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Let $T=7$. Given the sequence $u_{n}$ such that $u_{3}=5, u_{6}=89$, and $u_{n+2}=3 u_{n+1}-u_{n}$ for integers $n \geq 1$, compute $u_{T}$. | 233 | OLYMPIADBENCH | test |
In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.
It so happens th... | 136 | OLYMPIADBENCH | test |
In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.
It so happens th... | $d=\frac{1+\sqrt{1+8 n}}{2}$ | OLYMPIADBENCH | test |
In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.
It so happens th... | 1680 | OLYMPIADBENCH | test |
In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.
It so happens th... | $\frac{d !}{2 k(d-k) !}$ | OLYMPIADBENCH | test |
A student computed the repeating decimal expansion of $\frac{1}{N}$ for some integer $N$, but inserted six extra digits into the repetend to get $.0 \overline{0231846597}$. Compute the value of $N$. | 606 | OLYMPIADBENCH | test |
Let $n$ be a four-digit number whose square root is three times the sum of the digits of $n$. Compute $n$. | 2916 | OLYMPIADBENCH | test |
Compute the sum of the reciprocals of the positive integer divisors of 24. | $\frac{5}{2}$ | OLYMPIADBENCH | test |
There exists a digit $Y$ such that, for any digit $X$, the seven-digit number $\underline{1} \underline{2} \underline{3} \underline{X} \underline{5} \underline{Y} \underline{7}$ is not a multiple of 11. Compute $Y$. | 4 | OLYMPIADBENCH | test |
A point is selected at random from the interior of a right triangle with legs of length $2 \sqrt{3}$ and 4 . Let $p$ be the probability that the distance between the point and the nearest vertex is less than 2. Then $p$ can be written in the form $a+\sqrt{b} \pi$, where $a$ and $b$ are rational numbers. Compute $(a, b)... | $(\frac{1}{4}, \frac{1}{27})$ | OLYMPIADBENCH | test |
The square $A R M L$ is contained in the $x y$-plane with $A=(0,0)$ and $M=(1,1)$. Compute the length of the shortest path from the point $(2 / 7,3 / 7)$ to itself that touches three of the four sides of square $A R M L$. | $\frac{2}{7} \sqrt{53}$ | OLYMPIADBENCH | test |
For each positive integer $k$, let $S_{k}$ denote the infinite arithmetic sequence of integers with first term $k$ and common difference $k^{2}$. For example, $S_{3}$ is the sequence $3,12,21, \ldots$ Compute the sum of all $k$ such that 306 is an element of $S_{k}$. | 326 | OLYMPIADBENCH | test |
Compute the sum of all values of $k$ for which there exist positive real numbers $x$ and $y$ satisfying the following system of equations.
$$
\left\{\begin{aligned}
\log _{x} y^{2}+\log _{y} x^{5} & =2 k-1 \\
\log _{x^{2}} y^{5}-\log _{y^{2}} x^{3} & =k-3
\end{aligned}\right.
$$ | $\frac{43}{48}$ | OLYMPIADBENCH | test |
Let $W=(0,0), A=(7,0), S=(7,1)$, and $H=(0,1)$. Compute the number of ways to tile rectangle $W A S H$ with triangles of area $1 / 2$ and vertices at lattice points on the boundary of WASH. | 3432 | OLYMPIADBENCH | test |
Compute $\sin ^{2} 4^{\circ}+\sin ^{2} 8^{\circ}+\sin ^{2} 12^{\circ}+\cdots+\sin ^{2} 176^{\circ}$. | $\frac{45}{2}$ | OLYMPIADBENCH | test |
Compute the area of the region defined by $x^{2}+y^{2} \leq|x|+|y|$. | $2+\pi$ | OLYMPIADBENCH | test |
The arithmetic sequences $a_{1}, a_{2}, a_{3}, \ldots, a_{20}$ and $b_{1}, b_{2}, b_{3}, \ldots, b_{20}$ consist of 40 distinct positive integers, and $a_{20}+b_{14}=1000$. Compute the least possible value for $b_{20}+a_{14}$. | 10 | OLYMPIADBENCH | test |
Compute the ordered triple $(x, y, z)$ representing the farthest lattice point from the origin that satisfies $x y-z^{2}=y^{2} z-x=14$. | $(-266,-3,-28)$ | OLYMPIADBENCH | test |
The sequence $a_{1}, a_{2}, a_{3}, \ldots$ is a geometric sequence with $a_{20}=8$ and $a_{14}=2^{21}$. Compute $a_{21}$. | 1 | OLYMPIADBENCH | test |
Let $T=1$. Circles $L$ and $O$ are internally tangent and have radii $T$ and $4 T$, respectively. Point $E$ lies on circle $L$ such that $\overline{O E}$ is tangent to circle $L$. Compute $O E$. | $2 \sqrt{2}$ | OLYMPIADBENCH | test |
Let $T=2 \sqrt{2}$. In a right triangle, one leg has length $T^{2}$ and the other leg is 2 less than the hypotenuse. Compute the triangle's perimeter. | 40 | OLYMPIADBENCH | test |
$\quad$ Let $T=40$. If $x+9 y=17$ and $T x+(T+1) y=T+2$, compute $20 x+14 y$. | 8 | OLYMPIADBENCH | test |
Let $T=8$. Let $f(x)=a x^{2}+b x+c$. The product of the roots of $f$ is $T$. If $(-2,20)$ and $(1,14)$ lie on the graph of $f$, compute $a$. | $\frac{8}{5}$ | OLYMPIADBENCH | test |
Let $T=\frac{8}{5}$. Let $z_{1}=15+5 i$ and $z_{2}=1+K i$. Compute the smallest positive integral value of $K$ such that $\left|z_{1}-z_{2}\right| \geq 15 T$. | 25 | OLYMPIADBENCH | test |
Let $T=25$. Suppose that $T$ people are standing in a line, including three people named Charlie, Chris, and Abby. If the people are assigned their positions in line at random, compute the probability that Charlie is standing next to at least one of Chris or Abby. | $\frac{47}{300}$ | OLYMPIADBENCH | test |
Let $A$ be the number you will receive from position 7 and let $B$ be the number you will receive from position 9. Let $\alpha=\sin ^{-1} A$ and let $\beta=\cos ^{-1} B$. Compute $\sin (\alpha+\beta)+\sin (\alpha-\beta)$. | $\frac{94}{4225}$ | OLYMPIADBENCH | test |
Let $T=13$. If $r$ is the radius of a right circular cone and the cone's height is $T-r^{2}$, let $V$ be the maximum possible volume of the cone. Compute $\pi / V$. | $\frac{12}{169}$ | OLYMPIADBENCH | test |
Let $T=650$. If $\log T=2-\log 2+\log k$, compute the value of $k$. | 13 | OLYMPIADBENCH | test |
Let $T=100$. Nellie has a flight from Rome to Athens that is scheduled to last for $T+30$ minutes. However, owing to a tailwind, her flight only lasts for $T$ minutes. The plane's speed is 1.5 miles per minute faster than what it would have been for the originally scheduled flight. Compute the distance (in miles) that ... | 650 | OLYMPIADBENCH | test |
Let $T=9$. Compute $\sqrt{\sqrt{\sqrt[T]{10^{T^{2}-T}}}}$. | 100 | OLYMPIADBENCH | test |
Let $T=3$. Regular hexagon $S U P E R B$ has side length $\sqrt{T}$. Compute the value of $B E \cdot S U \cdot R E$. | 9 | OLYMPIADBENCH | test |
Let $T=70$. Chef Selma is preparing a burrito menu. A burrito consists of: (1) a choice of chicken, beef, turkey, or no meat, (2) exactly one of three types of beans, (3) exactly one of two types of rice, and (4) exactly one of $K$ types of cheese. Compute the smallest value of $K$ such that Chef Selma can make at leas... | 3 | OLYMPIADBENCH | test |
Compute the smallest positive integer $N$ such that $20 N$ is a multiple of 14 and $14 N$ is a multiple of 20 . | 70 | OLYMPIADBENCH | test |
Call a positive integer fibbish if each digit, after the leftmost two, is at least the sum of the previous two digits. Compute the greatest fibbish number. | 10112369 | OLYMPIADBENCH | test |
An ARMLbar is a $7 \times 7$ grid of unit squares with the center unit square removed. A portion of an ARMLbar is a square section of the bar, cut along the gridlines of the original bar. Compute the number of different ways there are to cut a single portion from an ARMLbar. | 96 | OLYMPIADBENCH | test |
Regular hexagon $A B C D E F$ and regular hexagon $G H I J K L$ both have side length 24 . The hexagons overlap, so that $G$ is on $\overline{A B}, B$ is on $\overline{G H}, K$ is on $\overline{D E}$, and $D$ is on $\overline{J K}$. If $[G B C D K L]=\frac{1}{2}[A B C D E F]$, compute $L F$. | 18 | OLYMPIADBENCH | test |
Compute the largest base-10 integer $\underline{A} \underline{B} \underline{C} \underline{D}$, with $A>0$, such that $\underline{A} \underline{B} \underline{C} \underline{D}=B !+C !+D !$. | 5762 | OLYMPIADBENCH | test |
Let $X$ be the number of digits in the decimal expansion of $100^{1000^{10,000}}$, and let $Y$ be the number of digits in the decimal expansion of $1000^{10,000^{100,000}}$. Compute $\left\lfloor\log _{X} Y\right\rfloor$. | 13 | OLYMPIADBENCH | test |
Compute the smallest possible value of $n$ such that two diagonals of a regular $n$-gon intersect at an angle of 159 degrees. | 60 | OLYMPIADBENCH | test |
Compute the number of quadratic functions $f(x)=a x^{2}+b x+c$ with integer roots and integer coefficients whose graphs pass through the points $(0,0)$ and $(15,225)$. | 8 | OLYMPIADBENCH | test |
A bubble in the shape of a hemisphere of radius 1 is on a tabletop. Inside the bubble are five congruent spherical marbles, four of which are sitting on the table and one which rests atop the others. All marbles are tangent to the bubble, and their centers can be connected to form a pyramid with volume $V$ and with a s... | $\frac{1}{54}$ | OLYMPIADBENCH | test |
Compute the smallest positive integer base $b$ for which $16_{b}$ is prime and $97_{b}$ is a perfect square. | 53 | OLYMPIADBENCH | test |
For a positive integer $n$, let $C(n)$ equal the number of pairs of consecutive 1's in the binary representation of $n$. For example, $C(183)=C\left(10110111_{2}\right)=3$. Compute $C(1)+C(2)+$ $C(3)+\cdots+C(256)$. | 448 | OLYMPIADBENCH | test |
A set $S$ contains thirteen distinct positive integers whose sum is 120 . Compute the largest possible value for the median of $S$. | 11 | OLYMPIADBENCH | test |
Let $T=11$. Compute the least positive integer $b$ such that, when expressed in base $b$, the number $T$ ! ends in exactly two zeroes. | 5 | OLYMPIADBENCH | test |
Let $T=5$. Suppose that $a_{1}=1$, and that for all positive integers $n, a_{n+1}=$ $\left\lceil\sqrt{a_{n}^{2}+34}\right\rceil$. Compute the least value of $n$ such that $a_{n}>100 T$. | 491 | OLYMPIADBENCH | test |
Compute the smallest $n$ such that in the regular $n$-gon $A_{1} A_{2} A_{3} \cdots A_{n}, \mathrm{~m} \angle A_{1} A_{20} A_{13}<60^{\circ}$. | 37 | OLYMPIADBENCH | test |
Let $T=37$. A cube has edges of length $T$. Square holes of side length 1 are drilled from the center of each face of the cube through the cube's center and across to the opposite face; the edges of each hole are parallel to the edges of the cube. Compute the surface area of the resulting solid. | 8640 | OLYMPIADBENCH | test |
Let $T=8640$. Compute $\left\lfloor\log _{4}\left(1+2+4+\cdots+2^{T}\right)\right\rfloor$. | 4320 | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | $\frac{1}{3},1,\frac{1}{9}$ | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | 81 | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | 27,54,108,135,189,216,270,297,351,378 | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | 1 | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | $1/3$ | OLYMPIADBENCH | test |
In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0... | $\frac{1}{3}, 1, 9$ | OLYMPIADBENCH | test |
Let $A R M L$ be a trapezoid with bases $\overline{A R}$ and $\overline{M L}$, such that $M R=R A=A L$ and $L R=$ $A M=M L$. Point $P$ lies inside the trapezoid such that $\angle R M P=12^{\circ}$ and $\angle R A P=6^{\circ}$. Diagonals $A M$ and $R L$ intersect at $D$. Compute the measure, in degrees, of angle $A P D$... | 48 | OLYMPIADBENCH | test |
A regular hexagon has side length 1. Compute the average of the areas of the 20 triangles whose vertices are vertices of the hexagon. | $\frac{9 \sqrt{3}}{20}$ | OLYMPIADBENCH | test |
Paul was planning to buy 20 items from the ARML shop. He wanted some mugs, which cost $\$ 10$ each, and some shirts, which cost $\$ 6$ each. After checking his wallet he decided to put $40 \%$ of the mugs back. Compute the number of dollars he spent on the remaining items. | 120 | OLYMPIADBENCH | test |
Let $x$ be the smallest positive integer such that $1584 \cdot x$ is a perfect cube, and let $y$ be the smallest positive integer such that $x y$ is a multiple of 1584 . Compute $y$. | 12 | OLYMPIADBENCH | test |
Emma goes to the store to buy apples and peaches. She buys five of each, hands the shopkeeper one $\$ 5$ bill, but then has to give the shopkeeper another; she gets back some change. Jonah goes to the same store, buys 2 apples and 12 peaches, and tries to pay with a single $\$ 10$ bill. But that's not enough, so Jonah ... | 1525 | OLYMPIADBENCH | test |
Circle $O$ has radius 6. Point $P$ lies outside circle $O$, and the shortest distance from $P$ to circle $O$ is 4. Chord $\overline{A B}$ is parallel to $\overleftrightarrow{O P}$, and the distance between $\overline{A B}$ and $\overleftrightarrow{O P}$ is 2 . Compute $P A^{2}+P B^{2}$. | 272 | OLYMPIADBENCH | test |
A palindrome is a positive integer, not ending in 0 , that reads the same forwards and backwards. For example, 35253,171,44, and 2 are all palindromes, but 17 and 1210 are not. Compute the least positive integer greater than 2013 that cannot be written as the sum of two palindromes. | 2019 | OLYMPIADBENCH | test |
Positive integers $x, y, z$ satisfy $x y+z=160$. Compute the smallest possible value of $x+y z$. | 50 | OLYMPIADBENCH | test |
Compute $\cos ^{3} \frac{2 \pi}{7}+\cos ^{3} \frac{4 \pi}{7}+\cos ^{3} \frac{8 \pi}{7}$. | $-\frac{1}{2}$ | OLYMPIADBENCH | test |
In right triangle $A B C$ with right angle $C$, line $\ell$ is drawn through $C$ and is parallel to $\overline{A B}$. Points $P$ and $Q$ lie on $\overline{A B}$ with $P$ between $A$ and $Q$, and points $R$ and $S$ lie on $\ell$ with $C$ between $R$ and $S$ such that $P Q R S$ is a square. Let $\overline{P S}$ intersect... | 14 | OLYMPIADBENCH | test |
Compute the sum of all real numbers $x$ such that
$$
\left\lfloor\frac{x}{2}\right\rfloor-\left\lfloor\frac{x}{3}\right\rfloor=\frac{x}{7}
$$ | -21 | OLYMPIADBENCH | test |
Let $S=\{1,2, \ldots, 20\}$, and let $f$ be a function from $S$ to $S$; that is, for all $s \in S, f(s) \in S$. Define the sequence $s_{1}, s_{2}, s_{3}, \ldots$ by setting $s_{n}=\sum_{k=1}^{20} \underbrace{(f \circ \cdots \circ f)}_{n}(k)$. That is, $s_{1}=f(1)+$ $\cdots+f(20), s_{2}=f(f(1))+\cdots+f(f(20)), s_{3}=f(... | 140 | OLYMPIADBENCH | test |
Compute the smallest positive integer $n$ such that $n^{2}+n^{0}+n^{1}+n^{3}$ is a multiple of 13 . | 5 | OLYMPIADBENCH | test |
Let T be any number you want. Compute $2^{\log _{T} 8}-8^{\log _{T} 2}$. | 0 | OLYMPIADBENCH | test |
Let $T=0$. At some point during a given week, a law enforcement officer had issued $T+2$ traffic warnings, 20 tickets, and had made $T+5$ arrests. How many more tickets must the officer issue in order for the combined number of tickets and arrests to be 20 times the number of warnings issued that week? | 15 | OLYMPIADBENCH | test |
Let $T=15$. In parallelogram $A R M L$, points $P$ and $Q$ trisect $\overline{A R}$ and points $W, X, Y, Z$ divide $\overline{M L}$ into fifths (where $W$ is closest to $M$, and points $X$ and $Y$ are both between $W$ and $Z$ ). If $[A R M L]=T$, compute $[P Q W Z]$. | 7 | OLYMPIADBENCH | test |
Let $T=7$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$. | 36 | OLYMPIADBENCH | test |
Let $T=36$. The graph of $y=x^{2}+2 x-T$ intersects the $x$-axis at points $A$ and $M$, which are diagonally opposite vertices of square $A R M L$. Compute $[A R M L]$. | 74 | OLYMPIADBENCH | test |
Let $S$ be the set of prime factors of the numbers you receive from positions 7 and 9 , and let $p$ and $q$ be the two least distinct elements of $S$, with $p<q$. Hexagon HEXAGO is inscribed in circle $\omega$, and every angle of $H E X A G O$ is $120^{\circ}$. If $H E=X A=G O=p$ and $E X=A G=O H=q$, compute the area o... | $\frac{67 \pi}{3}$ | OLYMPIADBENCH | test |
Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting the four helpers. | 7 | OLYMPIADBENCH | test |
Let $T=8$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2 T)^{2}+b$. Compute $b$. | 184 | OLYMPIADBENCH | test |
Let $T=19$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the sequence. | 8 | OLYMPIADBENCH | test |
Let $T=17$. A regular $n$-gon has exactly $T$ more diagonals than a regular $(n-1)$-gon. Compute the value of $n$. | 19 | OLYMPIADBENCH | test |
Let $T=27/2$. The sequence $a_{1}, a_{2}, a_{3}, \ldots$, is arithmetic with $a_{16}=13$ and $a_{30}=20$. Compute the value of $k$ for which $a_{k}=T$. | 17 | OLYMPIADBENCH | test |
Let $T=114$. A rectangular prism has a length of 1 , a width of 3 , a height of $h$, and has a total surface area of $T$. Compute the value of $h$. | $\frac{27}{2}$ | OLYMPIADBENCH | test |
The zeros of $x^{2}+b x+93$ are $r$ and $s$. If the zeros of $x^{2}-22 x+c$ are $r+1$ and $s+1$, compute $c$. | 114 | OLYMPIADBENCH | test |
Let $N=888,888 \times 9,999,999$. Compute the sum of the digits of $N$. | 63 | OLYMPIADBENCH | test |
Five equilateral triangles are drawn in the plane so that no two sides of any of the triangles are parallel. Compute the maximum number of points of intersection among all five triangles. | 60 | OLYMPIADBENCH | test |
$\quad$ Let $S$ be the set of four-digit positive integers for which the sum of the squares of their digits is 17 . For example, $2023 \in S$ because $2^{2}+0^{2}+2^{2}+3^{2}=17$. Compute the median of $S$. | 2302 | OLYMPIADBENCH | test |
Let $E U C L I D$ be a hexagon inscribed in a circle of radius 5 . Given that $E U=U C=L I=I D=6$, and $C L=D E$, compute $C L$. | $\frac{14}{5}$ | OLYMPIADBENCH | test |
The ARMLLexicon consists of 10 letters: $\{A, R, M, L, e, x, i, c, o, n\}$. A palindrome is an ordered list of letters that read the same backwards and forwards; for example, MALAM, n, oncecno, and MoM are palindromes. Compute the number of 15-letter palindromes that can be spelled using letters in the ARMLLexicon, amo... | 99956 | OLYMPIADBENCH | test |
Let $10^{y}$ be the product of all real numbers $x$ such that $\log x=\frac{3+\left\lfloor(\log x)^{2}\right\rfloor}{4}$. Compute $y$. | 8 | OLYMPIADBENCH | test |
The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute
$$
\frac{r_{1}}{\sqrt[3]{r_{2}}}+\frac{r_{2}}{\sqrt[3]{r_{1}}}
$$ | 508 | OLYMPIADBENCH | test |
Circle $\omega$ is tangent to parallel lines $\ell_{1}$ and $\ell_{2}$ at $A$ and $B$ respectively. Circle $\omega_{1}$ is tangent to $\ell_{1}$ at $C$ and to $\omega$ externally at $P$. Circle $\omega_{2}$ is tangent to $\ell_{2}$ at $D$ and to $\omega$ externally at $Q$. Circles $\omega_{1}$ and $\omega_{2}$ are also... | $5 \sqrt{10}$ | OLYMPIADBENCH | test |
Given quadrilateral $A R M L$ with $A R=20, R M=23, M L=25$, and $A M=32$, compute the number of different integers that could be the perimeter of $A R M L$. | 49 | OLYMPIADBENCH | test |
Let $\mathcal{S}$ denote the set of all real polynomials $A(x)$ with leading coefficient 1 such that there exists a real polynomial $B(x)$ that satisfies
$$
\frac{1}{A(x)}+\frac{1}{B(x)}+\frac{1}{x+10}=\frac{1}{x}
$$
for all real numbers $x$ for which $A(x) \neq 0, B(x) \neq 0$, and $x \neq-10,0$. Compute $\sum_{A \i... | 46760 | OLYMPIADBENCH | test |
Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greatest digit in $T$. In square $N O R M, N O=b$, and points $P_{1}$ and $P_{2}$ lie on $\overline{N O}$ and $\overline{O R}$, respectively, so that $O P_{1}=O P_{2}=a$. A circle centered at $O$ has radius $a$, and quarter-circular arc $\wideha... | 36 | OLYMPIADBENCH | test |
Let $T=36$. Square $A B C D$ has area $T$. Points $M, N, O$, and $P$ lie on $\overline{A B}$, $\overline{B C}, \overline{C D}$, and $\overline{D A}$, respectively, so that quadrilateral $M N O P$ is a rectangle with $M P=2$. Compute $M N$. | $6 \sqrt{2}-2$ | OLYMPIADBENCH | test |
In a game, a player chooses 2 of the 13 letters from the first half of the alphabet (i.e., A-M) and 2 of the 13 letters from the second half of the alphabet (i.e., N-Z). Aditya plays the game, and then Ayesha plays the game. Compute the probability that Aditya and Ayesha choose the same set of four letters. | $\frac{1}{6084}$ | OLYMPIADBENCH | test |
Let $T=\frac{1}{6084}$. Compute the least positive integer $n$ such that when a fair coin is flipped $n$ times, the probability of it landing heads on all $n$ flips is less than $T$. | 13 | OLYMPIADBENCH | test |
Let $T=13$. Compute the least integer $n>2023$ such that the equation $x^{2}-T x-n=0$ has integer solutions. | 2028 | OLYMPIADBENCH | test |
In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relatively prime to each other.
Find the maximu... | 1,2 | OLYMPIADBENCH | test |
In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relatively prime to each other.
Find the maximu... | 1,3 | OLYMPIADBENCH | test |
For an integer $n \geq 4$, define $a_{n}$ to be the product of all real numbers that are roots to at least one quadratic polynomial whose coefficients are positive integers that sum to $n$. Compute
$$
\frac{a_{4}}{a_{5}}+\frac{a_{5}}{a_{6}}+\frac{a_{6}}{a_{7}}+\cdots+\frac{a_{2022}}{a_{2023}} .
$$ | -2019 | OLYMPIADBENCH | test |
Suppose that $u$ and $v$ are distinct numbers chosen at random from the set $\{1,2,3, \ldots, 30\}$. Compute the probability that the roots of the polynomial $(x+u)(x+v)+4$ are integers. | $\frac{17}{145}$ | OLYMPIADBENCH | test |
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