ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Consider real numbers $A$, $B$, \dots, $Z$ such that:
\[\begin{align*}
  EVIL &= \frac{5}{31}, \\
  LOVE &= \frac{6}{29}, \\
  IMO  &= \frac{7}{3}.
\end{align*}\]
If $OMO = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
Answer: 579
Problem: Find the smallest positive multiple of $77$ such that the last four digits of the number are $2020$.
Answer: 482020
Problem: The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
Answer: 256
Problem: On a $5 \times 5$ grid, cover the area with several S-Tetrominos (also known as Z-Tetrominos) such that in every square, there are two or fewer tiles covering it (tiles can overlap). Determine the maximum possible number of squares that can be covered by at least one tile.
Answer: 24
Problem: $P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1) = 5$ and $P(P(1)) = 177$, what is the sum of all possible values of $P(10)$?
Answer: 1202
Problem: The sequence of letters **TAGC** is written in succession 55 times on a strip, as shown below. The strip is to be cut into segments between letters, leaving strings of letters on each segment, which we call words. For example, a cut after the first G, after the second T, and after the second C would yield the words **TAG**, **CT**, and **AGC**. At most, how many distinct words could be found if the entire strip were cut? Justify your answer.

\[ \boxed{\textbf{T A G C T A G C T A G}} \ldots \boxed{\textbf{C T A G C}} \]
Answer: 16
Problem: Let $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$. Then there are rational numbers $A$ and $B$ such that:

$$\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}.$$ 

Find $A + B$.
Answer: 34
Problem: Nathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \times 1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w \times h$ (where $w$ is the number of columns and $h$ is the number of rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20 \times 20$?
Answer: 5339
Problem: The cross-section of a prism with an index of refraction $1.5$ is an equilateral triangle, as shown in the diagram. A ray of light enters horizontally from the air into the prism and exits at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number.
Answer: 13
Problem: For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 11
Problem: There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?
Answer: 510
Problem: Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.
Answer: 86
Problem: Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.
Answer: 904
Problem: Let $\varphi$ be the positive solution to the equation $x^2 = x + 1$. For $n \ge 0$, let $a_n$ be the unique integer such that $\varphi^n - a_n \varphi$ is also an integer. Compute $\sum_{n=0}^{10} a_n$.
Answer: 143
Problem: Compute the sum \[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \] where every three consecutive additions (\(+\)) are followed by two subtractions (\(-\)).
Answer: 401799
Problem: Triangle $ABC$ has side lengths $AB=75$, $BC=100$, and $CA=125$. Point $D$ is the foot of the altitude from $B$, and $E$ lies on segment $BC$ such that $DE \perp BC$. Find the area of triangle $BDE$.
Answer: 864
Problem: Determine the largest natural number such that all its decimal digits are different and it is divisible by each of its digits.
Answer: 9867312
Problem: In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
Answer: 2014
Problem: Suppose $x$ and $y$ are nonzero real numbers that simultaneously satisfy the equations:

\[ x + \frac{2018}{y} = 1000 \]

and

\[ \frac{9}{x} + y = 1 \]

Find the maximum possible value of $x + 1000y$.
Answer: 1991
Problem: An equilateral triangle with side length $20$ is divided by a series of parallel lines into $400$ equilateral triangles, each with side length $1$. What is the maximum number of these small triangles that can be crossed (internally) by one line?
Answer: 39
Problem: Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?
Answer: 200
Problem: Determine the number of 3-digit numbers in base 10 having at least one 5 and at most one 3.
Answer: 249
Problem: If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression:
\[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]
Answer: 6
Problem: On the party, every boy gave $1$ candy to every girl, and every girl gave $1$ candy to every boy. Then every boy ate $2$ candies, and every girl ate $3$ candies. It is known that $\frac{1}{4}$ of all candies were eaten. Find the greatest possible number of children at the party.
Answer: 35
Problem: An *access pattern* $\pi$ is a permutation of $\{1,2,\dots,50\}$ describing the order in which some 50 memory addresses are accessed. We define the *locality* of $\pi$ to be how much the program jumps around the memory, or numerically, 
\[ \sum_{i=2}^{50}\left\lvert\pi(i)-\pi(i-1)\right\rvert. \]
If $\pi$ is a uniformly randomly chosen access pattern, what is the expected value of its locality?
Answer: 833
Problem: Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are considered the same if one can be rotated or flipped to become the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?
Answer: 198
Problem: For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geq f(j)$ for $i > j$ and $\sum_{i=1}^{n} (i + f(i)) = 2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is minimized. How many such functions exist in that case?
Answer: 15
Problem: Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
Answer: 6
Problem: For the NEMO, Kevin needs to compute the product:
\[ 9 \times 99 \times 999 \times \cdots \times 999999999. \]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
Answer: 870
Problem: In a tournament bracket, a player must be defeated three times to be eliminated. If there are 512 contestants entering the tournament, what is the greatest number of games that could be played?
Answer: 1535
Problem: Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \leq p < q < r \leq 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
Answer: 7
Problem: How many ordered pairs of integers \((x, y)\) satisfy the equation:
\[
8(x^3 + x^2y + xy^2 + y^3) = 15(x^2 + y^2 + xy + 1)
\]
Answer: 2
Problem: Call a positive integer \textit{prime-simple} if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple?
Answer: 6
Problem: How many ways are there to rearrange the letters of \( CCARAVEN \)?
Answer: 10080
Problem: Find the number of integers $a$ with $1 \le a \le 2012$ for which there exist nonnegative integers $x, y, z$ satisfying the equation:
\[ x^2(x^2 + 2z) - y^2(y^2 + 2z) = a. \]
Answer: 1256
Problem: Sixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints:

- An arc between a red and a green point is assigned the number $1$.
- An arc between a red and a blue point is assigned the number $2$.
- An arc between a blue and a green point is assigned the number $3$.
- An arc between two points of the same color is assigned the number $0$.

What is the greatest possible sum of all the numbers assigned to the arcs?
Answer: 100
Problem: Find the largest natural number $n$ such that $2^n + 2^{11} + 2^8$ is a perfect square.
Answer: 12
Problem: On the evening, more than $\frac{1}{3}$ of the students of a school are going to the cinema. On the same evening, more than $\frac{3}{10}$ are going to the theatre, and more than $\frac{4}{11}$ are going to the concert. At least how many students are there in this school?
Answer: 173
Problem: Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB$, $\angle BFC$, $\angle CGD$, $\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
Answer: 98
Problem: Find the smallest positive multiple of 999 that does not contain the digit 9.
Answer: 111888
Problem: Steven the troll submits problems using silly names instead of his own. On day $1$, he gives no name at all. Every subsequent day, he alternates between adding $2$ words and $4$ words to his name. For example, on day $4$, he submits an $8\text{-word}$ name. On day $n$, he submits a $44\text{-word}$ name: "Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice". Compute $n$.
Answer: 16
Problem: Consider a random permutation of the set $\{1, 2, \ldots, 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, \ldots, 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j)$ with $i - j > 155$ and $a_i - a_j > 266$?
Answer: 651222
Problem: Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside the grid, in how many ways can the squares be chosen so that the final $1 \times 1$ square is the center of the original $5 \times 5$ grid?
Answer: 36
Problem: Suppose that positive integers $m, n, k$ satisfy the equations $$m^2 + 1 = 2n^2,$$ $$2m^2 + 1 = 11k^2.$$ Find the residue when $n$ is divided by $17$.
Answer: 5
Problem: Find a $7$-digit integer divisible by $128$, whose decimal representation contains only the digits $2$ and $3$.
Answer: 2223232
Problem: Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941$. What is the largest possible number whose digits Terry could have swapped in the summation?
Answer: 59
Problem: Let \[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left[ \prod_{n=1}^{2011} a_n \right]. \]

Find the remainder when \( N \) is divided by 1000.
Answer: 95
Problem: A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?
Answer: 350
Problem: Identical spherical tennis balls of radius $1$ are placed inside a cylindrical container of radius $2$ and height $19$. Compute the maximum number of tennis balls that can fit entirely inside this container.
Answer: 9
Problem: Around a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their difference is $1$ or $2$. We will say that a number is *huge* if it is greater than its two neighbors, and that it is *tiny* if it is less than its two neighbors. The sum of all the huge numbers is equal to the sum of all the tiny numbers plus $1810$. Determine how many odd numbers there can be around the circumference.
Answer: 1005
Problem: Let $S$ be the set of all ordered pairs $(x, y)$ of integer solutions to the equation:
\[ 6x^2 + y^2 + 6x = 3xy + 6y + x^2y. \]
$S$ contains a unique ordered pair $(a, b)$ with a maximal value of $b$. Compute $a + b$.
Answer: 34
Problem: There are several pairs of integers \((a, b)\) satisfying the equation:
\[ a^2 - 4a + b^2 - 8b = 30 \]
Find the sum of the sum of the coordinates of all such points.
Answer: 72
Problem: On a straight line lie $100$ points and another point outside the line. What is the maximum number of isosceles triangles that can be formed using these $101$ points as vertices?
Answer: 50
Problem: Alan is assigning values to lattice points on the 3D coordinate plane. First, Alan computes the roots of the cubic $20x^3 - 22x^2 + 2x + 1$ and finds that they are $\alpha$, $\beta$, and $\gamma$. He finds out that each of these roots satisfies $|\alpha|, |\beta|, |\gamma| \leq 1$. On each point $(x, y, z)$ where $x, y,$ and $z$ are all nonnegative integers, Alan writes down $\alpha^x \beta^y \gamma^z$. What is the value of the sum of all numbers he writes down?
Answer: 20
Problem: The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known that there are at most $7$ prophetic words of length $3$. Find the maximum number of prophetic words of length $10$.
Answer: 504
Problem: Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n = 1234567891011\ldots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? In other words, compute $1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 1 + 0 - 1 + \ldots$.
Answer: -1008
Problem: $A_1, A_2, \ldots, A_n$ are subsets of a set $S$ with $|S| = 2019$ such that the union of any three of them equals $S$, but the union of any two does not equal $S$. Find the maximum value of $n$.
Answer: 64
Problem: In the future, each country in the world produces its Olympic athletes via cloning and strict training programs. Therefore, in the finals of the 200 m free, there are two indistinguishable athletes from each of the four countries. How many ways are there to arrange them into eight lanes?
Answer: 2520
Problem: Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$), then inverts each, and sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
Answer: 3
Problem: Edward starts at his house located at \((0,0)\) and needs to reach the point \((6,4)\), which is the coordinate for his school. However, there is a park shaped as a square with coordinates \((2,1), (2,3), (4,1), (4,3)\). Edward cannot walk inside the park, but there is a road along the perimeter of the square. Determine how many different shortest routes exist from Edward's house to his school.
Answer: 110
Problem: Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the expression $(bcd + acd + abd + abc)^7$.
Answer: 630
Problem: How many positive three-digit integers $abc$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $bc$ corresponds to the day in that month, or $ab$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.
Answer: 273
Problem: The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?
Answer: 630
Problem: Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.
Answer: 190
Problem: Let $A$ be the set of the first $16$ positive integers. Find the smallest positive integer $k$ such that in every $k$-subset of $A$, there exist two distinct elements $a, b \in A$ for which $a^2 + b^2$ is a prime number.
Answer: 9
Problem: Suppose real numbers $a, b, c, d$ satisfy the following conditions:

1. $a + b + c + d = 17$
2. $ab + bc + cd + da = 46$

Determine the minimum possible value of $a^2 + b^2 + c^2 + d^2$. If this value can be expressed as a rational number $\frac{p}{q}$ in its simplest form, find the sum $p + q$.
Answer: 199
Problem: Three congruent circles of radius $2$ are drawn in the plane such that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs. If $K = \sqrt{a} - b$ for positive integers $a, b$, find $100a+b$.
Answer: 4806
Problem: In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$, compute the area of $\triangle ABF$.
Answer: 3
Problem: Find the largest integer $k$ such that:
\[
k \leq \sqrt{2} + \sqrt[3]{\frac{3}{2}} + \sqrt[4]{\frac{4}{3}} + \sqrt[5]{\frac{5}{4}} + \cdots + \sqrt[2015]{\frac{2015}{2014}}.
\]
Answer: 2014
Problem: Find the minimum positive value of $1 \times 2 \times 3 \times 4 \times \ldots \times 2020 \times 2021 \times 2022$ where you can replace $\times$ with $+$ or $-$.
Answer: 1
Problem: A fair $20$-sided die has faces numbered $1$ through $20$. The die is rolled three times and the outcomes are recorded. If $a$ and $b$ are relatively prime integers such that $\frac{a}{b}$ is the probability that the three recorded outcomes can be the sides of a triangle with positive area, find $a + b$.
Answer: 1201
Problem: A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
Answer: 80
Problem: A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes.
Answer: 258
Problem: Given an isosceles triangle $\triangle ABC$ where $AB = BC$. A point $K$ is inside $\triangle ABC$ such that $CK = AB = BC$ and $\angle KAC = 30^\circ$. Find $\angle AKB$.
Answer: 150
Problem: The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8, 15, 8, 8, 8, 6, 8,$ and $29$ as shown. All of its angles are right angles. Transform this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$.
Answer: 80
Problem: Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is an exact square.
Answer: 1972
Problem: Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$, such as $\{3, 31\}$.
Answer: 9
Problem: Let $M=\{0,1,2,\dots,2022\}$ and let $f:M\times M\to M$ such that for any $a,b\in M$,  
\[f(a,f(b,a))=b\]  
and $f(x,x)\neq x$ for each $x\in M$. How many possible functions $f$ are there $\pmod{1000}$?
Answer: 0
Problem: Let $f(x) = x + \frac{1}{2x + \frac{1}{2x + \frac{1}{2x + \cdots}}}$. Find $f(99)f'(99)$.
Answer: 99
Problem: Jerry is mowing a rectangular lawn which measures $77$ feet from north to south and $83$ feet from east to west. His lawn mower cuts a path that is $18$ inches wide. Jerry starts mowing by cutting a path from west to east across the north side of the lawn, then makes a right turn to cut a path along the east side of the lawn. He continues mowing by making right turns to mow a path along the next side each time he completes mowing a side. 

How many right turns will he make?
Answer: 102
Problem: An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\frac{1}{a_3a_4}\cdots=\frac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$.
Answer: 102
Problem: The distance between cities $A$ and $B$ is $30$ km. A bus departs from $A$ and makes a stop every $5$ km for $2$ minutes. The bus travels between stops at a speed of $80$ km/h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet the bus, traveling at a speed of $27$ km/h. How far from $A$ will the cyclist meet the bus?
Answer: 20
Problem: Compute
$$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$
Answer: 1
Problem: The ship *Meridiano do Bacalhau* conducts its fishing business over a period of $64$ days. Each day, the captain chooses a direction, either north or south, and the ship sails in that direction. On the first day, the ship sails $1$ mile; on the second day, it sails $2$ miles; generally, on the $n$-th day, it sails $n$ miles. After the $64$-th day, the ship is $2014$ miles north of its initial position. What is the greatest number of days that the ship could have sailed south?
Answer: 7
Problem: Find the minimum of \( \sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2 \) where \( x \) is a real number.
Answer: 1435