ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number of cubic feet of dirt needed to form the pile can be written as $m + n \pi$ where $m$ and $n$ are positive integers. Find $m + n.$
Answer: 280
Problem: How many ordered triplets $(a, b, c)$ of positive integers satisfy the inequality $30a + 50b + 70c \leq 343$?
Answer: 30
Problem: Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that:
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$.
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.

Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
Answer: 5152
Problem: Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$.
Answer: -13
Problem: The diagram below shows a $18 \times 35$ rectangle with eight points marked that divide each side into three equal parts. Four triangles are removed from each of the corners of the rectangle, leaving the shaded region. Find the area of this shaded region.
Answer: 490
Problem: Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying:
- $f(1) = 1$,
- For $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$.

Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.
Answer: 1889
Problem: Five teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled. 

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?
Answer: 6
Problem: Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked "clear". When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the "clear" button, the display is completely erased. If Emma starts with an empty display and presses five (not necessarily distinct) buttons at random, where all ten buttons have equal probability of being chosen, the expected value of the number produced is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. (Take an empty display to represent the number 0.)
Answer: 6642902
Problem: Let $a_1, a_2, \ldots, a_{100}$ be non-negative integers such that:

1. There exists a positive integer $k \leq 100$ such that $a_1 \leq a_2 \leq \cdots \leq a_k$ and $a_i = 0$ for $i > k$.
2. $a_1 + a_2 + a_3 + \cdots + a_{100} = 100$.
3. $a_1 + 2a_2 + 3a_3 + \cdots + 100a_{100} = 2022$.

Find the minimum value of $a_1 + 2^2a_2 + 3^2a_3 + \cdots + 100^2a_{100}$.
Answer: 40940
Problem: Evaluate $ \int_{\sqrt{2}-1}^{\sqrt{2}+1} \frac{x^4+x^2+2}{(x^2+1)^2}\, dx.$
Answer: 2
Problem: Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.
Answer: 1997
Problem: What is the smallest possible value of \( x^2 + y^2 - x - y - xy \)?
Answer: -1
Problem: Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave smudges that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. 

For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. 

For example, if there are smudges on $3$ and $9$, two possible codes are $393939$ and $993999$. Which number of distinct digits in the access code offers the most combinations?
Answer: 5
Problem: Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1 \le x_2 \le \cdots \le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.
Answer: 601
Problem: How many seven-digit positive integers do not either start or end with \(7\)?
Answer: 7200000
Problem: Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?
Answer: 905
Problem: Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.
Answer: 804
Problem: Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$, and $\angle BAD = 110^{\circ}$. Determine the measure of $\angle CAB$ (in degrees).
Answer: 70
Problem: There exist two triples of real numbers \((a,b,c)\) such that \(a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}\) are the roots to the cubic equation \(x^3-5x^2-15x+3\) listed in increasing order. Denote those \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\). If \(a_1, b_1,\) and \(c_1\) are the roots to the monic cubic polynomial \(f\) and \(a_2, b_2,\) and \(c_2\) are the roots to the monic cubic polynomial \(g\), find \(f(0)^3+g(0)^3\).
Answer: -14
Problem: Find the largest positive integer $n$ such that for each prime $p$ with $2 < p < n$, the difference $n - p$ is also a prime number.
Answer: 10
Problem: A $70$ foot pole stands vertically in a plane and is supported by three $490$ foot wires. All wires are attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. Determine the distance, in feet, between any two of those anchor points.
Answer: 840
Problem: At a party, there are $2011$ people, each with a glass of fruit juice, sitting around a circular table. Every second, they clink glasses according to the following two rules:
1. They do not clink glasses crosswise.
2. At each point in time, everyone can clink glasses with at most one other person.

How many seconds are needed, at a minimum, for everyone to have clinked glasses with everyone else?
Answer: 2011
Problem: A circle with area $40$ is tangent to a circle with area $10$. Let $R$ be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
Answer: 240
Problem: Suppose $a$, $b$, and $c$ are positive integers with the property that $ab$, $bc$, and $ac$ are pairwise distinct perfect squares. What is the smallest possible value of $a + b + c$?
Answer: 14
Problem: Find all prime numbers $p$ such that $p^2 - p + 1$ is a perfect cube.
Answer: 19
Problem: Let $N = 1890 \times 1930 \times 1970$. Find the number of divisors of $N$ which are not divisors of $45$.
Answer: 506
Problem: Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor $.
Answer: 3
Problem: Compute the remainder when $1^{2547} + 2^{2547} + \ldots + 2547^{2547}$ is divided by $25$.
Answer: 4
Problem: Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
Answer: 40
Problem: In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$.
Answer: 84
Problem: Determine the number of $8$-tuples of nonnegative integers $(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $0 \le a_k \le k$, for each $k = 1,2,3,4$, and $a_1 + a_2 + a_3 + a_4 + 2b_1 + 3b_2 + 4b_3 + 5b_4 = 19$.
Answer: 1540
Problem: Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.
Answer: 15
Problem: The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
Answer: 336
Problem: 12. What is the sum of all possible $\binom{i}{j}$ subject to the restrictions that $i \geq 10$, $j \geq 0$, and $i+j \leq 20$? Count different $i, j$ that yield the same value separately - for example, count both $\binom{10}{1}$ and $\binom{10}{9}$.
Answer: 27633
Problem: A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.
Answer: 1333
Problem: For how many ordered pairs of positive integers \((x, y)\) is the least common multiple of \(x\) and \(y\) equal to \(1,003,003,001\)?
Answer: 343
Problem: For a positive integer $n$, let $\sigma(n)$ be the sum of the divisors of $n$ (for example, $\sigma(10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2, \ldots, 100\}$ do we have $\sigma(n) < n + \sqrt{n}$?
Answer: 26
Problem: For some integers $a$ and $b$, the function $f(x) = ax + b$ has the properties that $f(f(0)) = 0$ and $f(f(f(4))) = 9$. Find $f(f(f(f(10))))$.
Answer: 10
Problem: Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
Answer: 163542
Problem: Suppose you have a $6$-sided die with $3$ faces colored red, $2$ faces colored blue, and $1$ face colored green. You roll this die $20$ times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces?
Answer: 190
Problem: Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the large circle. Calculate the area of quadrilateral $ABCD$.
Answer: 24
Problem: An integer $n$ satisfies the following conditions:

1. $\Bigl\lfloor \frac{n}{9} \Bigr\rfloor$ is a three-digit number with equal digits.
2. $\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor$ is a four-digit number with the digits $2, 0, 2, 4$ in some order.

Determine the remainder when $n$ is divided by $100$.
Answer: 91
Problem: Let $x, y$ be distinct positive real numbers satisfying:
\[
\frac{1}{\sqrt{x + y} - \sqrt{x - y}} + \frac{1}{\sqrt{x + y} + \sqrt{x - y}} = \frac{x}{\sqrt{y^3}}.
\]
If \( \frac{x}{y} = \frac{a + \sqrt{b}}{c} \) for positive integers \( a, b, c \) with \( \gcd(a, c) = 1 \), find \( a + b + c \).
Answer: 8
Problem: Find the smallest positive three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$.
Answer: 102
Problem: Determine the smallest natural number that can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
Answer: 10010
Problem: Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.
Answer: 707
Problem: Find the number of triples of nonnegative integers $(x, y, z)$ such that 
\[x^2 + 2xy + y^2 - z^2 = 9.\]
Answer: 10
Problem: A rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of the existing surfaces is cut by a straight line, resulting in two new areas. What is the minimum number of cuts needed such that among the obtained polygons, there are $251$ polygons with $11$ sides?
Answer: 2007
Problem: The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
Answer: 13
Problem: Find an odd natural number not exceeding $1000$ such that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.
Answer: 729
Problem: Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.
Answer: 3
Problem: A circle with radius $r$ has an area of $505$. Compute the area of a circle with a diameter of $2r$.
Answer: 2020
Problem: Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.
Answer: 70
Problem: On triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ such that $AE$ bisects angle $BAC$. Let $G$ be the intersection of $AE$ and $CD$, and let point $F$ be the intersection of side $AC$ and the ray $BG$. If $AB$ has length $28$, $AC$ has length $14$, and $CD$ has length $10$, then the length of $CF$ can be written as $\frac{k-m\sqrt{p}}{n}$ where $k$, $m$, $n$, and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p$.
Answer: 623
Problem: In the coordinate plane, the line passing through the points $(2023, 0)$ and $(-2021, 2024)$ also passes through the point $(1, c)$ for a constant $c$. Find the value of $c$.
Answer: 1012
Problem: Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
Answer: -1002
Problem: Leo the fox has a $5 \times 5$ checkerboard grid with alternating red and black squares. He fills in the grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.
Answer: 169
Problem: Consider all pairs of points $(a, b, c)$ and $(d, e, f)$ in the $3$-D coordinate system such that $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?
Answer: 90
Problem: A book is published in three volumes, with the pages numbered consecutively starting from $1$. The numbering continues from the first volume to the second and then to the third. The second volume contains $50$ more pages than the first volume, and the third volume contains one and a half times as many pages as the second volume. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?
Answer: 17
Problem: Find the number of pairs of integers $(x, y)$ such that $x^2 + 2y^2 < 25$.
Answer: 55
Problem: For real numbers $B$, $M$, and $T$, we have the following conditions:  
1. $B^2 + M^2 + T^2 = 2022$  
2. $B + M + T = 72$

Compute the sum of the minimum and maximum possible values of $T$.
Answer: 48
Problem: Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?
Answer: 72
Problem: How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?
Answer: 28
Problem: The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.
Answer: 20
Problem: $N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find its 26th digit.
Answer: 3
Problem: A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?
Answer: 13
Problem: For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n! + 1$ and $(n+1)!$. For $n < 100$, find the largest value of $h_n$.
Answer: 97
Problem: Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points.
Answer: 1964
Problem: Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , \ldots, \mathbb{S}_n$ with the following properties:

1. $|\mathbb{S}_i \cup \mathbb{S}_j| \leq 2004$ for any two indices $1 \leq i, j \leq n$.
2. $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1, 2, \ldots, 2008 \}$ for any $1 \leq i < j < k \leq n$.
Answer: 32
Problem: Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be part of multiple groups.

On day $101$, Princess Pear summons all $100$ jesters to court one final time. Every group of $6$ jesters with a median height of $50.5$ inches presents one more candied cherry to the Princess. How many candied cherries does Princess Pear receive in total?

Please provide a numerical answer (with justification).
Answer: 384160000
Problem: Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
Answer: 2077
Problem: From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime. How many numbers remain without erasing?
Answer: 431
Problem: We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.

Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n \ge 5$), with all digits different from $0$. What is the remainder of $k(n)$ when divided by $8$?
Answer: 1
Problem: The sum
\[
\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}
\]
can be expressed as a rational number \(N\). Find the last 3 digits of \(2021! \cdot N\).
Answer: 977
Problem: How many positive integers less than 2005 are relatively prime to 1001?
Answer: 1442
Problem: There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.
Answer: 2017
Problem: Let $f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \cdots + f(2016)$ is divided by $100$.
Answer: 24
Problem: There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron where you wait in a group with five other souls, each of whom had already come through one of the entrances as well, to get a ride across. In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.
Answer: 270
Problem: Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) \cdots (1 - 2008x)$.
Answer: -1004
Problem: The integer $n$, between 10000 and 99999, is represented as $abcde$ in decimal notation. The digits $a$, $b$, $c$, $d$, and $e$ are defined as follows:

- $a$ is the remainder when $n$ is divided by 2.
- $b$ is the remainder when $n$ is divided by 3.
- $c$ is the remainder when $n$ is divided by 4.
- $d$ is the remainder when $n$ is divided by 5.
- $e$ is the remainder when $n$ is divided by 6.

Find the value of $n$.
Answer: 11311
Problem: Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.
Answer: 167
Problem: In the $3$-dimensional coordinate space, find the distance from the point $(36, 36, 36)$ to the plane that passes through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$.
Answer: 200
Problem: How many solutions does the equation \( \frac{\lfloor x \rfloor}{\{ x \}} = \frac{2007x}{2008} \) have?
Answer: 2006
Problem: $\text{HOW}$, $\text{BOW}$, and $\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given that $D$, $A$, $B$ are collinear in that order, find the length of $BA$.
Answer: 11
Problem: In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L$, $M$, and $T$ such that consecutive multiples of $n$, for any positive integer $n$, are not assigned the same letter?
Answer: 24
Problem: Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.
Answer: 556
Problem: A $1 \times 5$ rectangle is divided into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second, it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?
Answer: 364
Problem: Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0 \leq n < 16} 2^n (-1)^{s(n)}. \]
Answer: 45