ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: $f$ is a function defined on integers and satisfies $f(x) + f(x+3) = x^2$ for every integer $x$. If $f(19) = 94$, then calculate $f(94)$.
Answer: 4335
Problem: Define $p(n)$ to be the product of all non-zero digits of $n$. For instance, $p(5) = 5$, $p(27) = 14$, $p(101) = 1$, and so on. Find the greatest prime divisor of the following expression:
\[
p(1) + p(2) + p(3) + \ldots + p(999).
\]
Answer: 103
Problem: After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a score of 100 and guessed the average would be 97. Each of the other 16 students guessed $30 + \frac{n}{2}$, where $n$ was that student's score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam?
Answer: 62
Problem: A function $f$ is defined on the positive integers by \[ \left\{ \begin{array}{rcl} f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array} \right. \] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.
Answer: 92
Problem: Consider a function $f:\mathbb{N}\to\mathbb{N}$ defined by the following rules: 

- $f(1) = 1$
- $f(2n) = f(n)$
- $f(2n+1) = f(2n) + 1$

Determine the number of integer values $n$ for which the maximum value of $f(n)$ is achieved, where $0 < n < 2014$.
Answer: 2
Problem: Let $x, y$ be two positive integers, with $x > y$, such that $2n = x + y$, where $n$ is a two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$.
Answer: 66
Problem: Find the largest possible value of $a + b$ that is less than or equal to $2007$, where $a$ and $b$ are relatively prime, and there exists a positive integer $n$ such that:
\[
\frac{2^3 - 1}{2^3 + 1} \cdot \frac{3^3 - 1}{3^3 + 1} \cdot \frac{4^3 - 1}{4^3 + 1} \cdots \frac{n^3 - 1}{n^3 + 1} = \frac{a}{b}.
\]
Answer: 1891
Problem: Let $101$ marbles be numbered from $1$ to $101$. The marbles are divided over two baskets $A$ and $B$. The marble numbered $40$ is in basket $A$. When this marble is removed from basket $A$ and put in $B$, the averages of the numbers in $A$ and $B$ both increase by $\frac{1}{4}$. How many marbles were there originally in basket $A$?
Answer: 73
Problem: A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\{ 0, 1, \dots, 10 \}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\{ 1,2,\dots,10 \}$. The probability that the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 111
Problem: Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$.
Answer: 11
Problem: The ex-radii of a triangle are $10\frac{1}{2}$, $12$, and $14$. If the sides of the triangle are the roots of the cubic equation $x^3 - px^2 + qx - r = 0$, where $p$, $q$, and $r$ are integers, find the nearest integer to $\sqrt{p + q + r}$.
Answer: 58
Problem: The vertices of a regular nonagon are colored such that:
1. Adjacent vertices are different colors.
2. If three vertices form an equilateral triangle, they are all different colors.

Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
Answer: 54
Problem: Find the total length of the set of real numbers satisfying the inequality:

\[
\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.
\]
Answer: 25
Problem: Half the volume of a 12-foot high cone-shaped pile is grade A ore, while the other half is grade B ore. The pile is worth $62. 

One-third of the volume of a similarly shaped 18-foot pile is grade A ore, while the other two-thirds is grade B ore. The second pile is worth $162. 

Two-thirds of the volume of a similarly shaped 24-foot pile is grade A ore, while the other one-third is grade B ore. 

What is the value in dollars ($) of the 24-foot pile?
Answer: 608
Problem: Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers?
Answer: 101
Problem: Suppose we choose two numbers $x, y \in [0,1]$ uniformly at random. If the probability that the circle with center $(x, y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1] \times [0,1]$ is written as $\frac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?
Answer: 10
Problem: Let $T$ be a finite set of positive integers, satisfying the following conditions:
1. For any two elements of $T$, their greatest common divisor and their least common multiple are also elements of $T$.
2. For any element $x$ of $T$, there exists an element $x'$ of $T$ such that $x$ and $x'$ are relatively prime, and their least common multiple is the largest number in $T$.

For each such set $T$, denote by $s(T)$ its number of elements. It is known that $s(T) < 1990$; find the largest value $s(T)$ may take.
Answer: 1024
Problem: Let $ABC$ be a triangle with $AB = 6$, $AC = 8$, $BC = 7$. Let $H$ be the orthocenter of $\triangle ABC$. Let $D \neq H$ be a point on $\overline{AH}$ such that $\angle HBD = \frac{3}{2} \angle CAB + \frac{1}{2} \angle ABC - \frac{1}{2} \angle BCA$. Find $DH$.
Answer: 3
Problem: We will call one of the cells of a rectangle $11 \times 13$ "peculiar", if after removing it the remaining figure can be cut into squares $2 \times 2$ and $3 \times 3$. How many of the 143 cells are "peculiar"?
Answer: 4
Problem: What is the largest integer that must divide $n^5 - 5n^3 + 4n$ for all integers $n$?
Answer: 120
Problem: On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of 
\[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
Answer: 494
Problem: Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form \( SWEET \), in which the two \( E \)'s represent the same digit but otherwise the digits are distinct. (The leading digit \( S \) can't be 0.) How many such integers are divisible by 16?
Answer: 282
Problem: Determine the number of ways to assign each integer from $1$ to $11$ one of the letters $L$, $M$, or $T$ such that no two consecutive multiples of any positive integer $n$ are assigned the same letter.
Answer: 24
Problem: For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.
Answer: 504
Problem: Moor has $3$ different shirts, labeled $T$, $E$, and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?
Answer: 72
Problem: How many integers $n$ with $10 \leq n \leq 500$ have the property that the hundreds digit of $17n$ and $17n + 17$ are different?
Answer: 84
Problem: Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$:
1. $(x + 1)\star 0 = (0\star x) + 1$
2. $0\star (y + 1) = (y\star 0) + 1$
3. $(x + 1)\star (y + 1) = (x\star y) + 1$.

If $123\star 456 = 789$, find $246\star 135$.
Answer: 579
Problem: Find the largest real number $x$ such that $$\left(\frac{x}{x-1}\right)^2 + \left(\frac{x}{x+1}\right)^2 = \frac{325}{144}.$$
Answer: 5
Problem: Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.
Answer: 600
Problem: There are 7 different numbers on the board, their sum is $10$. For each number on the board, Petya wrote the product of this number and the sum of the remaining 6 numbers in his notebook. It turns out that the notebook only has 4 distinct numbers in it. Determine one of the numbers that is written on the board.
Answer: -20
Problem: A triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle?
Answer: 76
Problem: A dot is marked at each vertex of a triangle $ABC$. Then, 2, 3, and 7 more dots are marked on the sides $AB$, $BC$, and $CA$, respectively. How many triangles have their vertices at these dots?
Answer: 357
Problem: How many real numbers $x$ are solutions to the following equation?
\[ |x-1| = |x-2| + |x-3| \]
Answer: 2
Problem: In triangle $ABC$, let $O$ be the circumcenter and $G$ be the centroid. The line perpendicular to $OG$ at $O$ intersects $BC$ at $M$ such that $M$, $G$, and $A$ are collinear, and $OM = 3$. Compute the area of $\triangle ABC$, given that $OG = 1$.
Answer: 12
Problem: The natural numbers from $1$ up to $300$ are evenly located around a circle. We say that such an ordering is "alternate" if each number is either less than its two neighbors or greater than its two neighbors. We will call a pair of neighboring numbers a "good" pair if, by removing that pair from the circumference, the remaining numbers form an alternate ordering. Determine the least possible number of good pairs in which there can be an alternate ordering of the numbers from $1$ to $300$ inclusive.
Answer: 150
Problem: \( \triangle PQR \) is isosceles and right-angled at \( R \). Point \( A \) is inside \( \triangle PQR \), such that \( PA = 11 \), \( QA = 7 \), and \( RA = 6 \). Legs \( \overline{PR} \) and \( \overline{QR} \) have length \( s = \sqrt{a+b\sqrt{2}} \), where \( a \) and \( b \) are positive integers. What is \( a+b \)?
Answer: 127
Problem: Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA = 90^{\circ}$. Given that $AI = 97$ and $BC = 144$, compute the area of $\triangle ABC$.
Answer: 14040
Problem: Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \).

Note: The function \( d(i) \) outputs the number of divisors of \( i \), \( \sigma (i) \) outputs the sum of the factors of \( i \), and \( \phi (i) \) outputs the number of positive integers less than or equal to \( i \) that are relatively prime to \( i \).
Answer: 350
Problem: Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties, or dining hall food quality. 

- If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$ hours per week studying. 
- If the university invests $b$ million dollars in better food, the students will spend an additional $3b$ hours per week studying. 
- Finally, if the $c$ million dollars are invested in parties, students will be more relaxed and spend $11c - c^2$ more hours per week studying. 

The university wants to invest its $5$ million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study?
Answer: 34
Problem: Michael is celebrating his fifteenth birthday today. Calculate the total number of Sundays that have occurred in his lifetime.
Answer: 783
Problem: Find the second smallest positive integer $n$ such that:
1. When $n$ is divided by $5$, the remainder is $3$.
2. When $n$ is divided by $7$, the remainder is $4$.
Answer: 53
Problem: There exist primes $p$ and $q$ such that \[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633 \times 134217730 \times 2^{2202} + 1. \]
Find the remainder when $p+q$ is divided by $1000$.
Answer: 358
Problem: Two fair six-sided dice are rolled. The probability that the positive difference between the two rolls is at least $4$ can be written in simplest form as $\frac{m}{n}$. Compute $m + n$.
Answer: 7
Problem: How many nonempty subsets of \( \{1, 2, \dots, 15\} \) are there such that the sum of the squares of each subset is a multiple of \( 5 \)?
Answer: 7583
Problem: Given that $a, b \in \mathbb{R}^+$ and the equations $a^{\log b} = 2$ and $a^{\log a} b^{\log b} = 5$, find the value of $(ab)^{\log(ab)}$.
Answer: 20
Problem: Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.
Answer: 240
Problem: A positive integer is called \textit{superprime} if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers.
Answer: 3
Problem: Find the sum of the four smallest prime divisors of $2016^{239} - 1$.
Answer: 528
Problem: On Monday, a school library was attended by 5 students; on Tuesday, by 6; on Wednesday, by 4; on Thursday, by 8; and on Friday, by 7. None of them attended the library two days in a row. What is the least possible number of students who visited the library during the week?
Answer: 15
Problem: Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first prime, the remainder is $1$. When she divides $N$ by the second prime, the remainder is $2$. When she divides $N$ by the third prime, the remainder is $3$. When she divides $N$ by the fourth prime, the remainder is $4$. Find the least possible value for $N$.
Answer: 53
Problem: Find the smallest positive integer $k$ such that there exist positive integers $M, O > 1$ satisfying:
\[
 (O \cdot M \cdot O)^k = (O \cdot M) \cdot \underbrace{(N \cdot O \cdot M) \cdot (N \cdot O \cdot M) \cdot \ldots \cdot (N \cdot O \cdot M)}_{2016 \text{\ (N \cdot O \cdot M)s}},
\]
where $N = O^M$.
Answer: 2823
Problem: How many ordered pairs of positive integers $(m, n)$ satisfy the system

\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2,\\
\text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
Answer: 2
Problem: The phrase "COLORFUL TARTAN" is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different colors such that neither bag contains more than one of the same letter?
Answer: 16
Problem: How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Answer: 600
Problem: Let $n$ be the 200th smallest positive real solution to the equation $x - \frac{\pi}{2} = \tan x$. Find the greatest integer that does not exceed \(\frac{n}{2}\).
Answer: 314
Problem: In a forest, there are five trees labeled $A$, $B$, $C$, $D$, and $E$ that are aligned in that order on a straight line. 

- At the midpoint of segment $AB$, there is a daisy.
- At the midpoint of segment $BC$, there is a rose bush.
- At the midpoint of segment $CD$, there is a jasmine.
- At the midpoint of segment $DE$, there is a carnation.

The distance between tree $A$ and tree $E$ is $28$ meters. Additionally, the distance between the daisy and the carnation is $20$ meters. Calculate the distance between the rose bush and the jasmine.
Answer: 6
Problem: A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in $3$ seconds. Each adjacent golden coin reduces this time by $1$ second. What is the minimal time required for the wizard to convert all coins to gold?
Answer: 201
Problem: The cells of an $8 \times 8$ table are colored either black or white such that each row has a distinct number of black squares, and each column also has a distinct number of black squares. Determine the maximum number of pairs of adjacent cells that have different colors.
Answer: 62
Problem: Suppose $a, b$ are integers and $a+b$ is a root of $x^2 + ax + b = 0$. What is the maximum possible value of $b^2$?
Answer: 81
Problem: Let $a_1 = 2021$ and for $n \geq 1$, let $a_{n+1} = \sqrt{4 + a_n}$. Then $a_5$ can be written as \[ \sqrt{\frac{m+\sqrt{n}}{2}} + \sqrt{\frac{m-\sqrt{n}}{2}} \] where $m$ and $n$ are positive integers. Find $10m + n$.
Answer: 45
Problem: If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$, \[ \sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1, \] find the smallest $n$ such that $a_n < \frac{1}{2018}$.
Answer: 4036
Problem: Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[ \sum_{i=1}^7 n_i^6 = 96957. \]
Answer: 2688
Problem: 13 fractions are formed using each of the numbers $1, 2, \ldots, 26$ exactly once. For example: $\frac{12}{5}, \frac{18}{26}, \ldots$. What is the maximum number of these fractions that can be integers?
Answer: 12
Problem: Evaluate \( \sum_{k=1}^{2007}(-1)^{k}k^{2} \)
Answer: -2015028
Problem: The diameter $[AC]$ of a circle is divided into four equal segments by points $P$, $M$, and $Q$. Consider a segment $[BD]$ that passes through $P$ and intersects the circle at points $B$ and $D$, such that $PD = \frac{3}{2} AP$. Given that the area of triangle $[ABP]$ is $1$ cm$^2$, calculate the area of quadrilateral $[ABCD]$.
Answer: 7
Problem: For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
Answer: 1
Problem: Let $ABCD$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $AB$ at point $E$. Then $DE = \frac{m}{n}$, where $m$, $n$ are positive integers and gcd $(m, n) = 1$. Find $100m + n$.
Answer: 807
Problem: A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$.
Answer: 8
Problem: Let \( \text{Revolution}(x) = x^3 + Ux^2 + Sx + A \), where \( U \), \( S \), and \( A \) are all integers and \( U + S + A + 1 = 1773 \). Given that \( \text{Revolution} \) has exactly two distinct nonzero integer roots \( G \) and \( B \), find the minimum value of \( |GB| \).
Answer: 392
Problem: We have two positive integers, both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference between the two integers.
Answer: 120
Problem: How many letters in the word \text{UNCOPYRIGHTABLE} have at least one line of symmetry?
Answer: 9
Problem: Twelve $1$'s and ten $-1$'s are written on a chalkboard. You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard. What sum do you get?
Answer: -42
Problem: Compute the sum of all positive integers $n < 200$ such that $\text{gcd}(n, k) \neq 1$ for every $k \in \{2 \cdot 11 \cdot 19, 3 \cdot 13 \cdot 17, 5 \cdot 11 \cdot 13, 7 \cdot 17 \cdot 19\}$.
Answer: 539
Problem: Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). Determine the number of possible DNA sequences on Rthea.
Answer: 28812
Problem: Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of 2.
Answer: 15
Problem: Ten test papers are to be prepared for the National Olympiad. Each paper contains 4 problems, and no two papers have more than one problem in common. What is the minimum number of problems needed?
Answer: 13
Problem: A four-digit number $n$ is said to be "literally 1434" if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is "literally 1434" because $1 \mod 5$ is $1$, $9 \mod 5$ is $4$, $8 \mod 5$ is $3$, and $4 \mod 5$ is $4$. Find the sum of all four-digit positive integers that are "literally 1434".
Answer: 67384
Problem: Two knights placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of an $8 \times 8$ chessboard such that they do NOT attack each other?
Answer: 1848
Problem: Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15 \text{ and } \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.
Answer: 286
Problem: Let $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x$, where $\lfloor a \rfloor$ is the greatest integer not exceeding $a$.
Answer: 652400
Problem: In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point on the $CD$ side such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.
Answer: 8
Problem: We take $100$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{100}$. Determine the last two digits of the number $a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}$.
Answer: 30
Problem: Mrs. Toad has a class of $2017$ students, with unhappiness levels $1, 2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class into exactly $15$ groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all $15$ groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into $15$ groups?
Answer: 1121
Problem: Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28$ degrees. What is the smallest possible value of $n$?
Answer: 45
Problem: For positive integers $n$, let the numbers $c(n)$ be determined by the rules:  
- $c(1) = 1$  
- $c(2n) = c(n)$  
- $c(2n+1) = (-1)^n c(n)$  

Find the value of \( \sum_{n=1}^{2013} c(n) c(n+2) \).
Answer: -1
Problem: The graph of \( (x^2 + y^2 - 1)^3 = x^2 y^3 \) is a heart-shaped curve. 

For how many ordered pairs of integers \((x, y)\) is the point \((x, y)\) inside or on this curve?
Answer: 7
Problem: Compute the sum: $$\sum_{i=0}^{101} \frac{x_i^3}{1-3x_i+3x_i^2}$$ for $$x_i=\frac{i}{101}$$.
Answer: 51
Problem: The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4$. What is the difference between the cubes of the smaller two numbers?
Answer: 320
Problem: The positive integers $x_1, x_2, \ldots, x_7$ satisfy $x_6 = 144$ and $x_{n+3} = x_{n+2}(x_{n+1} + x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.
Answer: 3456
Problem: Find the sum of all integers $N > 1$ such that each prime factor of $N$ is either $2$, $3$, or $5$, and $N$ is not divisible by any perfect cube greater than $1$.
Answer: 2820