ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: 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 $c$.
Answer: 114
Problem: Let $N=888,888 \times 9,999,999$. Compute the sum of the digits of $N$.
Answer: 63
Problem: 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.
Answer: 60
Problem: $\quad$ 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 $S$.
Answer: 2302
Problem: 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 $C L$.
Answer: \frac{14}{5}
Problem: 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, among which there are four consecutive letters that spell out $A R M L$.
Answer: 99956
Problem: 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 $y$.
Answer: 8
Problem: The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute

$$
\frac{r_{1}}{\sqrt[3]{r_{2}}}+\frac{r_{2}}{\sqrt[3]{r_{1}}}
$$
Answer: 508
Problem: 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 externally tangent to each other. Given that $A Q=12$ and $D Q=8$, compute $C D$.
Answer: 5 \sqrt{10}
Problem: 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 $A R M L$.
Answer: 49
Problem: 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 $\sum_{A \in \mathcal{S}} A(10)$.
Answer: 46760
Problem: 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 $\widehat{P_{1} P_{2}}$ is drawn. There is a circle that is tangent to $\widehat{P_{1} P_{2}}$ and to sides $\overline{M N}$ and $\overline{M R}$. The radius of this circle can be written in the form $x-y \sqrt{2}$, where $x$ and $y$ are positive integers. Compute $x+y$.
Answer: 36
Problem: 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 $M N$.
Answer: 6 \sqrt{2}-2
Problem: 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.
Answer: \frac{1}{6084}
Problem: 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 $T$.
Answer: 13
Problem: Let $T=13$. Compute the least integer $n>2023$ such that the equation $x^{2}-T x-n=0$ has integer solutions.
Answer: 2028
Problem: 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 maximum and minimum possible number of cromulent elements in a sequence of $n$ consecutive positive integers with $n=6$;
Answer: 1,2
Problem: 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 maximum and minimum possible number of cromulent elements in a sequence of $n$ consecutive positive integers with $n=7$.
Answer: 1,3
Problem: 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

$$
\frac{a_{4}}{a_{5}}+\frac{a_{5}}{a_{6}}+\frac{a_{6}}{a_{7}}+\cdots+\frac{a_{2022}}{a_{2023}} .
$$
Answer: -2019
Problem: 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.
Answer: \frac{17}{145}
Problem: The degree-measures of the interior angles of convex hexagon TIEBRK are all integers in arithmetic progression. Compute the least possible degree-measure of the smallest interior angle in hexagon TIEBRK.
Answer: 65
Problem: A six-digit natural number is "sort-of-decreasing" if its first three digits are in strictly decreasing order and its last three digits are in strictly decreasing order. For example, 821950 and 631631 are sort-of-decreasing but 853791 and 911411 are not. Compute the number of sort-of-decreasing six-digit natural numbers.
Answer: 14400
Problem: For each positive integer $N$, let $P(N)$ denote the product of the digits of $N$. For example, $P(8)=8$, $P(451)=20$, and $P(2023)=0$. Compute the least positive integer $n$ such that $P(n+23)=P(n)+23$.
Answer: 34
Problem: Compute the least integer value of the function

$$
f(x)=\frac{x^{4}-6 x^{3}+2 x^{2}-6 x+2}{x^{2}+1}
$$

whose domain is the set of all real numbers.
Answer: -7
Problem: Suppose that noncongruent triangles $A B C$ and $X Y Z$ are given such that $A B=X Y=10, B C=$ $Y Z=9$, and $\mathrm{m} \angle C A B=\mathrm{m} \angle Z X Y=30^{\circ}$. Compute $[A B C]+[X Y Z]$.
Answer: 25 \sqrt{3}
Problem: The mean, median, and unique mode of a list of positive integers are three consecutive integers in some order. Compute the least possible sum of the integers in the original list.
Answer: 12
Problem: David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the other. For example, a table with the numbers $8,4,5$ (in clockwise order) is considered the same as a table with the numbers 4, 5,8 (in clockwise order), but both tables are different from a table with the numbers 8, 5, 4 (in clockwise order). Given that the numbers he carves sum to 17 , compute the number of different tables he can make.
Answer: 7711
Problem: In quadrilateral $A B C D, \mathrm{~m} \angle B+\mathrm{m} \angle D=270^{\circ}$. The circumcircle of $\triangle A B D$ intersects $\overline{C D}$ at point $E$, distinct from $D$. Given that $B C=4, C E=5$, and $D E=7$, compute the diameter of the circumcircle of $\triangle A B D$.
Answer: \sqrt{130}
Problem: Let $i=\sqrt{-1}$. The complex number $z=-142+333 \sqrt{5} i$ can be expressed as a product of two complex numbers in multiple different ways, two of which are $(57-8 \sqrt{5} i)(-6+5 \sqrt{5} i)$ and $(24+\sqrt{5} i)(-3+14 \sqrt{5} i)$. Given that $z=-142+333 \sqrt{5} i$ can be written as $(a+b \sqrt{5} i)(c+d \sqrt{5} i)$, where $a, b, c$, and $d$ are positive integers, compute the lesser of $a+b$ and $c+d$.
Answer: 17
Problem: Parallelogram $A B C D$ is rotated about $A$ in the plane, resulting in $A B^{\prime} C^{\prime} D^{\prime}$, with $D$ on $\overline{A B^{\prime}}$. Suppose that $\left[B^{\prime} C D\right]=\left[A B D^{\prime}\right]=\left[B C C^{\prime}\right]$. Compute $\tan \angle A B D$.
Answer: \sqrt{2}-1$,$\frac{3-\sqrt{2}}{7}
Problem: Compute the least integer greater than 2023 , the sum of whose digits is 17 .
Answer: 2069
Problem: Let $T$ = 2069, and let $K$ be the sum of the digits of $T$. Let $r$ and $s$ be the two roots of the polynomial $x^{2}-18 x+K$. Compute $|r-s|$.
Answer: 16
Problem: Let $T=$ 7, and let $K=9 T$. Let $A_{1}=2$, and for $n \geq 2$, let

$$
A_{n}= \begin{cases}A_{n-1}+1 & \text { if } n \text { is not a perfect square } \\ \sqrt{n} & \text { if } n \text { is a perfect square. }\end{cases}
$$

Compute $A_{K}$.
Answer: 21
Problem: Let $T=$ 21. The number $20^{T} \cdot 23^{T}$ has $K$ positive divisors. Compute the greatest prime factor of $K$.
Answer: 43
Problem: Let $T=43$. Compute the positive integer $n \neq 17$ for which $\left(\begin{array}{c}T-3 \\ 17\end{array}\right)=\left(\begin{array}{c}T-3 \\ n\end{array}\right)$.
Answer: 23
Problem: Let $T=23$ . Compute the units digit of $T^{2023}+T^{20}-T^{23}$.
Answer: 1
Problem: Let $T=$ 3. Suppose that $T$ fair coins are flipped. Compute the probability that at least one tails is flipped.
Answer: \frac{7}{8}
Problem: Let $T=$ $\frac{7}{8}$. The number $T$ can be expressed as a reduced fraction $\frac{m}{n}$, where $m$ and $n$ are positive integers whose greatest common divisor is 1 . The equation $x^{2}+(m+n) x+m n=0$ has two distinct real solutions. Compute the lesser of these two solutions.
Answer: -8
Problem: Let $T=$ -8, and let $i=\sqrt{-1}$. Compute the positive integer $k$ for which $(-1+i)^{k}=\frac{1}{2^{T}}$.
Answer: 16
Problem: Let $T=$ 16. Compute the value of $x$ that satisfies $\log _{4} T=\log _{2} x$.
Answer: 4
Problem: Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, whose area is $T$. Given that $L E=5 \sqrt{2}$, compute $[L E O]$.
Answer: 7
Problem: Let $T=$ 7. Compute the units digit of $T^{2023}+(T-2)^{20}-(T+10)^{23}$.
Answer: 5
Problem: Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\overline{B P}$ is tangent to the smaller circle. Compute $B P$.
Answer: \sqrt{15}
Problem: Compute the largest prime divisor of $15 !-13$ !.
Answer: 19
Problem: Three non-overlapping squares of positive integer side lengths each have one vertex at the origin and sides parallel to the coordinate axes. Together, the three squares enclose a region whose area is 41 . Compute the largest possible perimeter of the region.
Answer: 32
Problem: A circle with center $O$ and radius 1 contains chord $\overline{A B}$ of length 1 , and point $M$ is the midpoint of $\overline{A B}$. If the perpendicular to $\overline{A O}$ through $M$ intersects $\overline{A O}$ at $P$, compute $[M A P]$.
Answer: \frac{\sqrt{3}}{32}
Problem: $\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$.
Answer: 162
Problem: The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real numbers in arithmetic progression. Compute the value of $j$.
Answer: -50
Problem: Compute the smallest positive integer $n$ such that

$$
n,\lfloor\sqrt{n}\rfloor,\lfloor\sqrt[3]{n}\rfloor,\lfloor\sqrt[4]{n}\rfloor,\lfloor\sqrt[5]{n}\rfloor,\lfloor\sqrt[6]{n}\rfloor,\lfloor\sqrt[7]{n}\rfloor, \text { and }\lfloor\sqrt[8]{n}\rfloor
$$

are distinct.
Answer: 4096
Problem: If $n$ is a positive integer, then $n$ !! is defined to be $n(n-2)(n-4) \cdots 2$ if $n$ is even and $n(n-2)(n-4) \cdots 1$ if $n$ is odd. For example, $8 ! !=8 \cdot 6 \cdot 4 \cdot 2=384$ and $9 ! !=9 \cdot 7 \cdot 5 \cdot 3 \cdot 1=945$. Compute the number of positive integers $n$ such that $n !$ ! divides 2012!!.
Answer: 1510
Problem: On the complex plane, the parallelogram formed by the points $0, z, \frac{1}{z}$, and $z+\frac{1}{z}$ has area $\frac{35}{37}$, and the real part of $z$ is positive. If $d$ is the smallest possible value of $\left|z+\frac{1}{z}\right|$, compute $d^{2}$.
Answer: \frac{50}{37}
Problem: One face of a $2 \times 2 \times 2$ cube is painted (not the entire cube), and the cube is cut into eight $1 \times 1 \times 1$ cubes. The small cubes are reassembled randomly into a $2 \times 2 \times 2$ cube. Compute the probability that no paint is showing.
Answer: \frac{1}{16}
Problem: In triangle $A B C, A B=B C$. A trisector of $\angle B$ intersects $\overline{A C}$ at $D$. If $A B, A C$, and $B D$ are integers and $A B-B D=7$, compute $A C$.
Answer: 146
Problem: The rational number $r$ is the largest number less than 1 whose base-7 expansion consists of two distinct repeating digits, $r=0 . \underline{A} \underline{B} \underline{A} \underline{B} \underline{A} \underline{B} \ldots$ Written as a reduced fraction, $r=\frac{p}{q}$. Compute $p+q$ (in base 10).
Answer: 95
Problem: Let $T=95$. Triangle $A B C$ has $A B=A C$. Points $M$ and $N$ lie on $\overline{B C}$ such that $\overline{A M}$ and $\overline{A N}$ trisect $\angle B A C$, with $M$ closer to $C$. If $\mathrm{m} \angle A M C=T^{\circ}$, then $\mathrm{m} \angle A C B=U^{\circ}$. Compute $U$.
Answer: 75
Problem: Let $T=75$. At Wash College of Higher Education (Wash Ed.), the entering class has $n$ students. Each day, two of these students are selected to oil the slide rules. If the entering class had two more students, there would be $T$ more ways of selecting the two slide rule oilers. Compute $n$.
Answer: 37
Problem: Compute the least positive integer $n$ such that the set of angles

$$
\left\{123^{\circ}, 246^{\circ}, \ldots, n \cdot 123^{\circ}\right\}
$$

contains at least one angle in each of the four quadrants.
Answer: 11
Problem: Let $T=11$. In ARMLvania, license plates use only the digits 1-9, and each license plate contains exactly $T-3$ digits. On each plate, all digits are distinct, and for all $k \leq T-3$, the $k^{\text {th }}$ digit is at least $k$. Compute the number of valid ARMLvanian license plates.
Answer: 256
Problem: Let $T=256$. Let $\mathcal{R}$ be the region in the plane defined by the inequalities $x^{2}+y^{2} \geq T$ and $|x|+|y| \leq \sqrt{2 T}$. Compute the area of region $\mathcal{R}$.
Answer: 1024-256 \pi
Problem: Triangle $A B C$ has $\mathrm{m} \angle A>\mathrm{m} \angle B>\mathrm{m} \angle C$. The angle between the altitude and the angle bisector at vertex $A$ is $6^{\circ}$. The angle between the altitude and the angle bisector at vertex $B$ is $18^{\circ}$. Compute the degree measure of angle $C$.
Answer: 44^{\circ}
Problem: Compute the number of ordered pairs of integers $(b, c)$, with $-20 \leq b \leq 20,-20 \leq c \leq 20$, such that the equations $x^{2}+b x+c=0$ and $x^{2}+c x+b=0$ share at least one root.
Answer: 81
Problem: A seventeen-sided die has faces numbered 1 through 17, but it is not fair: 17 comes up with probability $1 / 2$, and each of the numbers 1 through 16 comes up with probability $1 / 32$. Compute the probability that the sum of two rolls is either 20 or 12.
Answer: \frac{7}{128}
Problem: Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \leq 50,1<b \leq 50$, and $\log _{b} a$ is rational.
Answer: 81
Problem: Suppose that 5-letter "words" are formed using only the letters A, R, M, and L. Each letter need not be used in a word, but each word must contain at least two distinct letters. Compute the number of such words that use the letter A more than any other letter.
Answer: 165
Problem: Positive integers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence. If $a_{1}=10$ and $a_{a_{2}}=100$, compute $a_{a_{a_{3}}}$.
Answer: 820
Problem: The graphs of $y=x^{2}-|x|-12$ and $y=|x|-k$ intersect at distinct points $A, B, C$, and $D$, in order of increasing $x$-coordinates. If $A B=B C=C D$, compute $k$.
Answer: 10+2 \sqrt{2}
Problem: The zeros of $f(x)=x^{6}+2 x^{5}+3 x^{4}+5 x^{3}+8 x^{2}+13 x+21$ are distinct complex numbers. Compute the average value of $A+B C+D E F$ over all possible permutations $(A, B, C, D, E, F)$ of these six numbers.
Answer: -\frac{23}{60}
Problem: Let $N=\left\lfloor(3+\sqrt{5})^{34}\right\rfloor$. Compute the remainder when $N$ is divided by 100 .
Answer: 47
Problem: Let $A B C$ be a triangle with $\mathrm{m} \angle B=\mathrm{m} \angle C=80^{\circ}$. Compute the number of points $P$ in the plane such that triangles $P A B, P B C$, and $P C A$ are all isosceles and non-degenerate. Note: the approximation $\cos 80^{\circ} \approx 0.17$ may be useful.
Answer: 6
Problem: If $\lceil u\rceil$ denotes the least integer greater than or equal to $u$, and $\lfloor u\rfloor$ denotes the greatest integer less than or equal to $u$, compute the largest solution $x$ to the equation

$$
\left\lfloor\frac{x}{3}\right\rfloor+\lceil 3 x\rceil=\sqrt{11} \cdot x
$$
Answer: \frac{189 \sqrt{11}}{11}
Problem: If $x, y$, and $z$ are positive integers such that $x y=20$ and $y z=12$, compute the smallest possible value of $x+z$.
Answer: 8
Problem: Let $T=8$. Let $A=(1,5)$ and $B=(T-1,17)$. Compute the value of $x$ such that $(x, 3)$ lies on the perpendicular bisector of $\overline{A B}$.
Answer: 20
Problem: Let T be a rational number. Let $N$ be the smallest positive $T$-digit number that is divisible by 33 . Compute the product of the last two digits of $N$.
Answer: 6
Problem: Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n}(z)=$ $f_{n-1}(\bar{z})$. If $f_{1}(z)+2 f_{2}(z)+3 f_{3}(z)+4 f_{4}(z)+5 f_{5}(z)=T+T i$, compute $|z|$.
Answer: \sqrt{26}
Problem: Let $T=\sqrt{26}$. Compute the number of ordered pairs of positive integers $(a, b)$ with the property that $a b=T^{20} \cdot 210^{12}$, and the greatest common divisor of $a$ and $b$ is 1 .
Answer: 32
Problem: Let $T=32$. Given that $\sin \theta=\frac{\sqrt{T^{2}-64}}{T}$, compute the largest possible value of the infinite series $\cos \theta+\cos ^{2} \theta+\cos ^{3} \theta+\ldots$.
Answer: \frac{1}{3}
Problem: Let $T=\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of square $A B C D$.
Answer: 450
Problem: Let $T=-14$, and let $d=|T|$. A person whose birthday falls between July 23 and August 22 inclusive is called a Leo. A person born in July is randomly selected, and it is given that her birthday is before the $d^{\text {th }}$ day of July. Another person who was also born in July is randomly selected, and it is given that his birthday is after the $d^{\text {th }}$ day of July. Compute the probability that exactly one of these people is a Leo.
Answer: \frac{9}{17}
Problem: Let $T=-10$. Given that $\log _{2} 4^{8 !}+\log _{4} 2^{8 !}=6 ! \cdot T \cdot x$, compute $x$.
Answer: -14
Problem: Let $T=20$. For some real constants $a$ and $b$, the solution sets of the equations $x^{2}+(5 b-T-a) x=T+1$ and $2 x^{2}+(T+8 a-2) x=-10 b$ are the same. Compute $a$.
Answer: -10
Problem: Let T be a rational number, and let $K=T-2$. If $K$ workers can produce 9 widgets in 1 hour, compute the number of workers needed to produce $\frac{720}{K}$ widgets in 4 hours.
Answer: 20
Problem: Let $T=2018$, and append the digits of $T$ to $\underline{A} \underline{A} \underline{B}$ (for example, if $T=17$, then the result would be $\underline{1} \underline{\underline{A}} \underline{A} \underline{B}$ ). If the resulting number is divisible by 11 , compute the largest possible value of $A+B$.
Answer: 14
Problem: Given that April $1^{\text {st }}, 2012$ fell on a Sunday, what is the next year in which April $1^{\text {st }}$ will fall on a Sunday?
Answer: 2018
Problem: Let $p$ be a prime number. If $p$ years ago, the ages of three children formed a geometric sequence with a sum of $p$ and a common ratio of 2 , compute the sum of the children's current ages.
Answer: 28
Problem: Define a reverse prime to be a positive integer $N$ such that when the digits of $N$ are read in reverse order, the resulting number is a prime. For example, the numbers 5, 16, and 110 are all reverse primes. Compute the largest two-digit integer $N$ such that the numbers $N, 4 \cdot N$, and $5 \cdot N$ are all reverse primes.
Answer: 79
Problem: Some students in a gym class are wearing blue jerseys, and the rest are wearing red jerseys. There are exactly 25 ways to pick a team of three players that includes at least one player wearing each color. Compute the number of students in the class.
Answer: 7
Problem: Point $P$ is on the hypotenuse $\overline{E N}$ of right triangle $B E N$ such that $\overline{B P}$ bisects $\angle E B N$. Perpendiculars $\overline{P R}$ and $\overline{P S}$ are drawn to sides $\overline{B E}$ and $\overline{B N}$, respectively. If $E N=221$ and $P R=60$, compute $\frac{1}{B E}+\frac{1}{B N}$.
Answer: \frac{1}{60}
Problem: $\quad$ Compute all real values of $x$ such that $\log _{2}\left(\log _{2} x\right)=\log _{4}\left(\log _{4} x\right)$.
Answer: \sqrt{2}
Problem: Let $k$ be the least common multiple of the numbers in the set $\mathcal{S}=\{1,2, \ldots, 30\}$. Determine the number of positive integer divisors of $k$ that are divisible by exactly 28 of the numbers in the set $\mathcal{S}$.
Answer: 23
Problem: Let $A$ and $B$ be digits from the set $\{0,1,2, \ldots, 9\}$. Let $r$ be the two-digit integer $\underline{A} \underline{B}$ and let $s$ be the two-digit integer $\underline{B} \underline{A}$, so that $r$ and $s$ are members of the set $\{00,01, \ldots, 99\}$. Compute the number of ordered pairs $(A, B)$ such that $|r-s|=k^{2}$ for some integer $k$.
Answer: 42
Problem: For $k \geq 3$, we define an ordered $k$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ to be special if, for every $i$ such that $1 \leq i \leq k$, the product $x_{1} \cdot x_{2} \cdot \ldots \cdot x_{k}=x_{i}^{2}$. Compute the smallest value of $k$ such that there are at least 2009 distinct special $k$-tuples.
Answer: 12
Problem: A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\frac{1}{r}+\frac{1}{h}$.
Answer: 6
Problem: If $6 \tan ^{-1} x+4 \tan ^{-1}(3 x)=\pi$, compute $x^{2}$.
Answer: \frac{15-8 \sqrt{3}}{33}
Problem: A rectangular box has dimensions $8 \times 10 \times 12$. Compute the fraction of the box's volume that is not within 1 unit of any of the box's faces.
Answer: \frac{1}{2}
Problem: Let $T=T N Y W R$. Compute the largest real solution $x$ to $(\log x)^{2}-\log \sqrt{x}=T$.
Answer: 10
Problem: Let $T=T N Y W R$. Kay has $T+1$ different colors of fingernail polish. Compute the number of ways that Kay can paint the five fingernails on her left hand by using at least three colors and such that no two consecutive fingernails have the same color.
Answer: 109890
Problem: Compute the number of ordered pairs $(x, y)$ of positive integers satisfying $x^{2}-8 x+y^{2}+4 y=5$.
Answer: 4
Problem: Let $T=T N Y W R$ and let $k=21+2 T$. Compute the largest integer $n$ such that $2 n^{2}-k n+77$ is a positive prime number.
Answer: 12
Problem: Let $T=T N Y W R$. In triangle $A B C, B C=T$ and $\mathrm{m} \angle B=30^{\circ}$. Compute the number of integer values of $A C$ for which there are two possible values for side length $A B$.
Answer: 5