question stringlengths 49 4.42k | answer stringlengths 1 202 | metadata dict |
|---|---|---|
Find the largest possible integer $k$, such that the following statement is true:
Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain
$... | 1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2082,
"problem_type": "Algebra",
"unit": null
} |
Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides of lengths
$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$ | f(z)=z | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2084,
"problem_type": "Algebra",
"unit": null
} |
For any integer $n \geq 2$, let $N(n)$ be the maximal number of triples $\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i}$ and $c_{i}$ such that the following two conditions are satisfied:
(1) $a_{i}+b_{i}+c_{i}=n$ for all $i=1, \ldots, N(n)$,
(2) If $i \neq j$, t... | N(n)=\left\lfloor\frac{2 n}{3}\right\rfloor+1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2091,
"problem_type": "Combinatorics",
"unit": null
} |
On a $999 \times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A nonintersecting route of the limp rook c... | 996000 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2095,
"problem_type": "Combinatorics",
"unit": null
} |
Let $A B C$ be a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet the sides $B C$ and $A C$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $A D C$. Suppose that $\angle B E K=45^{\circ}$. Find all possible values of $\angle B A C$. | 90^{\circ}$,$60^{\circ} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2099,
"problem_type": "Geometry",
"unit": null
} |
Find all positive integers $n$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying
$$
a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1
$$
for every $k$ with $2 \leq k \leq n-1$. | 1,2,3,4 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2111,
"problem_type": "Number Theory",
"unit": null
} |
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary.
Find the largest $n$ for which there exist $n$ boxes $B_{1}, \ldots, B_{n}$ such t... | 6 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2124,
"problem_type": "Combinatorics",
"unit": null
} |
In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every ... | 180180 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2126,
"problem_type": "Combinatorics",
"unit": null
} |
Let $n$ and $k$ be fixed positive integers of the same parity, $k \geq n$. We are given $2 n$ lamps numbered 1 through $2 n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off).
Let $N$ be t... | 2^{k-n} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2127,
"problem_type": "Combinatorics",
"unit": null
} |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
$$
f(1+x y)-f(x+y)=f(x) f(y) \text { for all } x, y \in \mathbb{R}
$$
and $f(-1) \neq 0$. | f(x)=x-1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2147,
"problem_type": "Algebra",
"unit": null
} |
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{1,2, \ldots, n\}$ such that the sums of the different pairs are different integers not exceeding $n$ ? | \lfloor\frac{2 n-1}{5}\rfloor | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2151,
"problem_type": "Combinatorics",
"unit": null
} |
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\left(C_{1}, C_{2}, C_{3}\right)$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain. | \frac{4 \cdot 999^{4}}{27} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2152,
"problem_type": "Combinatorics",
"unit": null
} |
Players $A$ and $B$ play a game with $N \geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules:
- On every move of his $B$ passes... | 4022 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2153,
"problem_type": "Combinatorics",
"unit": null
} |
Find all triples $(x, y, z)$ of positive integers such that $x \leq y \leq z$ and
$$
x^{3}\left(y^{3}+z^{3}\right)=2012(x y z+2) \text {. }
$$ | (2,251,252) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2165,
"problem_type": "Number Theory",
"unit": null
} |
Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
holds for all rational numbers $x$ and $y$.
$$
f(x f(x)+y)=f(y)+x^{2}
$$
Here, $\mathbb{Q}$ denotes the set of rational numbers. | f(x)=x,f(x)=-x | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2171,
"problem_type": "Algebra",
"unit": null
} |
A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is $f a t$, if $O$ is strictly inside the triangle. Find the maximum numb... | 2021 \cdot 505 \cdot 337 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2174,
"problem_type": "Geometry",
"unit": null
} |
Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) Ther... | k=3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2177,
"problem_type": "Combinatorics",
"unit": null
} |
Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine... | 6m | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2185,
"problem_type": "Combinatorics",
"unit": null
} |
Let $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by $a_{1}=a_{2}=1$, $a_{3}=4$, and for all $n \geq 4$,
$$
a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3} .
$$
Determine all integers $m$ such that every term of the sequence is a square. | 1,2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2193,
"problem_type": "Number Theory",
"unit": null
} |
The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.
- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
- If contestan... | 2^{n}-n-1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2198,
"problem_type": "Combinatorics",
"unit": null
} |
Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b \text {. }
$$ | (0,1,1),(0,-1,-1),(1,0,1),(-1,0,-1),(1,1,0)$,$(-1,-1,0),\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$,$\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2203,
"problem_type": "Algebra",
"unit": null
} |
Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$.... | \frac{n(n+1)}{2} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2204,
"problem_type": "Combinatorics",
"unit": null
} |
Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that
(1) $a_{0}+a_{1}=-\frac{1}{n}$, and
(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$. | N=n | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2212,
"problem_type": "Algebra",
"unit": null
} |
Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \ldots, 10$ in some order. | 11,13 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2216,
"problem_type": "Combinatorics",
"unit": null
} |
Let $k$ be a positive integer. Lexi has a dictionary $\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathcal{D}$ when read from t... | 2^{k-1} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2230,
"problem_type": "Combinatorics",
"unit": null
} |
In an increasing sequence of numbers with an odd number of terms, the difference between any two consecutive terms is a constant $d$, and the middle term is 302 . When the last 4 terms are removed from the sequence, the middle term of the resulting sequence is 296. What is the value of $d$ ? | 3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2234,
"problem_type": "Algebra",
"unit": null
} |
There are two increasing sequences of five consecutive integers, each of which have the property that the sum of the squares of the first three integers in the sequence equals the sum of the squares of the last two. Determine these two sequences. | 10,11,12,13,14,-2,-1,0,1,2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2235,
"problem_type": "Algebra",
"unit": null
} |
If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value? | 2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2236,
"problem_type": "Algebra",
"unit": null
} |
Determine all integer values of $x$ such that $\left(x^{2}-3\right)\left(x^{2}+5\right)<0$. | -1,0,1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2238,
"problem_type": "Algebra",
"unit": null
} |
At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Det... | 3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2239,
"problem_type": "Algebra",
"unit": null
} |
What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ? | 9 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2241,
"problem_type": "Algebra",
"unit": null
} |
Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$. | 210 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2242,
"problem_type": "Algebra",
"unit": null
} |
Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect. | (-\infty,-5) | {
"answer_type": "Interval",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2243,
"problem_type": "Algebra",
"unit": null
} |
If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ? | 13 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2247,
"problem_type": "Algebra",
"unit": null
} |
The functions $f$ and $g$ satisfy
$$
\begin{aligned}
& f(x)+g(x)=3 x+5 \\
& f(x)-g(x)=5 x+7
\end{aligned}
$$
for all values of $x$. Determine the value of $2 f(2) g(2)$. | -84 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2248,
"problem_type": "Algebra",
"unit": null
} |
Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$.
The numbers are arranged in increasing order.
What is the probability that the resulting sequence is an arithmetic sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding... | \frac{2}{5} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2249,
"problem_type": "Combinatorics",
"unit": null
} |
What is the largest two-digit number that becomes $75 \%$ greater when its digits are reversed? | 48 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2251,
"problem_type": "Combinatorics",
"unit": null
} |
Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, ... | 45 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2253,
"problem_type": "Geometry",
"unit": "minute"
} |
Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$, and $N$. Determine all the values of $a$ for which the area of the trapezoid $K L M N$ is 36 . | \frac{32}{9}$,$\frac{1}{2} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2254,
"problem_type": "Geometry",
"unit": null
} |
A 75 year old person has a $50 \%$ chance of living at least another 10 years.
A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 y... | 62.5% | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2255,
"problem_type": "Algebra",
"unit": null
} |
Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$. | 1000 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2256,
"problem_type": "Algebra",
"unit": null
} |
The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each ... | 4090 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2257,
"problem_type": "Algebra",
"unit": null
} |
The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each ... | 2RC+R+C | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2258,
"problem_type": "Algebra",
"unit": null
} |
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determ... | 4085 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2260,
"problem_type": "Number Theory",
"unit": null
} |
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determ... | 55 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2261,
"problem_type": "Number Theory",
"unit": null
} |
Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly selected and removed together. What is the probability that the smaller of the two numbers on the tickets selected is less than or equal to 4 ? | \frac{14}{15} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2263,
"problem_type": "Combinatorics",
"unit": null
} |
A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\... | (6,5) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2265,
"problem_type": "Geometry",
"unit": null
} |
Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9, 11 is an arithmetic sequence.) | 1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2266,
"problem_type": "Algebra",
"unit": null
} |
If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ? | 5,1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2268,
"problem_type": "Algebra",
"unit": null
} |
What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ? | 1002001 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2275,
"problem_type": "Algebra",
"unit": null
} |
A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cos... | 8 \% | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2276,
"problem_type": "Algebra",
"unit": null
} |
Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages. | 7,14,18 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2277,
"problem_type": "Algebra",
"unit": null
} |
The parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\triangle D E F$. | 48 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2280,
"problem_type": "Geometry",
"unit": null
} |
If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ? | \frac{58}{3} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2281,
"problem_type": "Algebra",
"unit": null
} |
For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$. | 1,-1,7,-7 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2282,
"problem_type": "Number Theory",
"unit": null
} |
Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ :
Step 1: Reflect the point in the $x$-axis.
Step 2: Translate the resulting point 2 units upwards.
Step 3: Reflect the resulting point in the $y$-axis.
As she does this, the point $(3,5)$ moves to $(3,-... | (-7,-1) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2283,
"problem_type": "Combinatorics",
"unit": null
} |
Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum of all $n$ terms is 1000 , what is the value of $n$ ?
(An arithmetic sequence is a sequence in which each term after the first is obtained fro... | 20 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2286,
"problem_type": "Algebra",
"unit": null
} |
Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is... | a=2$, $a=-6-4 \sqrt{2} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2287,
"problem_type": "Algebra",
"unit": null
} |
A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. Wh... | \frac{4}{7} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2288,
"problem_type": "Combinatorics",
"unit": null
} |
Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\log _{\frac{1}{2}} b$ for all $b>0$. Determine all $\theta$ with $0 \leq \theta \leq 2 \pi$ for which $f(g(\sin \theta))=0$. | \frac{1}{6} \pi, \frac{5}{6} \pi, \frac{1}{4} \pi, \frac{3}{4} \pi | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2289,
"problem_type": "Algebra",
"unit": null
} |
Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$. | (3,2),(-3,2),(3,-2),(-3,-2) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2292,
"problem_type": "Number Theory",
"unit": null
} |
Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$. | (-\sqrt{11},-2)\cup (\sqrt{11},9) | {
"answer_type": "Interval",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2298,
"problem_type": "Algebra",
"unit": null
} |
The numbers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence with $a_{1} \neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_{1}, a_{4}, a_{k}$ also form a geometric sequence in that order.
(A... | 34 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2300,
"problem_type": "Algebra",
"unit": null
} |
For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer. | 1,2,5,8,9 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2301,
"problem_type": "Number Theory",
"unit": null
} |
Consider the following system of equations in which all logarithms have base 10:
$$
\begin{aligned}
(\log x)(\log y)-3 \log 5 y-\log 8 x & =a \\
(\log y)(\log z)-4 \log 5 y-\log 16 z & =b \\
(\log z)(\log x)-4 \log 8 x-3 \log 625 z & =c
\end{aligned}
$$
If $a=-4, b=4$, and $c=-18$, solve the system of equations. | (10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2}) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2303,
"problem_type": "Algebra",
"unit": null
} |
Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ? | \frac{11}{36} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2312,
"problem_type": "Combinatorics",
"unit": null
} |
If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship. | (3,0),(-3,1) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2313,
"problem_type": "Algebra",
"unit": null
} |
Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions.
Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total of $(10 \times 7... | 60 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2315,
"problem_type": "Algebra",
"unit": null
} |
If $T=x^{2}+\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$. | -3,0 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2317,
"problem_type": "Algebra",
"unit": null
} |
A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions:
i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and
ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$.
For examp... | (4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2318,
"problem_type": "Combinatorics",
"unit": null
} |
A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions:
i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and
ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$.
For examp... | (7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2319,
"problem_type": "Combinatorics",
"unit": null
} |
The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what is $m$ ? | 623 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2321,
"problem_type": "Number Theory",
"unit": null
} |
Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ? | 40 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2322,
"problem_type": "Combinatorics",
"unit": null
} |
Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$. | 1, 3, 5, 15 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2323,
"problem_type": "Number Theory",
"unit": null
} |
Ada starts with $x=10$ and $y=2$, and applies the following process:
Step 1: Add $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 2: Multiply $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change.
Step 3: Add $y$ and 1. Let $y$ equal the result. The value of $x$ does not... | 340 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2325,
"problem_type": "Combinatorics",
"unit": null
} |
Determine all integers $k$, with $k \neq 0$, for which the parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts. | -2,-1,1,2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2326,
"problem_type": "Number Theory",
"unit": null
} |
The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ? | \frac{19}{34} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2327,
"problem_type": "Number Theory",
"unit": null
} |
A geometric sequence has first term 10 and common ratio $\frac{1}{2}$.
An arithmetic sequence has first term 10 and common difference $d$.
The ratio of the 6th term in the geometric sequence to the 4th term in the geometric sequence equals the ratio of the 6th term in the arithmetic sequence to the 4 th term in the a... | -\frac{30}{17} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2328,
"problem_type": "Algebra",
"unit": null
} |
For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \leq p \leq x+10$. What is the value of $f(f(20))$ ? | 5 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2329,
"problem_type": "Algebra",
"unit": null
} |
Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations:
$$
\begin{aligned}
(x-1)(y-2) & =0 \\
(x-3)(z+2) & =0 \\
x+y z & =9
\end{aligned}
$$ | (1,-4,-2),(3,2,3),(13,2,-2) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2330,
"problem_type": "Algebra",
"unit": null
} |
Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ? | 4 \pi^{2}-1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2331,
"problem_type": "Algebra",
"unit": null
} |
Determine all pairs of angles $(x, y)$ with $0^{\circ} \leq x<180^{\circ}$ and $0^{\circ} \leq y<180^{\circ}$ that satisfy the following system of equations:
$$
\begin{aligned}
\log _{2}(\sin x \cos y) & =-\frac{3}{2} \\
\log _{2}\left(\frac{\sin x}{\cos y}\right) & =\frac{1}{2}
\end{aligned}
$$ | (45^{\circ}, 60^{\circ}),(135^{\circ}, 60^{\circ}) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2332,
"problem_type": "Algebra",
"unit": null
} |
Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and C... | \frac{1-p^{2}}{3} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2333,
"problem_type": "Combinatorics",
"unit": null
} |
Three microphones $A, B$ and $C$ are placed on a line such that $A$ is $1 \mathrm{~km}$ west of $B$ and $C$ is $2 \mathrm{~km}$ east of $B$. A large explosion occurs at a point $P$ not on this line. Each of the three microphones receives the sound. The sound travels at $\frac{1}{3} \mathrm{~km} / \mathrm{s}$. Microphon... | \frac{41}{12} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2334,
"problem_type": "Geometry",
"unit": "km"
} |
Kerry has a list of $n$ integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$. Kerry calculates the pairwise sums of all $m=\frac{1}{2} n(n-1)$ possible pairs of integers in her list and orders these pairwise sums as $s_{1} \leq s_{2} \leq \ldots \leq s_{m}$. For example, if Kerry'... | (1,7,103, 105), (3, 5, 101, 107) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2336,
"problem_type": "Number Theory",
"unit": null
} |
Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$. | -1$,$2$,$-2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2338,
"problem_type": "Algebra",
"unit": null
} |
Determine the number of positive divisors of 900, including 1 and 900, that are perfect squares. (A positive divisor of 900 is a positive integer that divides exactly into 900.) | 8 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2339,
"problem_type": "Number Theory",
"unit": null
} |
Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\angle A B C=\angle A C B$, determine all possible values of $k$. | 8$,$4 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2340,
"problem_type": "Geometry",
"unit": null
} |
A chemist has three bottles, each containing a mixture of acid and water:
- bottle A contains $40 \mathrm{~g}$ of which $10 \%$ is acid,
- bottle B contains $50 \mathrm{~g}$ of which $20 \%$ is acid, and
- bottle C contains $50 \mathrm{~g}$ of which $30 \%$ is acid.
She uses some of the mixture from each of the bottl... | 17.5% | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2341,
"problem_type": "Combinatorics",
"unit": null
} |
Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$. | 10 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2342,
"problem_type": "Algebra",
"unit": null
} |
A bag contains 40 balls, each of which is black or gold. Feridun reaches into the bag and randomly removes two balls. Each ball in the bag is equally likely to be removed. If the probability that two gold balls are removed is $\frac{5}{12}$, how many of the 40 balls are gold? | 26 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2343,
"problem_type": "Combinatorics",
"unit": null
} |
The geometric sequence with $n$ terms $t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}$ has $t_{1} t_{n}=3$. Also, the product of all $n$ terms equals 59049 (that is, $t_{1} t_{2} \cdots t_{n-1} t_{n}=59049$ ). Determine the value of $n$.
(A geometric sequence is a sequence in which each term after the first is obtained from the... | 20 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2344,
"problem_type": "Algebra",
"unit": null
} |
If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ? | 4027 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2345,
"problem_type": "Algebra",
"unit": null
} |
Determine all real numbers $x$ for which
$$
\left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000
$$ | 10^{100}$,$10^{1 / 100} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2346,
"problem_type": "Algebra",
"unit": null
} |
Without using a calculator, determine positive integers $m$ and $n$ for which
$$
\sin ^{6} 1^{\circ}+\sin ^{6} 2^{\circ}+\sin ^{6} 3^{\circ}+\cdots+\sin ^{6} 87^{\circ}+\sin ^{6} 88^{\circ}+\sin ^{6} 89^{\circ}=\frac{m}{n}
$$
(The sum on the left side of the equation consists of 89 terms of the form $\sin ^{6} x^{\ci... | 221,$8 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2349,
"problem_type": "Algebra",
"unit": null
} |
Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers $f(1), f(2), \ldots, f(2014)$ have a units digit of 1 . | 202 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2350,
"problem_type": "Number Theory",
"unit": null
} |
If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ? | 1000 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2351,
"problem_type": "Algebra",
"unit": null
} |
If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$. | \frac{97}{36} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2352,
"problem_type": "Algebra",
"unit": null
} |
A die, with the numbers $1,2,3,4,6$, and 8 on its six faces, is rolled. After this roll, if an odd number appears on the top face, all odd numbers on the die are doubled. If an even number appears on the top face, all the even numbers are halved. If the given die changes in this way, what is the probability that a 2 wi... | \frac{2}{9} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2355,
"problem_type": "Combinatorics",
"unit": null
} |
The table below gives the final standings for seven of the teams in the English Cricket League in 1998. At the end of the year, each team had played 17 matches and had obtained the total number of points shown in the last column. Each win $W$, each draw $D$, each bonus bowling point $A$, and each bonus batting point $B... | 16,3,1,1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2356,
"problem_type": "Algebra",
"unit": null
} |
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