ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a \rfloor$.
Answer: 941
Problem: Each of the integers $1, 2, \ldots, 729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122\ldots$ How many times in this string does the substring $012$ appear?
Answer: 148
Problem: Consider the set $A = \{0, 1, 2, \dots , 9 \}$ and let $(B_1,B_2, \dots , B_k)$ be a collection of nonempty subsets of $A$ such that $B_i \cap B_j$ has at most two elements for $i \neq j$. What is the maximal value of $k$?
Answer: 175
Problem: Compute the number of natural numbers $1 \leq n \leq 10^6$ such that the least prime divisor of $n$ is $17$.
Answer: 11284
Problem: Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\).
Answer: 1940
Problem: How many pairs of positive integers \((a, b)\) satisfy the equation \(\log_a 16 = b\)?
Answer: 3
Problem: The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?
Answer: 1033
Problem: One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write:  
\[ \gcd\left(\frac{a}{b},\frac{c}{d}\right) = \frac{\gcd(ad,bc)}{bd}. \]  

For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that:  
\[ \gcd(m,n)=\frac{1}{K}. \]  

What is $f(2017)-f(2016)$?
Answer: 1152
Problem: Fedja used matches to form a parallelogram, ensuring that the sides are of equal length and the vertices do not lie on a single line. It is observed that exactly 7 or 9 matches fit into the diagonals. Determine the number of matches that make up the perimeter of the parallelogram.
Answer: 22
Problem: Given $1994$ points on a plane, it is known that any $100$ of them can be selected such that $98$ of these can be enclosed by a circle with a diameter of $1$. Determine the minimum number of circles with a radius of $1$ required to cover all $1994$ points.
Answer: 21
Problem: Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.
Answer: 247
Problem: How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group?
Answer: 18377
Problem: The expression $\circ 1 \circ 2 \circ 3 \circ \dots \circ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
Answer: 1006
Problem: Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \frac{193}{137}.\] If $a+b=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.
Answer: 28009
Problem: Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area.
Answer: 612
Problem: For how many integers $n$ between $1$ and $2021$ does the infinite nested expression $\sqrt{n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}}$ give a rational number?
Answer: 44
Problem: Consider all functions $f: \mathbb{N} \to \mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $\mathbb{N}$. Determine the least possible value of $f(1998)$.
Answer: 120
Problem: Let $a$ and $b$ be positive integers such that $(2a + b)(2b + a) = 4752$. Find the value of $ab$.
Answer: 520
Problem: Let $P$ be the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the areas of triangles $\triangle ABC$, $\triangle BCD$, and $\triangle DAP$ are $8 \text{ cm}^2$, $9 \text{ cm}^2$, and $10 \text{ cm}^2$, respectively. Find the area of the quadrilateral $ABCD$.
Answer: 24
Problem: Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$.  If 
\[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] 
can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are positive coprime integers, find $m+n$.
Answer: 240
Problem: The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.

\[\text{[asy]}
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{pair A = (0,10), B=origin, C=(12,0), D=(24,0), E=(24,10), F=(12,10), G=extension(C,E,D,F);}
\text{filldraw(A--C--G--F--cycle, gray(0.7));}
\text{draw(A--B--D--E--F^^E--G--D);}
\text{label("$A$", A, NW);}
\text{label("$B$", B, SW);}
\text{label("$C$", C, S);}
\text{label("$D$", D, SE);}
\text{label("$E$", E, NE);}
\text{label("$F$", F, N);}
\text{[/asy]}
\]
Answer: 90
Problem: In trapezoid $ABCD$, where $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.
Answer: 252
Problem: What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
Answer: 7
Problem: For every positive integer $n$, define $S_n$ to be the sum
\[
  S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach?
Answer: 1944
Problem: Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. What is the minimum number of distinct problems needed?
Answer: 13
Problem: There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students. 

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap: 
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$
Answer: 2013
Problem: How many solutions does the equation:
$$\left\lfloor \frac{x}{20} \right\rfloor = \left\lfloor \frac{x}{17} \right\rfloor$$
have over the set of positive integers?

$\left\lfloor a \right\rfloor$ denotes the largest integer that is less than or equal to $a$.
Answer: 56
Problem: How many cubic polynomials of the form $x^3 - ax^2 + (a+d)x - (a+2d)$ have roots that are all non-negative integers, where $a$ and $d$ are integers?
Answer: 3
Problem: Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
Answer: 20
Problem: How many seven-digit numbers are multiples of $388$ and end in $388$?
Answer: 93
Problem: Let $S$ be a subset of \( \{1, 2, \ldots, 500\} \) such that no two distinct elements of $S$ have a product that is a perfect square. Find, with proof, the maximum possible number of elements in $S$.
Answer: 306
Problem: Let $A$ be a positive integer which is a multiple of 3 but not a multiple of 9. If adding the product of each digit of $A$ to $A$ results in a multiple of 9, find the possible minimum value of $A$.
Answer: 138
Problem: Cat and Claire are having a conversation about Cat's favorite number.

Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"

Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"

Cat says, "That's cool! My favorite number is not among those four numbers, though."

Claire says, "Now I know your favorite number!"

What is Cat's favorite number?
Answer: 70
Problem: The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $\frac{2007}{24832}$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
Answer: 194
Problem: Determine all positive integers $n$ for which $2^{n-1}n + 1$ is a perfect square.
Answer: 5
Problem: Students are in a classroom with $n$ rows. In each row, there are $m$ tables. It is given that $m, n \geq 3$. At each table, there is exactly one student. We call the neighbors of a student those sitting one place to the right, left, in front, and behind him. Each student shook hands with his neighbors. In the end, there were $252$ handshakes. How many students were in the classroom?
Answer: 153
Problem: Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so that $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime, find $p + q$.
Answer: 1369
Problem: Find the integer part of \( \sum_{n=1}^{40000} \frac{1}{\sqrt{n}} \).
Answer: 398
Problem: There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N}$, including $A$ and $B$. What is the least number of colours which suffices?
Answer: 4
Problem: Let \( a^3 - a - 1 = 0 \). Find the exact value of the expression 
\[
\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.
\]
Answer: 2
Problem: For every $a \in \mathbb{N}$, denote by $M(a)$ the number of elements in the set
\[ \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \}. \]
Find $\max_{a \leq 1983} M(a).$
Answer: 121
Problem: Find the total number of different integer values the function 
\[ f(x) = \lfloor x \rfloor + \lfloor 2x \rfloor + \left\lfloor \frac{5x}{3} \right\rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor \]
takes for real numbers $x$ with $0 \leq x \leq 100$.
Answer: 734
Problem: I have a $2 \times 4$ grid of squares. How many ways can I shade at least one of the squares such that no two shaded squares share an edge?
Answer: 40
Problem: How many positive integers $n$ satisfy the inequality
\[\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \ ?\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
Answer: 15049
Problem: Given that $0.3010 < \log 2 < 0.3011$ and $0.4771 < \log 3 < 0.4772$, find the leftmost digit of $12^{37}$.
Answer: 8
Problem: What is the greatest common divisor of the set of numbers $\{16^n + 10n - 1 \mid n=1,2,\ldots \}$?
Answer: 25
Problem: A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
Answer: 1000
Problem: Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the number of galleons between the three of them is a multiple of $7$. What are all the possible numbers of galleons that Harry can have?
Answer: 6
Problem: It is given that the number $4^{11} + 1$ is divisible by some prime greater than $1000$. Determine this prime.
Answer: 2113
Problem: In triangle $ABC$, the lengths of all sides are integers, $\angle B = 2 \angle A$ and $\angle C > 90^\circ$. Find the smallest possible perimeter of this triangle.
Answer: 77
Problem: For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$ when $n$ is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6$. Find the sum of the two solutions to the equation $n - 3s(n) = 2022$.
Answer: 4107
Problem: What is the size of the largest subset $S'$ of $S = \{2^x 3^y 5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?
Answer: 19
Problem: Let $f_1(n)$ be the number of divisors of $n$, and define $f_k(n) = f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such that $f_k(2013^{2013}) = 2$.
Answer: 4
Problem: A polynomial whose roots are all equal to each other is called a unicorn. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns.
Answer: 5
Problem: Consider a triangle $ABC$ with $AB=AC$, and $D$ as the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE = \angle ECB = 18^{\circ}$. If $AD = 3$, find the length of the segment $CE$.
Answer: 6
Problem: Determine the values of the real parameter $a$, such that the equation
\[ \sin 2x \sin 4x - \sin x \sin 3x = a \]
has a unique solution in the interval $[0, \pi)$.
Answer: 1
Problem: Find all three-digit numbers $\overline{abc}$ ($a,b \neq 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$.
Answer: 627
Problem: There exist complex numbers $z_1, z_2, \dots, z_{10}$ which satisfy the condition: $$|z_k i^k + z_{k+1} i^{k+1}| = |z_{k+1} i^k + z_k i^{k+1}|$$ for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. Given that $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1| + |z_2| + \cdots + |z_{10}|$.
Answer: 103
Problem: A permutation \((a_1, a_2, a_3, \dots, a_{2012})\) of \((1, 2, 3, \dots, 2012)\) is selected at random. If \(S\) is the expected value of 
\[
\sum_{i = 1}^{2012} | a_i - i |
\],
then compute the sum of the prime factors of \(S\).
Answer: 2083
Problem: For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is:
\[ (-1)^{\alpha(i-1, j-1)}. \]
Compute the remainder when $\det M$ is divided by $1000$.
Answer: 208
Problem: Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.
Answer: 382
Problem: Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?
Answer: 4000
Problem: Find a positive integer $n$ such that if you place the digit $2$ on the left and the digit $1$ on the right of $n$, the resulting number is equal to $33n$.
Answer: 87
Problem: When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and lose 1 energy, where $n$ is her current energy, or rest (i.e., buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?
Answer: 4293
Problem: Say that a function $f : \{1, 2, \ldots, 1001\} \to \mathbb{Z}$ is *almost* polynomial if there exists a polynomial $p(x) = a_{200}x^{200} + \ldots + a_1x + a_0$ where each $a_n$ is an integer with $|a_n| \le 201$, and $|f(x) - p(x)| \le 1$ for all $x \in \{1, 2, \ldots, 1001\}$. Let $N$ be the number of such almost polynomial functions. Compute the remainder when $N$ is divided by $199$.
Answer: 0
Problem: A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $\overline{AB}$ for which $A$ and $B$ are distinct lattice points in this set, and the midpoint of $\overline{AB}$ is also a lattice point (not necessarily in the set).
Answer: 1200
Problem: Determine the largest integer $n$ such that $7^{2048} - 1$ is divisible by $2^n$.
Answer: 14
Problem: Evaluate the expression:
\[2^{7} \times 3^{0} + 2^{6} \times 3^{1} + 2^{5} \times 3^{2} + \cdots + 2^{0} \times 3^{7}.\]
Answer: 6305
Problem: Find the coefficient of $x^{7}y^{6}$ in $(xy + x + 3y + 3)^{8}$.
Answer: 672
Problem: Compute \( \gcd(5^{2547} - 1, 5^{2004} - 1) \).
Answer: 124
Problem: A lucky number is a number whose digits are only $4$ or $7$. What is the $17^{th}$ smallest lucky number?
Answer: 4474
Problem: Natural numbers $k, l, p$, and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$, then $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of $x^2 - px + q = 0$. What is the sum of all possible values of $q$?
Answer: 4
Problem: Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, and $97$. Suppose the curve $\mathcal{V}$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal{V}$ at each of $A$, $B$, and $C$.
Answer: 237
Problem: Maria has $14$ days to train for an olympiad. The conditions are:
1. She cannot train for $3$ consecutive days.
2. She cannot rest for $3$ consecutive days.

Determine how many configurations of training days she can achieve her goal.
Answer: 1220
Problem: The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook?
Answer: 57
Problem: A point is chosen uniformly at random from the interior of a unit square. Let $p$ be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that $p^2$ can be written as $m-\sqrt{n}$ for positive integers $m$ and $n$, what is $m+n$?
Answer: 11
Problem: How many subsets of the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \) have the property that no two of its elements differ by more than 5? For example, count the sets \( \{3\} \), \( \{2, 5, 7\} \), and \( \{5, 6, 7, 8, 9\} \) but not the set \( \{1, 3, 5, 7\} \).
Answer: 256
Problem: Compute the remainder when $2^{3^5} + 3^{5^2} + 5^{2^3}$ is divided by $30$.
Answer: 6
Problem: On the sides $BC$ and $CD$ of the square $ABCD$ with side length $1$, points $E$ and $F$ are chosen such that $\angle EAB = 20^\circ$ and $\angle EAF = 45^\circ$. Calculate the distance from point $A$ to the line $EF$.
Answer: 1
Problem: Yasha writes in the cells of a $99 \times 99$ table all positive integers from 1 to $99^2$, with each number appearing once. Grisha looks at the table and selects several cells such that no two selected cells share a common side. Grisha then sums up the numbers in all the selected cells. Determine the largest sum that Grisha can guarantee to achieve.
Answer: 24017351
Problem: Misha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transactions:
- Buy or sell one ore for $3.
- Buy or sell bundles of three wheat for $12 each.
- Sell one wheat for one ore.

His ultimate goal is to build a city, which requires:
- Three ore
- Two wheat

How many dollars must Misha begin with in order to build a city after three days of working?
Answer: 9
Problem: The 16 fields of a $4 \times 4$ checkerboard can be arranged in 18 lines as follows: the four rows, the four columns, the five diagonals from northwest to southeast, and the five diagonals from northeast to southwest. These diagonals consist of 2, 3, or 4 edge-adjacent fields of the same color; the corner fields of the chessboard alone do not form a diagonal. Now, we place a token in 10 of the 16 fields. Each of the 18 lines containing an even number of tokens earns a point. What is the highest possible point total that can be achieved by optimally placing the 10 tokens? Explain your answer.
Answer: 16
Problem: Find the number of arithmetic sequences $a_1, a_2, a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
Answer: 4
Problem: Equilateral triangle $ABC$ has a side length of $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and the minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the "lune" that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p} - \frac{q\pi}{r}$, where $p, q$, and $r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
Answer: 10
Problem: The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ such that $\sin^2\theta = \frac{m}{n}$. Find $m + n$.
Answer: 55
Problem: How many ordered pairs of integers \((x, y)\) satisfy the equation \(x^2 + y^2 + xy = 4(x + y)\)?
Answer: 3
Problem: $A = 1 \cdot 4 \cdot 7 \cdots 2014$. Find the last non-zero digit of $A$ if it is known that $A \equiv 1 \pmod{3}$.
Answer: 2
Problem: The subnumbers of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime.
Answer: 9
Problem: Let $ABC$ be an equilateral triangle with an area of $1998\ \text{cm}^2$. Points $K$, $L$, and $M$ divide the segments $[AB]$, $[BC]$, and $[CA]$, respectively, in the ratio $3:4$. The line $AL$ intersects the lines $CK$ and $BM$ at points $P$ and $Q$, respectively. The line $BM$ intersects the line $CK$ at point $R$. Find the area of triangle $PQR$.
Answer: 54
Problem: Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.
Answer: 8
Problem: Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
Answer: 4620
Problem: Let $ABCD$ and $AEFG$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle BAE < 45^\circ$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
Answer: 4940