sungyub/dapo-math-17k-verl huggingface dataset:
user: 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
user: 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
user: 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
user: How many ways are there to rearrange the letters of \( CCARAVEN \)?

Answer: 10080
user: 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
user: 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
user: Find the largest natural number $n$ such that $2^n + 2^{11} + 2^8$ is a perfect square.

Answer: 12
user: 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
user: 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
user: Find the smallest positive multiple of 999 that does not contain the digit 9.

Answer: 111888
user: 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
user: 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
user: 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
user: 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
user: Find a $7$-digit integer divisible by $128$, whose decimal representation contains only the digits $2$ and $3$.

Answer: 2223232
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: $A_1, A_2, \ldots, A_n$ 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
user: 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
user: 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
user: 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
user: Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the expression $(bcd + acd + abd + abc)^7$.

Answer: 630
user: 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
user: 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
user: 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
user: 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
user: Suppose real numbers $a, b, c, d$ satisfy the following conditions:

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

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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is an exact square.

Answer: 1972
user: 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
user: 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
user: Let $f(x) = x + \frac{1}{2x + \frac{1}{2x + \frac{1}{2x + \cdots}}}$. Find $f(99)f'(99)$.

Answer: 99
user: 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
user: 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
user: 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
user: Compute
$$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

Answer: 1
user: 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
user: Find the minimum of \( \sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2 \) where \( x \) is a real number.

Answer: 1435
user: The MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be divided into prize groups to potentially receive a prize. However, based on the number of contestants, the ideal number of groups doesn't work as expected. Consider the following scenarios:

- When dividing into $4$ groups, there are $3$ left over.
- When dividing into $5$ groups, there are $2$ left over.
- When dividing into $6$ groups, there is $1$ left over.
- When dividing into $7$ groups, there are $2$ left over.

Given that there are fewer than $300$ participants in the MIG, how many participants are there?

Answer: 247
user: A $1 \times n$ rectangle ($n \geq 1$) is divided into $n$ unit ($1 \times 1$) squares. Each square of this rectangle is colored red, blue, or green. Let $f(n)$ be the number of colorings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $\frac{f(9)}{f(3)}$? (The number of red squares can be zero.)

Answer: 37
user: Determine the greatest value of the sum $M = 11xy + 3x + 2012yz$, where $x, y, z$ are non-negative integers satisfying the condition $x + y + z = 1000$.

Answer: 503000000
user: Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting $470$ pounds. How many plates did Arnold have on the bar?

Answer: 13
user: Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n$, 
\[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]

Answer: 4
user: Let $x$, $y$, and $z$ be consecutive integers such that \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{45}. \] Find the maximum value of $x + y + z$.

Answer: 405
user: A circumference was divided into $n$ equal parts. On each of these parts, a number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$, and $17$ were in consecutive positions. In how many parts was the circumference divided?

Answer: 20
user: In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.

Answer: 80
user: At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem, you can earn an integer score from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are $8^6 = 262144$ participants, and no two get the same score on every problem. Find the score of the participant whose rank was $7^6 = 117649$.

Answer: 1
user: Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble.

You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 323
user: Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b \pmod{n}$ for a fixed $n \in \mathbb{Z}^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + \ldots$ Consider
$$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$
Then $P(n)$, $n \in \mathbb{Z}^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

Answer: 0
user: A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Answer: 100
user: A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

Answer: 3003
user: Find the number of degree $8$ polynomials $f(x)$ with nonnegative integer coefficients satisfying both $f(1) = 16$ and $f(-1) = 8$.

Answer: 47775
user: Find the number of three-digit palindromes that are divisible by $3$. Recall that a palindrome is a number that reads the same forward and backward, such as $727$ or $905509$.

Answer: 30
user: Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\frac{m}{\sqrt{n}}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$.

Answer: 9799
user: Calculate the sum $\sum_{n=1}^{1,000,000} \lfloor \sqrt{n} \rfloor$.

You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.

Answer: 666167500
user: Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x \le 100$ and $\text{lcm}(16,x) = 16x$.

Answer: 2500
user: Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \(\gcd(n,2^n - 1) = 3.\)

Answer: 360
user: Compute the positive integer $n$ such that $\log_3 n < \log_2 3 < \log_3 (n + 1)$.

Answer: 5
user: Find the sum of all prime numbers that can be expressed both as a sum of two prime numbers and as a difference of two prime numbers.

Answer: 5
user: The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.

Answer: 2224
user: Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$, a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 8
user: Compute the smallest positive integer $n$ for which $\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}$ is an integer.

Answer: 6156
user: Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\frac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137 \leq W \leq 1061$ is $\frac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

Answer: 743
user: At a concert, $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible nonzero values of $n$. Compute the largest nonnegative integer $k$ such that $2^k$ divides $p$.

Answer: 38
user: $P(x)$ is a polynomial of degree $3n$ such that:

\[
\begin{align*}
P(0) = P(3) = \cdots &= P(3n) = 2, \\
P(1) = P(4) = \cdots &= P(3n-2) = 1, \\
P(2) = P(5) = \cdots &= P(3n-1) = 0, \\
\text{and } & P(3n+1) = 730.
\end{align*}
\]

Determine $n$.

Answer: 4