meetforever/GRPO_CL huggingface dataset:
user: Given a table with numbers $1, 2, \ldots, 18$, determine the minimal number of numbers that must be erased so that the sum of any two remaining numbers is not a perfect square of a positive integer.
Answer: 9
user: Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.
Answer: 7208
user: Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
Answer: -54
user: Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$.
Answer: 1009
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. Determine the perimeter of this quadrilateral.
Answer: 11
user: What is the maximum possible value for the sum of the squares of the roots of \(x^4 + ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\), and \(d\) are \(2\), \(0\), \(1\), and \(7\) in some order?
Answer: 49
user: Let $Q$ be a set of permutations of $1, 2, \ldots, 100$ such that for all $1 \leq a, b \leq 100$, $a$ can be found to the left of $b$ and adjacent to $b$ in at most one permutation in $Q$. Find the largest possible number of elements in $Q$.
Answer: 100
user: The non-zero natural number $n$ is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223 - \frac{3}{2} \cdot n$. Find the quotient of the division.
Answer: 14
user: Let $a = 3^{\frac{1}{223}} + 1$ and for all $n \geq 3$, let:
\[
f(n) = \binom{n}{0} a^{n-1} - \binom{n}{1} a^{n-2} + \binom{n}{2} a^{n-3} - \ldots + (-1)^{n-1} \binom{n}{n-1} a^0.
\]
Find $f(2007) + f(2008).$
Answer: 19683
user: Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, \ldots , x_7) = f(x_1) + f(x_2) + \ldots + f(x_7)$. If $x_1, x_2, \ldots, x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, \ldots , x_7)$ does $g(x_1, x_2, \ldots , x_7)$ obtain its maximal value.
Answer: 1
user: How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20} = 0.05$ has two digits after the decimal point, so $20$ is a valid $N$.
Answer: 3997
user: Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_i b_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_i b_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
Answer: 257
user: Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?
Answer: 57
user: We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?
Answer: 24
user: The vertices of a regular nonagon are colored such that:
1. Adjacent vertices are different colors.
2. If 3 vertices form an equilateral triangle, they are all different colors.

Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
Answer: 54
user: A subset of $\{1, 2, 3, \ldots, 2017, 2018\}$ has the property that none of its members are 5 times another. What is the maximum number of elements that such a subset could have?
Answer: 1682
user: Let $x$, $y$, and $z$ be real numbers such that:
\[
12x - 9y^2 = 7 \\
6y - 9z^2 = -2 \\
12z - 9x^2 = 4
\]
Find the value of $6x^2 + 9y^2 + 12z^2$.
Answer: 9
user: The figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven triangles: 
- $P_{0}P_{1}P_{3}$
- $P_{0}P_{3}P_{6}$
- $P_{0}P_{6}P_{7}$
- $P_{0}P_{7}P_{8}$
- $P_{1}P_{2}P_{3}$
- $P_{3}P_{4}P_{6}$
- $P_{4}P_{5}P_{6}$

In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ such that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.
Answer: 1
user: Two circles with radius $2$ and radius $4$ have a common center at $P$. Points $A$, $B$, and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.
Answer: 192
user: Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt{5}$, $4\sqrt{1}$, and $1\sqrt{6}$, respectively. Compute the sum of $a+b$ across all expressions that Rishabh writes.
Answer: 3480
user: Karl likes the number $17$. His favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term has a coefficient of $1$.)
Answer: 8960
user: Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$.
Answer: 37
user: How many pairs $(m, n)$ of positive integers with $m < n$ fulfill the equation $\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}$?
Answer: 5
user: What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
Answer: 570
user: Compute the base 10 value of $14641_{99}$.
Answer: 100000000
user: When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.
Answer: 962963
user: Find the largest value of $a^b$ such that the positive integers $a,b > 1$ satisfy:
\[ a^b b^a + a^b + b^a = 5329 \]
Answer: 81
user: Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
Answer: 4
user: There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$, if $x_1 = ax + b$ and $y_1 = cy + d$, then $y_1^2 = x_1^3 - 36x_1$. What is $a + b + c + d$?
Answer: 90
user: Find the greatest integer less than the number

$$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000000}}$$
Answer: 1998
user: Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha}, \cos{2\alpha}, \ldots, \cos{2^n\alpha}, \ldots$ are negative.
Answer: 120
user: **Problem Statement:**

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is \(1000\) km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? 

*(Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).*
Answer: 99000
user: Determine the smallest non-negative integer $n$ such that \( \sqrt{(6n+11)(6n+14)(20n+19)} \in \mathbb{Q} \).
Answer: 219
user: Let $ABCD$ be a parallelogram. Let $G$ and $H$ be the feet of the altitudes from $A$ to $CD$ and $BC$, respectively. If $AD = 15$, $AG = 12$, and $AH = 16$, find the length of $AB$.
Answer: 20
user: Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x$, $y$, or $z$-directions. However, the evil David anticipated Justin's arrival and has placed a mine at the point $(2, 2, 2)$. If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin.

How many paths can Justin take to reach his destination safely?
Answer: 10350
user: What is the maximum number of subsets of size $5$, taken from the set $A = \{1, 2, 3, \ldots, 20\}$ such that any two of them share exactly one element?
Answer: 16
user: Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A = 7 \cdot 11^5$, the answer would be $7 + 11 + 5 = 23$.
Answer: 677
user: A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ such that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has a diameter of 1 and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
Answer: 3995
user: A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Answer: 61
user: $DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=1$, find the radius of the circle.
Answer: 8
user: In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after the transferring has occurred?
Answer: 170
user: For $n$ a positive integer, define $f_1(n) = n$. Then for $i$ a positive integer, define $f_{i+1}(n) = f_i(n)^{f_i(n)}$. Determine $f_{100}(75) \pmod{17}$. Justify your answer.
Answer: 1
user: There are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere to the left of the man. How many such arrangements are there?
Answer: 2520
user: How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
Answer: 1248
user: The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrives once every one-hour block at some uniformly random time (once at a random time between $9\text{am}$ and $10\text{am}$, once at a random time between $10\text{am}$ and $11\text{am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.
Answer: 525
user: We delete the four corners of an $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
Answer: 21600
user: Let $A_1A_2A_3\ldots A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set $\{A_1, A_2, A_3, \ldots, A_{12}\}$.
Answer: 60
user: What is the sum of the prime factors of \(20!\)?
Answer: 77
user: Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$.
Answer: 9
user: Distinct integers $x, y, z$ satisfy the equation $(x-y)(y-z)(z-x) = x+y+z$. Determine the smallest possible value of $|x+y+z|$.
Answer: 54
user: Let the complex number $z = \cos\frac{1}{1000} + i \sin\frac{1}{1000}$. Find the smallest positive integer $n$ such that $z^n$ has an imaginary part that exceeds $\frac{1}{2}$.
Answer: 524
user: Find all prime numbers $p$ for which there exists an integer $n$ such that there are no integers $x, y$ satisfying $x^3 + y^3 \equiv n \pmod{p}$ (i.e., not all residues are the sum of two cubes).

For example, for $p = 7$, one could set $n = \pm 3$ since $x^3, y^3 \equiv 0, \pm 1 \pmod{7}$, thus $x^3 + y^3 \equiv 0, \pm 1, \pm 2 \pmod{7}$ only.
Answer: 7
user: Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.
Answer: 2013
user: Determine all real numbers $q$ for which the equation $x^4 - 40x^2 + q = 0$ has four real solutions that form an arithmetic progression.
Answer: 144
user: Camila creates a pattern to write the following numbers:
\[
2, 4 \\
5, 7, 9, 11 \\
12, 14, 16, 18, 20, 22 \\
23, 25, 27, 29, 31, 33, 35, 37 \\
\ldots
\]
Following the same pattern, what is the sum of the numbers in the tenth row?
Answer: 3840
user: How many numbers appear in both the arithmetic sequence $10, 16, 22, 28, \ldots, 1000$ and the arithmetic sequence $10, 21, 32, 43, \ldots, 1000$?
Answer: 16
user: Let $X$ be a point inside convex quadrilateral $ABCD$ such that $\angle AXB + \angle CXD = 180^{\circ}$. Given $AX = 14$, $BX = 11$, $CX = 5$, $DX = 10$, and $AB = CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
Answer: 90
user: In a circle, let $AB$ and $BC$ be chords, with $AB = \sqrt{3}$, $BC = 3\sqrt{3}$, and $\angle ABC = 60^\circ$. Find the length of the circle chord that bisects the angle $\angle ABC$.
Answer: 4
user: Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$
Answer: 3999
user: Let $a_1$, $a_2$, \dots, $a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence "good" if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \] Find the minimum possible number of good subsequences across all possible $(a_i)$.
Answer: 19320
user: Suppose $A \subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b \in A$ (where $a$ and $b$ are allowed to be the same), $a + b + 30k$ is always not the product of two consecutive integers. Please find $A$ with the largest possible cardinality.
Answer: 10
user: Compute the minimum value of
\[ \frac{x^4 + 2x^3 + 3x^2 + 2x + 10}{x^2 + x + 1} \]
where \( x \) can be any real number.
Answer: 6
user: On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.
Answer: 43
user: The base-7 representation of the number \( n \) is \( \overline{abc}_{(7)} \), and the base-9 representation of the number \( n \) is \( \overline{cba}_{(9)} \). What is the decimal (base-10) representation of \( n \)?
Answer: 248
user: For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary. For example, $d(13) = 1 + 1 + 0 + 1 = 3$. Let
\[
S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3.
\]
Determine $S$ modulo $2020$.
Answer: 1990
user: Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.
Answer: 11
user: Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[ P(P(x)) - P(x)^2 = x^2 + x + 2016 \] for all real $x$. What is $P(1)$?
Answer: 1010
user: Let $n$ be the smallest positive integer such that the remainder of $3n + 45$, when divided by $1060$, is $16$. Find the remainder of $18n + 17$ upon division by $1920$.
Answer: 1043
user: Suppose $\overline{a_1a_2\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i = 1, 2, \ldots, 2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$.

(Note: $\overline{xyz\ldots}$ denotes an integer whose digits are $x, y, z, \ldots$.)
Answer: 6
user: One day while Tony plays in the backyard of the Kubik's home, he wonders about the width of the backyard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly $12$ feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around $15$ feet. He then measures $2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?
Answer: 82
user: Evaluate the following sum:

$$ \frac{1}{\log_2\left(\frac{1}{7}\right)} + \frac{1}{\log_3\left(\frac{1}{7}\right)} + \frac{1}{\log_4\left(\frac{1}{7}\right)} + \frac{1}{\log_5\left(\frac{1}{7}\right)} + \frac{1}{\log_6\left(\frac{1}{7}\right)} - \frac{1}{\log_7\left(\frac{1}{7}\right)} - \frac{1}{\log_8\left(\frac{1}{7}\right)} - \frac{1}{\log_9\left(\frac{1}{7}\right)} - \frac{1}{\log_{10}\left(\frac{1}{7}\right)} $$
Answer: 1
user: Let $a$ and $b$ be two positive integers satisfying the equation:
\[ 20\sqrt{12} = a\sqrt{b}. \]
Compute the sum of all possible distinct products $ab$.
Answer: 10800
user: Find the smallest positive integer $n$ such that $107n$ has the same last two digits as $n$.
Answer: 50
user: The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle DAC$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?
Answer: 90
user: The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0$, $P_{1}(x)=x$, and 
\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\]
For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
Answer: 0
user: In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.
Answer: 30
user: The remainder when $x^{100} - x^{99} + \cdots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a + b$.
Answer: -49
user: Numbers $1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2001}$ are written on a blackboard. A student erases two numbers $x, y$ and writes down the number $x + y + xy$ instead. Determine the number that will be written on the board after $2000$ such operations.
Answer: 2001
user: Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits.
Answer: 54
user: Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.
Answer: 1899
user: For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
Answer: 1009375
user: Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v \times w \times x \times y \times z$. What is the sum of the integers?
Answer: 5
user: Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
Answer: 2592
user: For a positive integer $n$, let $A(n)$ be the remainder when $n$ is divided by $11$. Define $T(n) = A(1) + A(2) + A(3) + \ldots + A(n)$. Find the value of $A(T(2021))$.
Answer: 3
user: Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such that $A$ and the square are on opposite sides of $BC$. Let the angle bisector of $\angle BAC$ intersect the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$.
Answer: 338
user: Let $k \ll n$ denote that $k < n$ and $k \mid n$. Let $f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\}$ be such that, if $n \leq 2013$ and $k \ll n$, then $f(k) \ll f(n)$. What is the least possible value of $M$?
Answer: 1024
user: A natural number $k > 1$ is called \textit{good} if there exist natural numbers \(a_1 < a_2 < \cdots < a_k\) such that \(\frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1\).

Let \(f(n)\) be the sum of the first \(n\) \textit{good} numbers, \(n \geq 1\). Find the sum of all values of \(n\) for which \(\frac{f(n+5)}{f(n)}\) is an integer.
Answer: 18
user: Consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $x, y$, the equation $f(x f(y)) = y f(x)$ holds. Find the smallest possible value of $f(2007)$.
Answer: 18
user: Each positive integer $n \geq 1$ is assigned the number $p_n$, which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_{32} = 6$, $p_{203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_{999}$. Find the largest prime that divides $S$.
Answer: 103
user: Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?
Answer: 280
user: For every pair of real numbers $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$:
\[
   x_n = (1 - a) x_{n - 1} + a y_{n - 1},
   \\ y_n = (1 - b) x_{n - 1} + b y_{n - 1}.
\]
The supermean of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying:
\[
\left (p - \frac{1}{2} \right)^2 + \left (q - \frac{1}{2} \right)^2 \le \left(\frac{1}{10}\right)^2,
\]
the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 307