ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: For integers $a$, $b$, $c$, and $d$, the polynomial $p(x) = ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
Answer: 47
Problem: There are 15 people at a party; each person has 10 friends. To greet each other, each person hugs all their friends. How many hugs are exchanged at this party?
Answer: 75
Problem: Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers?
Answer: 28
Problem: A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
Answer: 60
Problem: Given that $\log_{10} 2 \approx 0.30103$, find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.
Answer: 30
Problem: The first $9$ positive integers are placed into the squares of a $3 \times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest numbers. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest numbers. How many ways can we distribute the numbers into the chessboard such that $a = b = 4$?
Answer: 25920
Problem: A positive integer $n$ is considered \textit{magical} if the following condition holds: 
\[ \lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor = \lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil \]
Determine the number of magical integers within the range of $1$ to $10,000$, inclusive.
Answer: 1330
Problem: In the country of Óxisz, the minister of finance observed at the end of the tax census that the sum of properties of any two neighboring cities, counted in dinar, is divisible by $1000$. She also observed that the sum of properties of all cities is divisible by $1000$. What is the least sum of properties of all cities if the map of the cities looks as follows?

![Map of cities](https://cdn.artofproblemsolving.com/attachments/0/5/274730ebfdd52d0c3642dfbd0596fe587eb211.png)

Remark: The cities may have non-integer properties, but it is also positive. On the map, the points are the cities, and two cities are neighboring if there is a direct connection between them.
Answer: 1000
Problem: Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
Answer: 3
Problem: Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned into $3$ subsets with equal sums. Find the number of elements of $E$.
Answer: 64
Problem: There are $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd-numbered geese are standing behind them in the order $1,3,5,\dots,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
Answer: 55
Problem: Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials: 
\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \] 
and 
\[ Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] 
Let $S$ be the set of all complex numbers which are a root of either $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$
Answer: 2400
Problem: 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 $137 \le W \le 1061$ is $\frac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.
Answer: 743
Problem: Define the determinant $D_1 = |1|$, the determinant $D_2 = \begin{vmatrix} 1 & 1 \\ 1 & 3 \end{vmatrix}$, and the determinant $D_3 = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 3 & 3 \\ 1 & 3 & 5 \end{vmatrix}$.

In general, for a positive integer $n$, let the determinant $D_n$ have 1s in every position of its first row and first column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least $n$ such that $D_n \geq 2015$.
Answer: 12
Problem: Suppose $a_0, a_1, \ldots, a_{2018}$ are integers such that \[ (x^2 - 3x + 1)^{1009} = \sum_{k=0}^{2018} a_k x^k \] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$.
Answer: 9
Problem: For a given number $n$, let us denote by $p_n$ the probability that when randomly selecting a pair of integers $k, m$ satisfying the conditions $0 \leq k \leq m \leq 2^n$ (the selection of each pair is equally probable), the number $\binom{m}{k}$ will be even. Calculate $\lim_{n \to \infty} p_n$.
Answer: 1
Problem: Determine the smallest positive prime $p$ which satisfies the congruence $p + p^{-1} \equiv 25 \pmod{143}$. Here, $p^{-1}$ denotes the multiplicative inverse of $p$.
Answer: 269
Problem: In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
Answer: 878
Problem: Let $\triangle ABC$ be a triangle in which $\angle ACB = 40^\circ$ and $\angle BAC = 60^\circ$. Let $D$ be a point inside the segment $BC$ such that $CD = \frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?
Answer: 40
Problem: Define the sequences $a_n$, $b_n$, and $c_n$ as follows:  
- $a_0 = k$, $b_0 = 4$, $c_0 = 1$.

For each $n$:
- If $a_n$ is even, then:
  - $a_{n+1} = \frac{a_n}{2}$,
  - $b_{n+1} = 2b_n$,
  - $c_{n+1} = c_n$.
- If $a_n$ is odd, then:
  - $a_{n+1} = a_n - \frac{b_n}{2} - c_n$,
  - $b_{n+1} = b_n$,
  - $c_{n+1} = b_n + c_n$.

Find the number of positive integers $k < 1995$ such that for some $n$, $a_n = 0$.
Answer: 31
Problem: Consider the ten numbers $ar, ar^2, ar^3, \ldots, ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, determine their product.
Answer: 243
Problem: Let $P_1, P_2, \dots, P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1 = 123456$, $P_2 = 123465$, and $P_{720} = 654321$). What is $P_{144}$?
Answer: 216543
Problem: Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move, you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be *on* and two of them will be *off*. Find $m+n$.
Answer: 142
Problem: We define $N$ as the set of natural numbers $n < 10^6$ with the following property: There exists an integer exponent $k$ with $1 \leq k \leq 43$, such that $2012 \mid n^k - 1$. Find $|N|$.
Answer: 1989
Problem: Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x)$. There are relatively prime positive integers such that $x = -\frac{m}{n}$. Find $m + n$.
Answer: 8
Problem: Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?
Answer: 231
Problem: Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$). The expected value of $S(v_1)$ is of the form $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m+n$. 

**Note:** In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself).
Answer: 9901
Problem: Let $P$ be the portion of the graph of
\[ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} \]
located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$. Find $\lfloor 1000d \rfloor$.
Answer: 433
Problem: Determine the number of integers $a$ with $1 \leq a \leq 1007$ such that both $a$ and $a+1$ are quadratic residues modulo $1009$.
Answer: 251
Problem: Determine the smallest positive integer $m$ such that $m^3 - 3m^2 + 2m$ is divisible by both $79$ and $83$.
Answer: 1660
Problem: Let $P(x) = x + 1$ and $Q(x) = x^2 + 1$. Consider all sequences $\langle (x_k, y_k) \rangle_{k \in \mathbb{N}}$ such that $(x_1, y_1) = (1, 3)$ and $(x_{k+1}, y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k), P(y_k))$ for each $k$. We say that a positive integer $n$ is nice if $x_n = y_n$ holds in at least one of these sequences. Find all nice numbers.
Answer: 3
Problem: Find the sum of all positive integers $n$ such that:

$$ \frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}} $$

is an integer.
Answer: 1040
Problem: Anna and Berta play a game in which they take turns removing marbles from a table. Anna takes the first turn. At the beginning of each turn, if there are $n \geq 1$ marbles on the table, the player whose turn it is removes $k$ marbles, where $k \geq 1$ satisfies one of the following conditions:

1. $k$ is an even number and $k \leq \frac{n}{2}$.
2. $k$ is an odd number and $\frac{n}{2} \leq k \leq n$.

A player wins the game if she removes the last marble from the table.

Determine the smallest number $N \geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the table at the beginning.
Answer: 131070
Problem: Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.
Answer: 336
Problem: Let $T$ be a $30$-$60$-$90$ triangle with a hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.
Answer: 50
Problem: Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside the square, and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$.
Answer: 57
Problem: In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.
Answer: 14
Problem: Find the remainder when dividing the number \( A = 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + 2009 \cdot 2010 + 2010 \cdot 2011 \) by 2012.
Answer: 0
Problem: In a sequence of natural numbers $a_1$, $a_2$, $\dots$, $a_{1999}$, the expression $a_n - a_{n-1} - a_{n-2}$ is divisible by $100$ for $3 \leq n \leq 1999$. It is known that $a_1 = 19$ and $a_2 = 99$. Find the remainder when $a_1^2 + a_2^2 + \dots + a_{1999}^2$ is divided by $8$.
Answer: 1
Problem: In the convex quadrilateral $AEBC$, it is given that $\angle BEA = \angle CAE = 90^{\circ}$, $AB = 15$, $BC = 14$, and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to the line $\overline{AB}$. If the ray $CD$ intersects $\overline{AE}$ at $F$, compute the product $AE \cdot AF$.
Answer: 99
Problem: A line with a negative slope passing through the point $(18,8)$ intersects the $x$-axis at $(a,0)$ and the $y$-axis at $(0,b)$. What is the smallest possible value of $a+b$?
Answer: 50
Problem: Each of the 2009 distinct points in the plane is colored either blue or red. On every blue-centered unit circle, there are exactly two red points. Determine the greatest possible number of blue points.
Answer: 1964
Problem: Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey the following equations:
\[
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, \\
0.2 &= \sin \gamma \sin \theta \cos \alpha, \\
0.3 &= \cos \gamma \cos \theta \sin \beta, \\
0.4 &= \cos \gamma \sin \theta \cos \beta, \\
0.5 &\ge \left\lvert N-100 \cos 2\theta \right\rvert.
\end{align*}
\]
Answer: 54
Problem: An equilateral $12$-gon has a side length of $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon.
Answer: 500
Problem: Sristan Thin is walking around the Cartesian plane. From any point \((x, y)\), Sristan can move to \((x+1, y)\) or \((x+1, y+3)\). How many paths can Sristan take from \((0, 0)\) to \((9, 9)\)?
Answer: 84
Problem: Find the smallest positive integer $n$ such that for every prime number $p$, the expression $p^2 + n$ is never prime.
Answer: 5
Problem: Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB$ has length $1$, and $CD$ has length $41$. Let points $X$ and $Y$ lie on sides $AD$ and $BC$, respectively, such that $XY \parallel AB \parallel CD$, and $XY$ has length $31$. Let $m$ and $n$ be two relatively prime positive integers such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $\frac{m}{n}$. Find $m + 2n$.
Answer: 10
Problem: Brian starts at the point $(1,0)$ in the plane. Every second, he performs one of two moves: he can move from $(a,b)$ to $(a-b,a+b)$ or from $(a,b)$ to $(2a-b,a+2b)$. How many different paths can he take to end up at $(28,-96)$?
Answer: 70
Problem: Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle with center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$.
Answer: 30
Problem: Let \( f \) be a function from \( \{1, 2, \ldots, 22\} \) to the positive integers such that \( mn \mid f(m) + f(n) \) for all \( m, n \in \{1, 2, \ldots, 22\} \). If \( d \) is the number of positive divisors of \( f(20) \), compute the minimum possible value of \( d \).
Answer: 2016
Problem: Shayan is playing a game by himself. He picks **relatively prime** integers $a$ and $b$ such that $1 < a < b < 2020$. He wins if every integer $m \geq \frac{ab}{2}$ can be expressed in the form $ax + by$ for nonnegative integers $x$ and $y$. He hasn't been winning often, so he decides to write down all winning pairs $(a, b)$, from $(a_1, b_1)$ to $(a_n, b_n)$. What is $b_1 + b_2 + \ldots + b_n$?
Answer: 1020108
Problem: Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$th prime number, starting with $p_1 = 2$. Let $\tau(x)$ be equal to the number of divisors of $x$. Find the remainder when
$$\sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d)$$
is divided by 91 for positive integers $d$. Recall that $d|a_n$ denotes that $d$ divides $a_n$.
Answer: 40
Problem: Gus needs to create a list of $250$ positive integers, where each integer is equal to the number of other numbers in the list that are different from it. For instance, if $15$ is one of the numbers in the list, then there should be exactly $15$ numbers in the list that are different from $15$. Determine the maximum number of distinct numbers that Gus's list can contain.
Answer: 21
Problem: Let $f$ be a function from positive integers to positive integers where $f(n) = \frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. If $a$ is the smallest positive integer satisfying \[ \underbrace{f(f(\cdots f}_{2013\ f\text{'s}} (a)\cdots)) = 2013, \] find the remainder when $a$ is divided by $1000$.
Answer: 496
Problem: \( f(x) = x^3 + 3x^2 - 1 \). Find the number of real solutions of \( f(f(x)) = 0 \).
Answer: 7
Problem: Given a random string of 33 bits (0 or 1), how many occurrences of two consecutive 0's would you expect? (They can overlap.)

For example:
- The string "100101" has 1 occurrence.
- The string "0001" has 2 occurrences.
Answer: 8
Problem: Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1, \dots, 30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1 + A_2 + \dots + A_{30})$.
Answer: 0
Problem: How many positive integers less than 4000 are not divisible by 2, 3, 5, or 7?
Answer: 913
Problem: A pen costs $\text{Rs.}\, 13$ and a notebook costs $\text{Rs.}\, 17$. A school spends exactly $\text{Rs.}\, 10000$ in the year 2017-18 to buy $x$ pens and $y$ notebooks such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in 2018-19, the school spends a little more than $\text{Rs.}\, 10000$ and buys $y$ pens and $x$ notebooks. How much more did the school pay?
Answer: 40
Problem: Let $\triangle ABC$ be a triangle with $BA<AC$, $BC=10$, and $BA=8$. Let $H$ be the orthocenter of $\triangle ABC$. Let $F$ be the point on segment $AC$ such that $BF=8$. Let $T$ be the point of intersection of $FH$ and the extension of line $BC$. Suppose that $BT=8$. Find the area of $\triangle ABC$.
Answer: 40
Problem: Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal.
Answer: 1010
Problem: A palindrome is a word that reads the same backward as forward. Examples include OMO, lepel, and parterretrap. Determine how many palindromes can be constructed using the letters $a, b, c, d,$ and $e$ under the following conditions:

- Each letter may appear no more than twice in any palindrome.
- The length of each palindrome must be at least $3$ letters.

Note: Any possible combination of letters is considered a word.
Answer: 640
Problem: Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_2$ be the radius of $(AEF)$. If $\frac{R_1^2}{R_2^2} = a - \sqrt{b + \sqrt{c}}$ where $a, b, c$ are integers, find $a^{b^{c}}$.
Answer: 256
Problem: Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\frac{1}{3}a_{n-1}$ and is $\frac{1}{9}a_{n-1}$ with probability $\frac{1}{2}$. If the expected value of $\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\frac{p}{q}$, what is $p+q$?
Answer: 16
Problem: Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Answer: 2268
Problem: The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the $n^{\text{th}}$ problem is worth $F_n$ points, where $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$. The two problem writers are fair people, so they make sure that each of them is responsible for problems worth an equal number of total points. Compute the number of ways problem writing assignments can be distributed between the two writers.
Answer: 32
Problem: Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$, and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Answer: 503
Problem: An integer sequence $a_1, a_2, \ldots, a_n$ is defined with the following properties:

1. $a_1 = 0$
2. $a_n \leq 10$
3. $a_{i+1} - a_i \geq 2$ for $1 \leq i < n$

How many possible sequences are there given these conditions? The sequence may be of any length.
Answer: 89
Problem: Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$.
Answer: 169
Problem: On $\triangle ABC$, let $D$ be a point on side $\overline{AB}$, $F$ be a point on side $\overline{AC}$, and $E$ be a point inside the triangle such that $\overline{DE} \parallel  \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Given that $AF = 6$, $AC = 33$, $AD = 7$, $AB = 26$, and the area of quadrilateral $ADEF$ is $14$, find the area of $\triangle ABC$.
Answer: 143
Problem: A positive integer is considered **detestable** if the sum of its digits is a multiple of $11$. Determine how many positive integers below $10,000$ are detestable.
Answer: 908
Problem: Find the three-digit number for which the ratio of the number to the sum of its digits is minimal.
Answer: 199
Problem: Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive 1-digit integer. Find all possible 1-digit integers Ben can end with from this process.
Answer: 9
Problem: Alexis imagines a $2008 \times 2008$ grid of integers arranged sequentially in the following way:

\[
\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}
1, & 2, & 3, & \ldots, & 2008 \\
2009, & 2010, & 2011, & \ldots, & 4026 \\
4017, & 4018, & 4019, & \ldots, & 6024 \\
\vdots & & & & \vdots \\
2008^2-2008+1, & 2008^2-2008+2, & 2008^2-2008+3, & \ldots, & 2008^2
\end{array}
\]

She picks one number from each row so that no two numbers she picks are in the same column. She then proceeds to add them together and finds that $S$ is the sum. Next, she picks $2008$ of the numbers that are distinct from the $2008$ she picked the first time. Again, she picks exactly one number from each row and column, and again the sum of all $2008$ numbers is $S$. Find the remainder when $S$ is divided by $2008$.
Answer: 1004
Problem: Of all positive integral solutions \((x,y,z)\) to the equation 
\[x^3+y^3+z^3-3xyz=607,\] 
compute the minimum possible value of \(x+2y+3z.\)
Answer: 1213
Problem: It is known that unequal numbers $a$, $b$, and $c$ are successive members of an arithmetic progression. All of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.
Answer: 2500
Problem: Add one pair of brackets to the expression $1+2\times 3+4\times 5+6$ so that the resulting expression has a valid mathematical value. What is the largest possible value that one can make?
Answer: 77
Problem: The function $f(x) = x^5 - 20x^4 + ax^3 + bx^2 + cx + 24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.
Answer: -24
Problem: In an acute triangle $ABC$ with an angle $\angle ACB = 75^\circ$, altitudes $AA_3$ and $BB_3$ intersect the circumscribed circle at points $A_1$ and $B_1$, respectively. On the lines $BC$ and $CA$, select points $A_2$ and $B_2$, respectively, such that the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$. Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.
Answer: 150
Problem: Mary has a sequence $m_2, m_3, m_4, \ldots$, such that for each $b \ge 2$, $m_b$ is the least positive integer $m$ for which none of the base-$b$ logarithms $\log_b(m), \log_b(m+1), \ldots, \log_b(m+2017)$ are integers. Find the largest number in her sequence.
Answer: 2188