ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. For example, her expression could be $4 + 3 \times 3$, with a value of 13. Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
Answer: 50
Problem: Suppose that $x^{10} + x + 1 = 0$ and $x^{100} = a_0 + a_1x + \ldots + a_9x^9$. Find $a_5$.
Answer: 252
Problem: The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient 1, meets the x-axis at the points $(1, 0)$, $(2, 0)$, $(3, 0)$, $\ldots$, $(2020, 0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $\frac{a!}{b}$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a + b$.
Answer: 2023
Problem: Find the product of all possible real values for $k$ such that the system of equations
\[
x^2 + y^2 = 80
\]
\[
x^2 + y^2 = k + 2x - 8y
\]
has exactly one real solution $(x, y)$.
Answer: 960
Problem: A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i = 2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
Answer: 1018
Problem: For any finite non-empty set $X$ of integers, let $\max(X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that
\[
\begin{aligned}
& \max(A) \times |B| = 12, \\
& |A| \times \max(B) = 11
\end{aligned}
\]
and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than 100, find $a + b$.
Answer: 43
Problem: In the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find $36A + 6B + C + 6D$.

\[
\text{[asy]}
\text{size(3cm);}
\text{real[]} A = \{0,8,13\};
\text{real[]} B = \{0,7,12\};

\text{for (int i = 0; i < 3; ++i) \{}
    \text{draw((A[i],0)--(A[i],-B[2]));}
    \text{draw((0,-B[i])--(A[2],-B[i]));}
\}

\text{label("8", (4,0), N);}
\text{label("5", (10.5,0),N);}
\text{label("7", (0,-3.5),W);}
\text{label("5", (0,-9.5),W);}

\text{label("$A$", (4,-3.5));}
\text{label("$B$", (10.5,-3.5));}
\text{label("$C$", (10.5,-9.5));}
\text{label("$D$", (4,-9.5));}
\text{[/asy]}
\]
Answer: 2491
Problem: Let $S_P$ be the set of all polynomials $P$ with complex coefficients, such that $P(x^2) = P(x)P(x-1)$ for all complex numbers $x$. Suppose $P_0$ is the polynomial in $S_P$ of maximal degree such that $P_0(1) \mid 2016$. Find $P_0(10)$.
Answer: 12321
Problem: An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as $U$, $W$, $XW$, $ZZ$, $V$, $Y$, $ZV$, $ZW$, $ZY$, and $X$. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as $UZWX$. Compute this number (expressed in base ten, with the usual, human digits).
Answer: 5390
Problem: At the end of a soccer tournament, each pair of teams played exactly once, with no draws. It was observed that for any three teams $A$, $B$, and $C$, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Each team calculated the positive difference between the number of games it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.
Answer: 100
Problem: Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$, the following inequality holds:
\[ \frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2} \geq C. \]
Determine when equality occurs.
Answer: 18
Problem: Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that \( \frac{||1|| + ||2|| + \cdots + ||n||}{n} = 100 \).
Answer: 89800
Problem: Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2 a + 4 \tan^2 b$.
Answer: 35
Problem: Astrophysicists have discovered a minor planet with a radius of $30$ kilometers, whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water, causing an increase of $1$ centimeter to the height of the water on the planet. What is the radius of the meteor in meters?
Answer: 300
Problem: Calculate the number of ways to choose $4$ numbers from the set $\{1, 2, \ldots, 11\}$ such that at least $2$ of the numbers are consecutive.
Answer: 260
Problem: Given the set $S = \{ xy(x + y) \mid x, y \in \mathbb{N} \}$. Let $a$ and $n$ be natural numbers such that $a + 2^k \in S$ for all $k = 1, 2, 3, \ldots, n$. Find the greatest value of $n$.
Answer: 3
Problem: A car has a 4-digit integer price, which is displayed in digital numbers (like those on a digital watch). While the salesman isn't watching, the buyer turns the price upside down and manages to purchase the car for $1626$ less. What was the original price of the car?
Answer: 6815
Problem: For $k=1, 2, \ldots, 2011$, we denote $S_k = \frac{1}{k} + \frac{1}{k+1} + \cdots + \frac{1}{2011}$. Compute the sum $S_1 + S_1^2 + S_2^2 + \cdots + S_{2011}^2$.
Answer: 4022
Problem: How many ordered pairs $(a, b)$ of positive integers are there such that \[\gcd(a, b)^3 = \mathrm{lcm}(a, b)^2 = 4^6\] is true?
Answer: 2
Problem: In a school tennis tournament with $m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants $m$ for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
Answer: 6
Problem: Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point. Note that his first roll does not gain him a point. Find the expected number of points that Albert receives.
Answer: 5
Problem: Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$.
Answer: 36
Problem: Some vertices of a regular $2024$-gon are marked such that for any regular polygon, all of whose vertices are vertices of the $2024$-gon, at least one of its vertices is marked. Find the minimal possible number of marked vertices.
Answer: 506
Problem: The distinct positive integers $a$ and $b$ have the property that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are all positive integers. Find the smallest possible value of $|a-b|$.
Answer: 30
Problem: Let $a$, $b$, $c$, $d$ be complex numbers satisfying the following equations:

\[
\begin{align*}
  5 &= a + b + c + d \\
  125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
  1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
  25 &= a^4 + b^4 + c^4 + d^4
\end{align*}
\]

Compute $abcd$.
Answer: 70
Problem: Let $f$ be a non-constant polynomial such that 
\[ f(x-1) + f(x) + f(x+1) = \frac{f(x)^2}{2013x} \] 
for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
Answer: 6039
Problem: What is the units (i.e., rightmost) digit of \[ \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ? \]
Answer: 3
Problem: A number is between $500$ and $1000$ and satisfies the following conditions: it leaves a remainder of $6$ when divided by $25$, and a remainder of $7$ when divided by $9$. Find the only odd number that meets these requirements.
Answer: 781
Problem: 25 little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colors of the rainbow to each donkey, so that successive donkeys receive balloons of different colors, and so that at least one balloon of each color is given to some donkey. Eeyore wants to give to each of the 24 remaining donkeys a pot of one of six colors of the rainbow (except red), so that at least one pot of each color is given to some donkey (but successive donkeys can receive pots of the same color). Which of the two friends has more ways to get his plan implemented, and how many times more?
Answer: 7
Problem: A circle $C_0$ is inscribed in an equilateral triangle $XYZ$ with a side length of 112. For each positive integer $n$, circle $C_n$ is inscribed in the region bounded by $XY$, $XZ$, and an arc of circle $C_{n-1}$. This forms an infinite sequence of circles that are tangent to sides $XY$ and $XZ$ and approach vertex $X$. If these circles collectively have an area of $m\pi$, find $m$.
Answer: 1176
Problem: In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively, such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$, and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.
Answer: 2
Problem: There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 1009
Problem: 162 pluses and 144 minuses are placed in a $30 \times 30$ table such that each row and each column contains at most 17 signs. (No cell contains more than one sign.) For every plus, count the number of minuses in its row, and for every minus, count the number of pluses in its column. Find the maximum of the sum of these numbers.
Answer: 2592
Problem: Find the sum of all positive integers $1 \leq k \leq 99$ such that there exist positive integers $a$ and $b$ with the property that 
\[ x^{100} - ax^k + b = (x^2 - 2x + 1)P(x) \] 
for some polynomial $P(x)$ with integer coefficients.
Answer: 117
Problem: Let $a, b, c$, and $d$ be real numbers such that:
\[ a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018. \]
Evaluate the expression:
\[ 3b + 8c + 24d + 37a. \]
Answer: 1215
Problem: Find the smallest positive integer $n$ which cannot be expressed as \( n = \frac{2^a - 2^b}{2^c - 2^d} \) for some positive integers \( a, b, c, \) and \( d \).
Answer: 11
Problem: Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
Answer: 20
Problem: Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?
Answer: 509
Problem: Find the smallest positive real $t$ such that the system of equations
\[\begin{align*}
 x_1 + x_3 &= 2t x_2, \\
 x_2 + x_4 &= 2t x_3, \\
 x_3 + x_5 &= 2t x_4 \\
\end{align*}\]
has a solution $x_1, x_2, x_3, x_4, x_5$ in non-negative reals, not all zero.
Answer: 1
Problem: In a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?
Answer: 1010
Problem: Find the number of sets $A$ that satisfy the following conditions:

1. $A$ is a set of two positive integers.
2. Each number in $A$ is at least $22\%$ the size of the other number.
3. $A$ contains the number $30$.
Answer: 129
Problem: Compute the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.
Answer: 15
Problem: Quadrilateral $ABCD$ has integer side lengths, and angles $\angle ABC$, $\angle ACD$, and $\angle BAD$ are right angles. Compute the smallest possible value of $AD$.
Answer: 25
Problem: Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or the number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "very prime" number with justification.
Answer: 373
Problem: Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.
Answer: 22
Problem: Suppose $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.
Answer: 274
Problem: Compute the number of permutations of \(\{1,2,3\}\) with the property that there is some number that can be removed such that the remaining numbers are in increasing order. For example, \((2,1,3)\) has this property because removing \(1\) leaves \((2,3)\), which is in increasing order.
Answer: 5
Problem: Let $ABC$ be a triangle with $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m + \sqrt{n}$ for positive integers $m$ and $n$. What is $100m + n$?
Answer: 460
Problem: Let $a_1,a_2,\ldots, a_{2013}$ be a permutation of the numbers from $1$ to $2013$. Let $A_n = \frac{a_1 + a_2 + \cdots + a_n}{n}$ for $n = 1,2,\ldots, 2013$. If the smallest possible difference between the largest and smallest values of $A_1,A_2,\ldots, A_{2013}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Answer: 3
Problem: Find the greatest natural number $n$ such that there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural numbers $a_{1} < a_{2} < \ldots < a_{n-1}$ satisfying the following equations for $i = 1, 2, \ldots, n-1$:  
\[x_{1}x_{2}\ldots x_{n} = 1980 \quad \text{and} \quad x_{i} + \frac{1980}{x_{i}} = a_{i}.\]
Answer: 6
Problem: Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.
Answer: 186
Problem: Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
Answer: 729
Problem: What are the last two digits of the number $(11^2 + 15^2 + 19^2 + \ldots + 2007^2)^2$?
Answer: 0
Problem: Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\frac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.
Answer: 26
Problem: Let $f(x)$ be the function such that $f(x) > 0$ for $x \geq 0$ and $\{f(x)\}^{2006} = \int_{0}^{x} f(t) \, dt + 1.$

Find the value of $\{f(2006)\}^{2005}.$
Answer: 2006
Problem: Compute the number of ordered triples of integers $(a, b, c)$ between $1$ and $12$, inclusive, such that, if \[ q = a + \frac{1}{b} - \frac{1}{b + \frac{1}{c}}, \] then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
Answer: 132
Problem: A circle is divided into $n$ equal parts. Marceline sets out to assign whole numbers from $1$ to $n$ to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers $887$, $217$, and $1556$ occupy consecutive positions. How many parts was the circumference divided into?
Answer: 2009
Problem: Determine the minimum value of the sum of several distinct positive integers whose product is divisible by $2006^2$.
Answer: 228
Problem: Let $(p_1, p_2, \dots) = (2, 3, \dots)$ be the list of all prime numbers, and $(c_1, c_2, \dots) = (4, 6, \dots)$ be the list of all composite numbers, both in increasing order. Compute the sum of all positive integers $n$ such that $|p_n - c_n| < 3$.
Answer: 16
Problem: Compute $\log_2 6 \cdot \log_3 72 - \log_2 9 - \log_3 8$.
Answer: 5
Problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying the relation $4f(3-x) + 3f(x) = x^2$ for any real $x$. Find the value of $f(27) - f(25)$ to the nearest integer. (Here $\mathbb{R}$ denotes the set of real numbers.)
Answer: 8
Problem: Find the sum of all real solutions to $(x^2 - 10x - 12)^{x^2+5x+2} = 1$.
Answer: 15
Problem: The vertices of hexagon $ABCDEF$ lie on a circle. The sides of the hexagon are given as follows: $AB = CD = EF = 6$ and $BC = DE = FA = 10$. The area of the hexagon is $m\sqrt{3}$. Find the value of $m$.
Answer: 94
Problem: Wu starts out with exactly one coin. Wu flips every coin he has at once after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$.
Answer: 71622400
Problem: $x$, $y$, and $z$ are positive real numbers that satisfy $x^3 + 2y^3 + 6z^3 = 1$. Let $k$ be the maximum possible value of $2x + y + 3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n + n$.
Answer: 35
Problem: Let $S$ be the set of all pairs of positive integers $(x, y)$ for which $2x^2 + 5y^2 \le 5 + 6xy$. Compute $\displaystyle\sum_{(x,y) \in S} (x+y+100)$.
Answer: 839
Problem: Consider a $2 \times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
\[
\begin{bmatrix} 
1 & 2 & 0 \\
2 & 1 & 0 
\end{bmatrix}
\]
Answer: 9
Problem: If $2011^{2011^{2012}} = x^x$ for some positive integer $x$, how many positive integer factors does $x$ have?
Answer: 2012
Problem: Find the number of 6-tuples $(x_1, x_2, \ldots, x_6)$, where $x_i = 0, 1, \text{ or } 2$ and $x_1 + x_2 + \cdots + x_6$ is even.
Answer: 365
Problem: The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$  

Alice chooses two numbers $a,b,$ erases them, and writes the number $\frac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.

If the sum of all possible numbers she can end up with is $\frac{p}{q}$ where $p, q$ are coprime, then what is the value of $\log_{2}(p+q)$?
Answer: 100
Problem: By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?
Answer: 21600
Problem: Let $V$ be a rectangular prism with integer side lengths. The largest face has an area of $240$ and the smallest face has an area of $48$. A third face has an area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?
Answer: 260
Problem: Aida made three cubes with positive integer side lengths $a$, $b$, and $c$. She decided to dismantle these cubes into unit cubes and tried to construct a larger cube with side length $a+b+c$. Unfortunately, she was 648 blocks short. How many possible ordered triples $(a, b, c)$ exist?
Answer: 7
Problem: I have 6 friends and during a vacation, I met them during several dinners. I found that:

- I dined with all 6 friends exactly on 1 day.
- I dined with every 5 of them on 2 days.
- I dined with every 4 of them on 3 days.
- I dined with every 3 of them on 4 days.
- I dined with every 2 of them on 5 days.

Additionally, every friend was present at 7 dinners and absent at 7 dinners. How many dinners did I have alone?
Answer: 1
Problem: The product $20! \cdot 21! \cdot 22! \cdot \ldots \cdot 28!$ can be expressed in the form $m \cdot n^3$, where $m$ and $n$ are positive integers, and $m$ is not divisible by the cube of any prime. Find $m$.
Answer: 825
Problem: How many ways are there to color the vertices of a triangle using the colors red, green, blue, or yellow such that no two vertices have the same color? Note that rotations and reflections are considered distinct.
Answer: 24
Problem: Compute the area of the trapezoid $ABCD$ with right angles $\angle BAD$ and $\angle ADC$. The side lengths are given as $AB = 3$, $BC = 5$, and $CD = 7$.
Answer: 15
Problem: Find the $2019^{th}$ strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.
Answer: 37805
Problem: For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\frac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.
Answer: 11
Problem: Let $a_0 = 1$, $a_1 = 2$, and $a_n = 4a_{n-1} - a_{n-2}$ for $n \ge 2$. Find an odd prime factor of $a_{2015}$.
Answer: 181
Problem: Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round, a winner is determined on each court. After each round:

1. The player who lost on court $1$ and the player who won on court $20$ remain in their respective positions.
2. For the remaining $38$ players:
   - The winner on court $i$ moves to court $i + 1$.
   - The loser moves to court $i - 1$.

The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?
Answer: 39
Problem: Point $Y$ lies on line segment $XZ$ such that $XY = 5$ and $YZ = 3$. Point $G$ lies on line $XZ$ such that there exists a triangle $ABC$ with centroid $G$ where $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$. Compute the largest possible value of $XG$.
Answer: 2
Problem: Let $S = \{1, 2, 3, \ldots, 12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
Answer: 31
Problem: Find the value of $a$ such that:

\[101a = 6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]
Answer: 2012
Problem: Among the coordinates $(x,y)$ where $1 \leq x,y \leq 101$, choose some points such that there do not exist $4$ points which form an isosceles trapezium with its base parallel to either the $x$ or $y$ axis (including rectangles). Find the maximum number of coordinate points that can be chosen.
Answer: 201