musical, satire, zombie
- Pet: chinchilla, guinea-pig, horse, rabbit
- Sport: cricket, lacrosse, skiing, tennis
- Transport: motorbike, skateboard, subway, train
Given the following premises about the line of people:
- Movie-Genre:adventure is on the left of Sport:tennis
- Transport:motorbike is on the left or right of Movie-Genre:satire
- Pet:horse == Movie-Genre:musical
- Pet:rabbit is somewhere to the right of Sport:lacrosse
- Transport:subway == Pet:chinchilla
- Transport:train is on the left or right of Pet:rabbit
- Nationality:canadian is on the right of Pet:horse
- Pet:horse is somewhere to the left of Sport:cricket
- Pet:horse is somewhere to the right of Nationality:nigerian
- Nationality:nigerian is somewhere to the right of Transport:skateboard
- Movie-Genre:musical == Nationality:spanish
Answer the following question:
Question: Question: What is the Nationality of the person who has/is musical??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
spanish
|
3/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Transport.
The attributes have the following possible values:
- Job: accountant, dressmaker, engineer, police-officer
- Food: broccoli, cabbage, kiwi, peas
- Beverage: coffee, cola, fanta, hot-chocolate
- Transport: boat, motorbike, quad-bike, ship
Given the following premises about the line of people:
- Transport:quad-bike is somewhere to the right of Transport:motorbike
- Beverage:coffee is on the far left or far right
- Transport:motorbike == Beverage:hot-chocolate or Transport:motorbike == Job:police-officer, but not both
- Job:dressmaker != Transport:motorbike
- Food:broccoli is in an even position
- Beverage:cola and Food:cabbage have different parity positions
- Food:peas is not to the left of Beverage:hot-chocolate
- Transport:motorbike and Beverage:hot-chocolate have the same parity positions
- Transport:boat and Beverage:coffee have the same parity positions
- Transport:motorbike is somewhere to the right of Job:accountant
- Transport:quad-bike is somewhere between Food:kiwi and Job:accountant
- Beverage:cola and Job:dressmaker have the same parity positions
Answer the following question:
Question: Question: What is the Food of the person who is dressmaker??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
kiwi
|
0/8
|
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Music-Genre, Food.
The attributes have the following possible values:
- Nationality: american, french, malaysian, mexican, nigerian, polish
- Hobby: cooking, fishing, hiking, magic-tricks, singing, sudoku
- Music-Genre: blues, disco, metal, pop, reggae, trance
- Food: apricot, cucumber, garlic, lemon, pear, pomegranate
Given the following premises about the line of people:
- Hobby:cooking is on the left of Food:pear
- Hobby:fishing == Food:pomegranate
- Music-Genre:blues is on the left of Nationality:mexican
- Food:pear is between Hobby:magic-tricks and Hobby:cooking
- Food:pear is on the left of Music-Genre:trance
- Hobby:cooking is somewhere to the right of Nationality:polish
- Music-Genre:blues is somewhere to the left of Food:lemon
- Hobby:singing is on the left of Food:apricot
- Music-Genre:metal is somewhere to the right of Food:pomegranate
- Hobby:hiking is somewhere to the left of Nationality:nigerian
- Music-Genre:reggae is on the left of Music-Genre:blues
- Nationality:malaysian == Music-Genre:disco
- Food:cucumber is on the right of Music-Genre:pop
- Hobby:magic-tricks is in an odd position
- Nationality:american is on the left of Nationality:french
- Nationality:american is somewhere to the left of Music-Genre:reggae
Answer the following question:
Question: Question: What is the Nationality of the person who has/is blues??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
nigerian
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Movie-Genre, Music-Genre, Sport.
The attributes have the following possible values:
- Job: chef, journalist, software-developer, teacher
- Movie-Genre: adventure, sports, superhero, zombie
- Music-Genre: folk, gospel, reggae, soul
- Sport: badminton, biathlon, handball, parkour
Given the following premises about the line of people:
- Music-Genre:reggae and Sport:biathlon have different parity positions
- Sport:handball and Job:teacher have different parity positions
- Movie-Genre:adventure is on the right of Movie-Genre:zombie
- Music-Genre:gospel == Sport:biathlon
- Job:journalist is between Movie-Genre:adventure and Sport:badminton
- Sport:biathlon is on the right of Movie-Genre:sports
- Sport:badminton == Music-Genre:soul
- Movie-Genre:zombie is not to the right of Job:chef
- Sport:handball is on the left of Sport:badminton
Answer the following question:
Question: Question: What is the Music-Genre of the person who has/is sports??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
soul
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Movie-Genre, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Movie-Genre: action, comedy, fantasy, satire, thriller
- Music-Genre: folk, hip-hop, reggae, salsa, techno
- Pet: bird, fish, frog, goldfish, rabbit
- Transport: motorbike, ship, skateboard, snowmobile, trike
Given the following premises about the line of people:
- Pet:fish is somewhere to the right of Music-Genre:techno
- Movie-Genre:action is on the far right
- Pet:frog == Movie-Genre:fantasy
- Transport:trike is somewhere to the right of Transport:skateboard
- Movie-Genre:fantasy is on the right of Music-Genre:hip-hop
- Movie-Genre:satire is somewhere between Movie-Genre:thriller and Music-Genre:hip-hop
- Pet:bird is not to the left of Music-Genre:hip-hop
- Movie-Genre:comedy is somewhere to the left of Transport:snowmobile
- Transport:snowmobile is on the right of Pet:rabbit
- Movie-Genre:fantasy == Music-Genre:folk
- Transport:motorbike and Music-Genre:hip-hop have different parity positions
- Music-Genre:folk is somewhere to the left of Transport:ship
- Music-Genre:reggae is on the left or right of Transport:skateboard
Answer the following question:
Question: Question: What is the Movie-Genre of the person who has/is folk??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
fantasy
|
1/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Nationality, Pet.
The attributes have the following possible values:
- Job: dressmaker, freelancer, nurse, teacher
- Hobby: card-games, filmmaking, magic-tricks, woodworking
- Nationality: american, argentine, canadian, indonesian
- Pet: cat, hamster, mouse, rat
Given the following premises about the line of people:
- Job:teacher is in an even position
- Hobby:filmmaking is on the far left or far right
- Hobby:magic-tricks is in an even position
- Pet:cat is somewhere to the right of Hobby:woodworking
- Pet:cat is on the left or right of Nationality:argentine
- Pet:hamster != Hobby:filmmaking
- Pet:cat is on the right of Pet:mouse
- Hobby:woodworking is on the left or right of Hobby:card-games
- Job:nurse is somewhere to the right of Pet:cat
- Hobby:magic-tricks is on the right of Job:dressmaker
- Nationality:indonesian is between Nationality:canadian and Job:nurse
Answer the following question:
Question: Question: What is the Nationality of the person who has/is card-games??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
indonesian
|
1/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Nationality, Food, Hobby, Beverage, Pet, Transport.
The attributes have the following possible values:
- Nationality: dutch, german, italian, malaysian, mexican
- Food: apple, papaya, radish, spinach, zucchini
- Hobby: baking, filmmaking, hiking, reading, singing
- Beverage: coffee, cola, fanta, lemonade, tea
- Pet: chinchilla, ferret, fish, rat, snake
- Transport: car, helicopter, scooter, taxi, tram
Given the following premises about the line of people:
- Food:spinach is in the middle
- Pet:fish is somewhere to the right of Beverage:cola
- Beverage:cola is between Pet:ferret and Nationality:malaysian
- Food:papaya is somewhere to the right of Beverage:lemonade
- Nationality:mexican is somewhere between Pet:snake and Food:apple
- Hobby:reading is on the left or right of Pet:snake
- Pet:rat == Transport:taxi
- Transport:car is on the right of Hobby:hiking
- Food:apple is on the left of Nationality:dutch
- Hobby:hiking is on the right of Transport:helicopter
- Hobby:baking is somewhere to the left of Hobby:filmmaking
- Beverage:coffee is somewhere to the left of Beverage:fanta
- Beverage:cola is in an odd position
- Nationality:malaysian is on the left or right of Food:zucchini
- Transport:tram is on the right of Food:radish
- Nationality:german is on the left of Hobby:singing
- Pet:fish is in an odd position
- Pet:snake is somewhere to the right of Pet:chinchilla
- Food:apple == Beverage:tea
Answer the following question:
Question: Question: What is the Hobby of the person who has/is fish??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
reading
|
0/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Movie-Genre, Music-Genre, Hobby, Pet, Sport.
The attributes have the following possible values:
- Nationality: french, italian, russian, spanish
- Movie-Genre: family, satire, superhero, thriller
- Music-Genre: disco, metal, soul, techno
- Hobby: baking, chess, drawing, filmmaking
- Pet: goat, hamster, horse, rat
- Sport: rowing, skateboarding, snowboarding, soccer
Given the following premises about the line of people:
- Pet:hamster is on the right of Music-Genre:soul
- Hobby:chess is between Sport:skateboarding and Movie-Genre:satire
- Music-Genre:metal is somewhere to the left of Music-Genre:techno
- Hobby:chess is between Hobby:drawing and Sport:snowboarding
- Movie-Genre:superhero is somewhere to the right of Nationality:italian
- Music-Genre:techno is on the left or right of Nationality:french
- Sport:snowboarding == Hobby:baking
- Pet:hamster is somewhere to the left of Hobby:chess
- Nationality:french == Hobby:chess
- Pet:goat is on the right of Movie-Genre:family
- Music-Genre:disco is somewhere to the right of Sport:skateboarding
- Nationality:italian is somewhere to the right of Pet:horse
- Sport:rowing is somewhere to the right of Nationality:spanish
Answer the following question:
Question: Question: What is the Pet of the person who is spanish??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
horse
|
0/8
|
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Food, Movie-Genre, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Food: kale, orange, radish, raspberry, tomato
- Movie-Genre: horror, scientific, sports, time-travel, zombie
- Music-Genre: blues, gospel, r&b, salsa, techno
- Pet: chinchilla, dog, frog, mouse, pony
- Sport: baseball, climbing, golf, skateboarding, skiing
Given the following premises about the line of people:
- Pet:mouse is not to the left of Sport:skateboarding
- Food:orange is on the far left or far right
- Movie-Genre:sports is on the left or right of Music-Genre:blues
- Food:tomato is not to the left of Food:radish
- Pet:chinchilla == Music-Genre:r&b
- Music-Genre:gospel and Movie-Genre:scientific have the same parity positions
- Movie-Genre:zombie == Sport:baseball or Movie-Genre:zombie == Music-Genre:techno, but not both
- Food:raspberry is on the left or right of Sport:skiing
- Movie-Genre:scientific is somewhere to the left of Movie-Genre:time-travel
- Pet:frog is somewhere between Pet:pony and Pet:mouse
- Sport:skateboarding == Music-Genre:blues
- Sport:climbing == Music-Genre:r&b or Pet:frog == Sport:climbing, but not both
- Movie-Genre:zombie == Food:radish
- Food:radish == Music-Genre:gospel
- Music-Genre:gospel is in an even position
- Pet:dog is not to the right of Food:orange
- Food:raspberry is not to the right of Music-Genre:salsa
- Food:kale is on the left of Food:raspberry
Answer the following question:
Question: Question: What is the Movie-Genre of the person who has/is gospel??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
zombie
|
1/8
|
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Beverage, Food, Music-Genre, Sport.
The attributes have the following possible values:
- Beverage: 7up, fanta, sprite, water
- Food: peach, spinach, tomato, watermelon
- Music-Genre: d&b, disco, jazz, rock
- Sport: basketball, handball, rowing, surfing
Given the following premises about the line of people:
- Food:spinach is not to the left of Food:tomato
- Food:spinach == Sport:handball or Food:spinach == Music-Genre:disco or both
- Music-Genre:d&b is not to the left of Sport:rowing
- Music-Genre:jazz != Food:watermelon
- Food:spinach is somewhere between Food:peach and Food:tomato
- Music-Genre:disco is somewhere between Beverage:sprite and Music-Genre:rock
- Sport:basketball and Beverage:sprite have different parity positions
- Beverage:water is not to the right of Music-Genre:disco
- Beverage:7up and Food:tomato have the same parity positions
- Sport:rowing != Beverage:7up
- Beverage:7up is not to the right of Music-Genre:rock
- Beverage:fanta is somewhere between Food:watermelon and Beverage:sprite
- Sport:basketball is not to the left of Sport:handball
- Sport:handball and Beverage:7up have different parity positions
Answer the following question:
Question: Question: What is the Food of the person who has/is water??
Please provide your answer in the format: \boxed{X}, where X is the answer.
|
watermelon
|
1/8
|
A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$?
|
250
|
3/8
|
Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
|
111
|
3/8
|
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 $c_1 + c_2 + \dots + c_{210}$?
|
329
|
3/8
|
Let $n \geq 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0 \leq k \leq n-1$ and there exist integers $x, y, z \in \mathbb{Z}$ such that
\[
\begin{align*}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{align*}
\]
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
|
101
|
3/8
|
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\lceil \sqrt{s} \rceil \mid s \in S\}$. Find the number of sets $T$ such that $|f(T)| = 2$ and $f(f(T)) = \{2\}$.
|
5043
|
5/8
|
Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides, thus dividing the rectangle into an $m \times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\binom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$.
|
1289
|
3/8
|
Let $S = \{ A = (a_1, \ldots, a_s) \mid a_i = 0 \text{ or } 1, \ i = 1, \ldots, 8 \}$. For any two elements of $S$, $A = \{ a_1, \ldots, a_8 \}$ and $B = \{ b_1, \ldots, b_8 \}$. Let $d(A,B) = \sum_{i=1}^{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most, how many elements can $S$ have such that the distance between any two sets is at least 5?
|
4
|
2/8
|
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
|
160
|
4/8
|
For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n) = 3 \cdot S(2n)$.
|
14499
|
0/8
|
Let $\triangle ABC$ be a triangle with $BC = 4$, $CA= 5$, $AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
|
23
|
4/8
|
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$ .
We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$ , determine how and how many ways we can make this sequence!!
|
2
|
1/8
|
In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?
|
\frac{\sqrt{2}}{2}
|
2/8
|
On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the $n-ladder$ . For example, the following figure shows a $3-ladder$ :
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?
|
2^{98}
|
3/8
|
Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$ , find the number of subsets of $M$ with $k$ elements with the given property.
|
495
|
0/8
|
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.
*Proposed by Jeff Lin*
|
3004
|
2/8
|
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.