meetforever/GRPO_CL huggingface dataset:
user: The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.
Answer: 16184704896
user: Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees.
Answer: 19
user: A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
Answer: 135
user: The digits $1$, $4$, $9$, and $2$ are each used exactly once to form a $4$-digit number $N$. What is the sum of all possible values of $N$?
Answer: 106656
user: Find the sum of all the positive integers that have at most three (not necessarily distinct) prime factors, where the primes come from the set \( \{ 2, 3, 5, 7 \} \).
Answer: 1932
user: If $x$, $y$, and $z$ are real numbers such that $x^2 + z^2 = 1$ and $y^2 + 2y(x + z) = 6$, find the maximum value of $y(z - x)$.
Answer: 4
user: Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.
Answer: 1729
user: How many ways are there to color the edges of a hexagon orange and black, assuming that two hexagons are indistinguishable if one can be rotated into the other? Note that colorings such as $OOBBOB$ and $BOBBOO$ are considered distinct. Flips are ignored.
Answer: 14
user: Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$.
Answer: 36
user: Five distinct 2-digit numbers are in a geometric progression. Find the middle term.
Answer: 36
user: Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.
Answer: -8
user: You and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses $5$ "lucky" buckets by marking an "X" on the bottom that you cannot see. However, he tells you that they either form a vertical or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:

- Either $\{(a, b), (a, b + 1), (a, b + 2), (a, b + 3), (a, b + 4)\}$,
- Or $\{(b, a), (b + 1, a), (b + 2, a), (b + 3, a), (b + 4, a)\}$,

where $1 \le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a "lucky" bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is "lucky"?
Answer: 9
user: Let $a_n$ be the last digit of the sum of the digits of the number formed by repeating the block $2005$ exactly $n$ times. Calculate $a_1 + a_2 + \dots + a_{2005}$.
Answer: 9025
user: Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
Answer: 118
user: Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
Answer: 30
user: Call a number "Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$.
Answer: 135
user: Let $T$ be a trapezoid with two right angles and side lengths $4$, $4$, $5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.
Answer: 120
user: Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.
Answer: 90
user: Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b$ where $a, b \in S$ and $a \neq b$ are equal.
Answer: 5
user: In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.
Answer: 2012
user: For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:
\[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \]
Answer: 10025
user: Let $x$, $y$, $z$ be positive integers satisfying $x < y < z$ and $x + xy + xyz = 37$. Find the greatest possible value of $x + y + z$.
Answer: 20
user: The diagram below shows a parallelogram $ABCD$ with $AB = 36$ and $AD = 60$. Diagonal $BD$ is perpendicular to side $AB$. Points $E$ and $F$ bisect sides $AD$ and $BC$, respectively. Points $G$ and $H$ are the intersections of $BD$ with $AF$ and $CE$, respectively. Find the area of quadrilateral $EGFH$.
Answer: 288
user: A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2} = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
Answer: 108
user: Given the expression:
\[ A = 3\sum_{m=1}^{n^2}\left(\frac{1}{2} - \{\sqrt{m}\}\right) \]
where $n$ is a positive integer. Find the largest integer $k$ such that $n^k$ divides $\lfloor A \rfloor$ (the greatest integer less than or equal to $A$).
Answer: 1
user: Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
Answer: 403
user: In triangle $ABC$, $AB=13$, $BC=14$, $CA=15$. Let $\Omega$ and $\omega$ be the circumcircle and incircle of $\triangle ABC$ respectively. Among all circles that are tangent to both $\Omega$ and $\omega$, call those that contain $\omega$ inclusive and those that do not contain $\omega$ exclusive. Let $\mathcal{I}$ and $\mathcal{E}$ denote the set of centers of inclusive circles and exclusive circles respectively, and let $I$ and $E$ be the area of the regions enclosed by $\mathcal{I}$ and $\mathcal{E}$ respectively. The ratio $\frac{I}{E}$ can be expressed as $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Answer: 1558057
user: Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
Answer: 38
user: For a real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and 
\[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \] 
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
Answer: 4
user: Find the smallest prime $p$ such that the sum of the digits of $p$ (in base 10) is a prime number greater than $10$.
Answer: 29
user: A finite sequence of positive integers $m_i$ for $i = 1, 2, \ldots, 2006$ is defined such that $m_1 = 1$ and $m_i = 10m_{i-1} + 1$ for $i > 1$. How many of these integers are divisible by $37$?
Answer: 668
user: In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? 

Note: When light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.
Answer: 111
user: Find the remainder when $3^{2007}$ is divided by $2007$.
Answer: 1620
user: Consider the polynomial 
\[P(x) = x^3 + 3x^2 + 6x + 10.\]
Let its three roots be \(a\), \(b\), and \(c\). Define \(Q(x)\) to be the monic cubic polynomial with roots \(ab\), \(bc\), and \(ca\). Compute \(|Q(1)|\).
Answer: 75
user: Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2 \pmod{n}$. Compute the number of positive integers $k$ such that $k \mid f(k)$.
Answer: 24
user: How many ordered pairs of positive integers $(m, n)$ satisfy the system:

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

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
Answer: 2
user: Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, \( a_i = a_{i-1}^2 - 1 \) if $a_{i-1}$ is odd and \( a_i = \frac{a_{i-1}}{2} \) if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$.
Answer: 105
user: Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?
Answer: 5040
user: On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$th, and let $P$ be that perfect square. Find $N + P$.
Answer: 218
user: There are $2012$ distinct points in the plane, each of which is to be colored using one of $n$ colors, so that the numbers of points of each color are distinct. A set of $n$ points is said to be multi-colored if their colors are distinct. Determine $n$ that maximizes the number of multi-colored sets.
Answer: 61
user: In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational.
Answer: 54
user: Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$.
Answer: 50
user: How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)
Answer: 12
user: A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^2+y^2<1\) in its interior. Compute the largest possible radius of the circle.
Answer: 61
user: There are $100$ lightbulbs $B_1, B_2, \ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1, S_2, \ldots, S_{100}$ such that for all $1 \leq i \leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where $B_{101} = B_1$ and $B_0 = B_{100}$). Suppose David chooses whether to flick each switch with probability $\frac{1}{2}$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?
Answer: 50
user: A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates the sum of the digits for each card. How many cards does Paulo need to take out of the box to ensure finding at least three cards with the same digit sum?
Answer: 53
user: Let $MATH$ be a trapezoid with $MA = AT = TH = 5$ and $MH = 11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.
Answer: 62
user: Given a set $I=\{(x_1,x_2,x_3,x_4) \mid x_i \in \{1,2,\cdots,11\}\}$. 

$A \subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4), (y_1,y_2,y_3,y_4) \in A$, there exists $i, j \ (1 \leq i < j \leq 4)$, such that $(x_i - x_j)(y_i - y_j) < 0$. 

Find the maximum value of $|A|$.
Answer: 24
user: Positive integers $m$ and $n$ satisfy the equation $$(m + n)(24mn + 1) = 2023.$$ Find the value of $m + n + 12mn$.
Answer: 151
user: Sita and Geeta are two sisters. If Sita's age is written after Geeta's age, a four-digit perfect square number is obtained. If the same exercise is repeated after 13 years, another four-digit perfect square number will be obtained. What is the sum of the present ages of Sita and Geeta?
Answer: 55
user: How many positive integers $n$, with $n \leq 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n$?
Answer: 1339
user: There is a sequence defined by the following conditions:

- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + a\left(\left\lceil \frac{n}{2} \right\rceil \right)$

Find the value of $a(2014)$.

*Note:*
- $\left\lfloor \frac{n}{2} \right\rfloor$ is the floor function, representing the largest integer less than or equal to $\frac{n}{2}$.
- $\left\lceil \frac{n}{2} \right\rceil$ is the ceiling function, representing the smallest integer greater than or equal to $\frac{n}{2}$.
Answer: 34
user: Find the number of positive integers less than or equal to $2017$ that have at least one pair of adjacent digits that are both even. For example, count the numbers $24$, $1862$, and $2012$, but not $4$, $58$, or $1276$.
Answer: 738
user: In the number arrangement:
\[
\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}
\]
What is the number that will appear directly below the number $2010$?
Answer: 2073
user: Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.
Answer: 11
user: For every integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and 
\[
(n-k) f_n(k-1) + kf_n(k+1) = nf_n(k)
\] 
for each $1 \le k < n$.  Let 
\[
S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N)
\]
Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$.  (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
Answer: 26
user: Let $a$, $b$, $c$ be the roots of the polynomial equation $x^3 - 7x^2 - 6x + 5 = 0$. Compute $(a+b)(a+c)(b+c)$.
Answer: -37
user: A semicircle with a radius of $5$ and a quarter circle with a radius of $8$ are tangent to each other and are situated inside a square as shown in the figure. Find the length of the part of the common tangent that is enclosed within the same square. 

![Diagram](https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png)
Answer: 13
user: Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$.
Answer: 12
user: A quadrilateral is given where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Find the perimeter of this quadrilateral.
Answer: 11
user: Consider the $10$-digit number $M = 9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. 

For example, from $M = 9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1} = 9786453210$. Note that any number of (disjoint) pairs can be interchanged. 

Find the number of new numbers that can be so obtained from $M$.
Answer: 88
user: Consider an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$. 

Upon the sequence $1, 2, 3, \ldots, 1024$, the operation is performed successively $5$ times. Show that at the end only one number remains and find this number.
Answer: 683
user: Let $O$ be the center of the circumcircle of triangle $ABC$. We know that $AB = 1$ and $AO = AC = 2$. Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively, such that $OD = OE$ and $BD = \sqrt{2} \cdot EC$. Find $OD^2$.
Answer: 4
user: Determine the greatest natural number $n$ such that for every set $S$ of 2015 different integers, there exist two subsets of $S$ (each subset can have at least one element and the subsets do not need to be disjoint) such that the sum of the elements in each subset is divisible by $n$.
Answer: 2015
user: Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.
Answer: 17
user: Let $ABC$ be a triangle with $\angle BAC = 117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD \sim \triangle ACB$. Compute the measure of $\angle ABC$, in degrees.
Answer: 42
user: In an equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $36\pi$, find the perimeter of the triangle.
Answer: 36
user: Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.
Answer: 20
user: The area of the region bound by the equations $y = 2\cos(4x)$, $y = \sin(2x) + 10$, $x=0$, and $x = 5\pi$ can be expressed as $x\pi$ for some value of $x$. Evaluate $x$.
Answer: 50
user: The equation
$$ (x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016) $$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Answer: 2016
user: Find the natural number $A$ such that there are $A$ integer solutions to $x+y \geq A$ where $0 \leq x \leq 6$ and $0 \leq y \leq 7$.
Answer: 10
user: In triangle $ABC$, point $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD$ and $BE$ are perpendicular to $CF$. Let $M$ be the midpoint of $AB$. If $ME = 13$, $AD = 15$, and $BE = 25$, find $AC + CB$.
Answer: 104
user: Seongcheol has $3$ red shirts and $2$ green shirts. He cannot distinguish between the red shirts or between the green shirts. In how many ways can he arrange them in a row in his closet such that the two green shirts are not adjacent?
Answer: 6
user: Let $\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$?
Answer: 29
user: Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.
Answer: 997
user: Let $a > 1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n \geq 1}$ is defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct.
Answer: 46
user: How many triples \((a,b,c)\) of positive integers strictly less than 51 are there such that \(a+b+c\) is divisible by \(a\), \(b\), and \(c\)?
Answer: 221
user: For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $\sum_{n=1}^{210} nf(n)$ is divided by $211$.
Answer: 48
user: Hapi, the god of the annual flooding of the Nile, is preparing for this year's flooding. The shape of the channel of the Nile can be described by the function $y = \frac{-1000}{x^2+100}$, where the $x$ and $y$ coordinates are in meters. The current depth of the river is $5$ meters. Hapi plans to increase the water level by $3$ meters. How many meters wide will the river be after the flooding? The depth of the river is always measured at its deepest point.
Answer: 40
user: How many positive integers $k$ are there such that $\frac{k}{2013}(a+b) = \text{lcm}(a,b)$ has a solution in positive integers $(a,b)$?
Answer: 1006
user: Let $x_1, x_2, \dots, x_{42}$ be real numbers such that $5x_{i+1} - x_i - 3x_ix_{i+1} = 1$ for each $1 \le i \le 42$, with $x_1 = x_{43}$. Find the product of all possible values for $x_1 + x_2 + \dots + x_{42}$.
Answer: 588
user: For a positive integer $n$, let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999488$ and $a_7 = 9999999488$. Find the value of $n$ such that $a_n$ is divisible by the highest power of $2$.
Answer: 6
user: Determine the least possible value of $f(1998)$, where $f:\mathbb{N} \to \mathbb{N}$ is a function such that for all $m,n \in \mathbb{N}$,  \[ f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}. \]
Answer: 120
user: In a $4 \times 4$ table, we place $k$ blocks such that:

1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we remove a block, there is at least one cell that is not covered.

Find the maximum value of $k$.

Note: The blocks can overlap.
Answer: 12
user: Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:

\[
1 < a < b+2 < 10.
\]
Answer: 28
user: Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.
Answer: 865
user: Triangle $ABC$ has $AB = BC = 10$ and $CA = 16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$.
Answer: 24
user: Consider trapezoid $ABCD$ with $AB \parallel CD$, where $AB = 5$ and $CD = 9$. Additionally, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$?
Answer: 2