ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $a, b$ be two real numbers such that \(\sqrt[3]{a} - \sqrt[3]{b} = 10\) and \(ab = \left( \frac{8 - a - b}{6}\right)^3\). Find \(a - b\).
Answer: 280
Problem: The initial number of inhabitants in a city, which is greater than 150, is a perfect square. With an increase of 1000 inhabitants, it becomes a perfect square plus one. After another increase of 1000 inhabitants, it becomes a perfect square again. Determine the initial number of inhabitants in the city.
Answer: 249001
Problem: Given $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$ in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is $8102$?
Answer: 1991
Problem: How many multiples of $12$ divide $12!$ and have exactly $12$ divisors?
Answer: 6
Problem: Freddy, the king of flavortext, has an infinite chest of coins. For each number \( p \) in the interval \([0, 1]\), Freddy has a coin that has a probability \( p \) of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is \( \frac{p}{q} \) for two integers \( p, q \), find \( p + q \).
Answer: 23
Problem: Mr. Schwartz has been hired to paint a row of 7 houses. Each house must be painted red, blue, or green. However, to make it aesthetically pleasing, he doesn't want any three consecutive houses to be the same color. Find the number of ways he can fulfill his task.
Answer: 1344
Problem: Find the number of addition problems in which a two-digit number is added to a second two-digit number to give a two-digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,, \begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end{tabular}\,, \begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}42\\23\\\hline 65\end{tabular}\,.
Answer: 3240
Problem: Find the integer $n$ such that:
\[
n + \lfloor \sqrt{n} \rfloor + \lfloor \sqrt{\sqrt{n}} \rfloor = 2017.\]

Here, $\lfloor \cdot \rfloor$ denotes the floor function.
Answer: 1967
Problem: Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $\triangle ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$.
Answer: 1814
Problem: Three fair six-sided dice are labeled with the numbers $\{1, 2, 3, 4, 5, 6\},$ $\{1, 2, 3, 4, 5, 6\},$ and $\{1, 2, 3, 7, 8, 9\},$ respectively. All three dice are rolled. The probability that at least two of the dice have the same value is $\frac{m}{n},$ where $m, n$ are relatively prime positive integers. Find $100m + n.$
Answer: 1136
Problem: Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2 + 4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$
Answer: 14
Problem: Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
Answer: 1186
Problem: There are several teacups in the kitchen, some with handles and others without handles. The number of ways to select two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?
Answer: 29
Problem: Find the number of pairs of integer solutions \((x,y)\) that satisfy the equation:
\[
(x-y+2)(x-y-2) = -(x-2)(y-2)
\]
Answer: 6
Problem: Compute the smallest integer $n \geq 4$ such that $\binom{n}{4}$ ends in 4 or more zeroes (i.e., the rightmost four digits of $\binom{n}{4}$ are $0000$).
Answer: 8128
Problem: Find all positive integers $k$ such that there exist positive integers $a$ and $b$ satisfying the equation:
\[
a^2 + 4 = (k^2 - 4)b^2.
\]
Answer: 3
Problem: Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod{p}$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod{p}?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
Answer: 24
Problem: What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$
Answer: 361
Problem: Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.
Answer: -2
Problem: We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define 
\[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \]
Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
Answer: 5
Problem: Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{2520}$.
Answer: 90
Problem: Circles $C_1$, $C_2$, $C_3$ have radius $1$ and centers $O$, $P$, $Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^\circ$, $\angle BQC = 36^\circ$, and $\angle COA = 72^\circ$. Find angle $\angle ABC$ (degrees).
Answer: 12
Problem: For what positive integer $k$ is $\binom{100}{k} \binom{200}{k}$ maximal?
Answer: 67
Problem: Consider a sequence of eleven squares with side lengths $3, 6, 9, 12, \ldots, 33$. Eleven copies of a single square, each with area $A$, have the same total area as the total area of the eleven squares in the sequence. Find $A$.
Answer: 414
Problem: Three vertices of a parallelogram are \((2, -4), (-2, 8),\) and \((12, 7)\). Determine the sum of the three possible x-coordinates of the fourth vertex.
Answer: 12
Problem: Let $\odot$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ and $c$, $(a + b) \odot c = (a \odot c) + (b \odot c)$ and $a \odot (b + c) = (a \odot b) \odot c.$ Given that $5 \odot 5 = 160$, find the value of $7 \odot 7$.
Answer: 896
Problem: Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$.
Answer: 62208
Problem: Find $\lfloor \sqrt{19992000} \rfloor$ where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
Answer: 4471
Problem: Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$.
Answer: 60
Problem: Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a, b, c)$ such that $2a + 3b + 5c = 200$ and $a + b + c = N$.
Answer: 280
Problem: Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \dots, P_n$ such that $n \ge 2$ and:

- The directed line segments $P_0P_1, P_1P_2, \dots, P_{n-1}P_n$ follow successive coordinate directions: east (for $P_0P_1$), north, west, south, east, etc.
- The lengths of these line segments are positive and strictly increasing.

**Question:**

How many of the points $(x,y)$ with integer coordinates $0 \le x \le 2011, 0 \le y \le 2011$ *cannot* be the last point, $P_n$, of any growing spiral?
Answer: 10053
Problem: Calculate the value of $\sqrt{\frac{11^4 + 100^4 + 111^4}{2}}$ and provide the answer as an integer.
Answer: 11221
Problem: Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.
Answer: 420
Problem: Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^\circ$. Find the degree measure of $\angle BAC$.
Answer: 84
Problem: Suppose that $m$ and $n$ are relatively prime positive integers with $A = \frac{m}{n}$, where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] 
Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form.
Answer: 2015
Problem: If $U, S, M, C, A$ are distinct (not necessarily positive) integers such that $U \cdot S \cdot M \cdot C \cdot A = 2020$, what is the greatest possible value of $U + S + M + C + A$?
Answer: 505
Problem: What is the maximal number of crosses that can fit in a $10 \times 11$ board without overlapping?
Answer: 14
Problem: Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$.
Answer: 1
Problem: You are given that \(17! = 355687ab8096000\) for some digits \(a\) and \(b\). Find the two-digit number \(\overline{ab}\) that is missing above.
Answer: 42
Problem: Find the least positive integer such that the product of its digits is $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$.
Answer: 257889
Problem: Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt{3}$, find the length of $AB$.
Answer: 80
Problem: Let $f(x) = x^4 - 4x^3 - 3x^2 - 4x + 1$. Compute the sum of the real roots of $f(x)$.
Answer: 5
Problem: Darwin takes an $11 \times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10 \times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew?
Answer: 220
Problem: Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of \(\frac{P^2}{A}\).
Answer: 45
Problem: You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be seamlessly covered with these dominoes without overlapping, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same way, i.e., their long sides must be parallel to each other. How many such coverings are possible?
Answer: 257
Problem: For real numbers $x$ and $y$, define \[ \nabla(x,y)=x-\frac{1}{y}. \] If \[ \underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \frac{m}{n} \] for relatively prime positive integers $m$, $n$, compute $100m + n$.
Answer: 203817
Problem: For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \frac{p}{q}$ for coprime positive integers $p$, $q$, find $p + q$.
Answer: 91
Problem: Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
Answer: 24
Problem: Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Answer: 100
Problem: Two points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $P(x) > \frac{2}{3}$.
Answer: 174
Problem: We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
Answer: 233
Problem: Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$:  
\[ 
\cos^5(9x) + \cos^5(x) = 32\cos^5(5x)\cos^5(4x) + 5\cos^2(9x)\cos^2(x)(\cos(9x) + \cos(x)). 
\]
Answer: 4064
Problem: Let $k$ be the smallest positive integer such that the binomial coefficient $\binom{10^9}{k}$ is less than the binomial coefficient $\binom{10^9 + 1}{k - 1}$. Let $a$ be the first (from the left) digit of $k$ and let $b$ be the second (from the left) digit of $k$. What is the value of $10a + b$?
Answer: 38
Problem: Let $f(n) = \frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c) = 2$. (Here, $f^5(x)$ means $f(f(f(f(f(x)))))$.)
Answer: 748
Problem: A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits. If you permute its digits, you will get 5 other numbers. Interestingly, the sum of these 5 numbers is exactly 2017. That's all." After a while, B can determine A's house number. Can you find his house number?
Answer: 425
Problem: Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n \ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
Answer: 6423
Problem: In triangle $ABC$, angle $B$ is obtuse and equal to $a$. The bisectors of angles $A$ and $C$ intersect the opposite sides at points $P$ and $M$, respectively. On the side $AC$, points $K$ and $L$ are taken such that $\angle ABK = \angle CBL = 2a - 180^\circ$. What is the angle between the straight lines $KP$ and $LM$?
Answer: 90
Problem: A tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. If the volume of the tetrahedron is $V$, compute $V^2$.
Answer: 252
Problem: Positive rational numbers $x < y < z$ sum to $1$ and satisfy the equation:
\[ (x^2 + y^2 + z^2 - 1)^3 + 8xyz = 0. \]
Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m + n < 1000$, find the maximum value of $m + n$.
Answer: 999
Problem: A disk with radius $10$ and a disk with radius $8$ are drawn such that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles. The radii of the smaller circles are both $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 19
Problem: Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1 \text{ and } d_7 = 3d_4 - 1$$
Answer: 2024
Problem: Define a search algorithm called $\texttt{powSearch}$. Assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0, 2^1, \ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point $A[2^k] = b$, we stop and return $2^k$. Once $A[2^k] > b > A[2^{k-1}]$, we discard the first $2^{k-1}$ elements of $A$ and recursively search in the same manner. For example, for an integer at position $3$, we will search the locations $1, 2, 4, 3$.

Define $g(x)$ as a function that returns the number of (not necessarily distinct) indices examined when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find
\[g(1) + g(2) + \ldots + g(64).\]
Answer: 808
Problem: The sequence of digits \(1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots\) is obtained by writing the positive integers in order. If the \(10^n\)-th digit in this sequence occurs in the part of the sequence in which the \(m\)-digit numbers are placed, define \(f(n)\) to be \(m\). For example, \(f(2)=2\) because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, \(f(1987)\).
Answer: 1984
Problem: Bill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the two circles, and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3$ and $4$, and the distance between the centers of these two circles is $5$, the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Answer: 4807
Problem: For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that
\[
r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4.
\]
Answer: 120
Problem: Quadrilateral $ABCD$ satisfies the following conditions: 
- $AB = 8$, 
- $BC = 5$, 
- $CD = 17$, 
- $DA = 10$. 

Let $E$ be the intersection of diagonals $AC$ and $BD$. It is given that the ratio $BE : ED = 1 : 2$. 

Find the area of quadrilateral $ABCD$.
Answer: 60
Problem: A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n \leq 1000$. Given that 
\[p(1001) = a \cdot 2^b - c,\]
where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.
Answer: 500
Problem: Let $f_1(x) = x^2 - 1$, and for each positive integer $n \geq 2$, define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have?
Answer: 2005
Problem: 30 segments of lengths $1, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{9}, \ldots, \sqrt{59}$ have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After 29 steps, only one segment remains. Find the possible values of its length.
Answer: 30
Problem: A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly $2012$ rocks from the table.
Answer: 18
Problem: At a movie theater, tickets for adults cost $4 more than tickets for children. One afternoon, the theater sold $100 more child tickets than adult tickets, for a total sales amount of $1475. How many dollars would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?
Answer: 1875
Problem: At a party, every guest is a friend of exactly fourteen other guests (not including themselves). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof.
Answer: 64
Problem: In a right trapezoid $ABCD$ where $AB \parallel CD$, the angle at vertex $B$ measures $75^\circ$. Point $H$ is the foot of the perpendicular from point $A$ to the line $BC$. If $BH = DC$ and $AD + AH = 8$, find the area of $ABCD$.
Answer: 8
Problem: Find the sum of all the digits in the decimal representations of all the positive integers less than $1000$.
Answer: 13500
Problem: Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$, $(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points.
Answer: 1247
Problem: How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
Answer: 256
Problem: Let $a = \sqrt[401]{4} - 1$ and for each $n \ge 2$, let $b_n = \binom{n}{1} + \binom{n}{2} a + \ldots + \binom{n}{n} a^{n-1}$. Find $b_{2006} - b_{2005}$.
Answer: 1024
Problem: Dilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A}$, and $f_{C,B}$. Each function $f_{X,Y}$ takes an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his 6 functions, without repeats, in such a way that the resulting expression is well-typed. This means an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. Determine the number of possible orders in which he can compose his 6 functions to satisfy this constraint.
Answer: 18
Problem: Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
Answer: 1112
Problem: Let $\alpha, \beta$ be the distinct positive roots of the equation $2x = \tan x$. Evaluate the integral:

$$\int_0^1 \sin(\alpha x) \sin(\beta x)\, dx$$
Answer: 0
Problem: We are given 5040 balls in \(k\) different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest \(k\) such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.
Answer: 6
Problem: A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?
Answer: 4068434225296
Problem: Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its side lengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^\circ$. Find the area of $ABCD$.
Answer: 30
Problem: The volume of a certain rectangular solid is $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.
Answer: 96
Problem: For a positive integer $n$, define a function $f_n(x)$ on the interval $[0, n+1]$ as follows:
\[ f_n(x) = \left( \sum_{i=1}^{n} |x-i| \right)^2 - \sum_{i=1}^{n} (x-i)^2. \]
Let $a_n$ be the minimum value of $f_n(x)$. Find the value of:
\[ \sum_{n=1}^{11} (-1)^{n+1} a_n. \]
Answer: 450
Problem: Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB = 5$, $BC = 9$, $CD = 10$, and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Answer: 90011
Problem: Determine the smallest value of $(a+5)^2 + (b-2)^2 + (c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2 + b^2 + c^2 - ab - bc - ca = 3$.
Answer: 72