sungyub/dapo-math-17k-verl huggingface dataset:
user: The value of the infinite series $$\sum_{n=2}^\infty \dfrac{n^4+n^3+n^2-n+1}{n^6-1}$$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive numbers. Compute $p+q$.

Answer: 23
user: Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $\triangle ABC$ and let $M$ be the midpoint of $BC$. If $BC = 18$ and $\angle MOA = 150^\circ$, find the area of $\triangle OMG$.

Answer: 15
user: In a simple graph with 300 vertices, no two vertices of the same degree are adjacent. What is the maximal possible number of edges in such a graph?

Answer: 42550
user: Find the number of positive integers $n$ such that $1 \leq n \leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.

Answer: 172
user: A graph has $30$ points, and each point has $6$ edges. Find the total number of triples such that each pair of points is either joined or not joined.

Answer: 4060
user: Find the integer part of the number \( \frac{2^1}{1!} + \frac{2^2}{2!} + \cdots + \frac{2^{2018}}{2018!} \).

Answer: 6
user: Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that for all $x, y \in \mathbb{R}^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$.

Answer: 47
user: What is the greatest three-digit divisor of $111777$?

Answer: 703
user: Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is chocolate if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?

Answer: 50
user: For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?

Answer: 205
user: Evaluate $\int_{1}^{4} \frac{x-2}{(x^2+4)\sqrt{x}}\,dx$

Answer: 0
user: On each non-boundary unit segment of an $8 \times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?

Answer: 0
user: Determine the smallest positive integer $n$ with the following property: for every triple of positive integers $x, y, z$, with $x$ dividing $y^3$, $y$ dividing $z^3$, and $z$ dividing $x^3$, it also holds that $(xyz)$ divides $(x + y + z)^n$.

Answer: 13
user: What is the greatest positive integer $m$ such that $n^2(1+n^2-n^4) \equiv 1 \pmod{2^m}$ for all odd integers $n$?

Answer: 7
user: Consider a box with dimensions $10\, \text{cm} \times 16\, \text{cm} \times 1\, \text{cm}$. Determine the maximum number of balls with a diameter of $1\, \text{cm}$ that the box can contain.

Answer: 160
user: A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find the sum of all possible values of $A$.

Answer: 1168
user: There are six lines in the plane. No two of them are parallel, and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines?

Answer: 7
user: In trapezoid $ABCD$, $AD$ is parallel to $BC$, with the side lengths $AB = 6$, $BC = 7$, $CD = 8$, and $AD = 17$. If the sides $AB$ and $CD$ are extended to meet at point $E$, find the measure of the angle at $E$ (in degrees).

Answer: 90
user: Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\substack{abc=2310 \\ a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.

Answer: 49140
user: Find the number of odd three-digit positive integers $x$ satisfying $x^2 \equiv 1 \pmod{8}$ and the product of the digits of $x$ is odd.

Answer: 125
user: Let $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?

Answer: 167
user: On a table near the sea, there are $N$ glass boxes where $N < 2021$, each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, they can throw out all of the balls in the selected box. The player who throws out the last ball wins. 

Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy, and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$?

Answer: 101
user: How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?

Answer: 250
user: Ria writes down the numbers $1, 2, \ldots, 101$ using red and blue pens. The largest blue number is equal to the number of numbers written in blue, and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with the red pen?

Answer: 68
user: Let $f$ be a real-valued function such that $4f(x) + x f\left(\frac{1}{x}\right) = x + 1$ for every positive real number $x$. Find $f(2014)$.

Answer: 403
user: There are 100 houses on the street, divided into 50 pairs. Each pair consists of houses directly opposite each other across the street. On the right side, houses have even numbers, while on the left side, houses have odd numbers. The numbers on both sides increase from the start to the end of the street but are not necessarily consecutive, meaning some numbers might be skipped.

For each house on the right side, the difference between its number and the number of the opposite house was calculated, and all these differences were unique. Let $n$ be the largest house number on this street. Find the smallest possible value of $n$.

Answer: 199
user: You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?

Answer: 1
user: Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$.

Answer: 16
user: Evaluate the following expression: 
\[ 0 - 1 - 2 + 3 - 4 + 5 + 6 + 7 - 8 + \ldots + 2000 \]
The terms with minus signs are exactly the powers of two.

Answer: 1996906
user: A regular $n$-gon $P_1P_2\ldots P_n$ satisfies $\angle P_1P_7P_8 = 178^\circ$. Compute $n$.

Answer: 630
user: There are five cities $A, B, C, D, E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? The order in which he visits the cities also matters: e.g., the routes $A \to B \to C \to A$ and $A \to C \to B \to A$ are different.

Answer: 60
user: The vertices of Durer's favorite regular decagon in clockwise order are labeled as $D_1, D_2, D_3, \ldots, D_{10}$. Determine the angle between the diagonals $D_1D_3$ and $D_2D_5$.

Answer: 54
user: Archit and Ayush are walking around on the set of points $(x,y)$ for all integers $-1 \leq x,y \leq 1$. Archit starts at $(1,1)$ and Ayush starts at $(1,0)$. Each second, they move to another point in the set chosen uniformly at random among the points with distance $1$ away from them. If the probability that Archit goes to the point $(0,0)$ strictly before Ayush does can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, compute $m+n$.

Answer: 7
user: An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle are $1$ and $4$, the side length of this equilateral triangle can be expressed as $\frac{a\sqrt{b}}{c}$, where $(a, c) = 1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a + b + c$.

Answer: 34
user: Find the largest positive integer $n$ such that the number $(2n)!$ ends with 10 more zeroes than the number $n!$.

Answer: 42
user: Two players play a game, starting with a pile of $N$ tokens. On each player's turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $1$ and $2019$ (inclusive) does the first player have a winning strategy?

Answer: 1346
user: Suppose that $a_1 = 1$, and for all $n \ge 2$, \[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1. \] Furthermore, suppose that \[ b_n = a_1 + a_2 + \ldots + a_n \] for all $n$. If \[ b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k \] for some $k$, find $k$.

Answer: 2022
user: Let triangle $ABC$ have $\angle A = 70^\circ$, $\angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ in degrees.

Answer: 50
user: Let $S(n)$ be the sum of digits for any positive integer $n$ (in decimal notation). Let $N=\sum_{k=10^{2003}}^{10^{2004}-1} S(k)$. Determine $S(N)$.

Answer: 27
user: An infinite sequence of integers, $a_0, a_1, a_2, \dots$, with $a_0 > 0$, has the property that for $n \ge 0$, $a_{n+1} = a_n - b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in reverse order. For example, if $a_0 = 1210$, $a_1 = 1089$, and $a_2 = -8712$, etc. Find the smallest value of $a_0$ such that $a_n \neq 0$ for all $n \ge 1$.

Answer: 1012
user: You are given that $17! = 355687ab8096000$ for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.

Answer: 42
user: Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB = 20$ and $CD = 14$, compute the area of $ABCD$.

Answer: 140
user: Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1 \leq a_i \leq i$ for all $1 \leq i \leq 2^{16}$. We say that a term $a_k$ in the sequence with $2 \leq k \leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.

Answer: 49153
user: Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

Answer: -7
user: Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum value of $m$?

Answer: 1012
user: For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime?

Answer: 22
user: For any positive integer $x$, let $f(x) = x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m))) = m^{m^{n+2020}}$. Compute the smallest possible value of $n$.

Answer: 13611
user: Let $S$ be the sum of all positive integers $n$ such that \( \frac{3}{5} \) of the positive divisors of \( n \) are multiples of \( 6 \) and \( n \) has no prime divisors greater than \( 3 \). Compute \( \frac{S}{36} \).

Answer: 2345
user: Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$. We say that $a$ is clearly bigger than $b$ if each digit of $a$ is larger than the corresponding digit of $b$. If the probability that $a$ is clearly bigger than $b$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, compute $m+n$.

Answer: 1061
user: Will the number $\frac{1}{1996}$ decrease or increase, and by how many times, if in the decimal notation of this number, the first non-zero digit after the decimal point is crossed out?

Answer: 50
user: Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?

Answer: 10
user: Let $f(n) = 9n^5 - 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18), \ldots ,f(2009)$.

Answer: 120
user: In $\triangle ABC$, let point $D$ be the foot of the altitude from $A$ to $BC$. Suppose that $\angle A = 90^{\circ}$, $AB - AC = 5$, and $BD - CD = 7$. Find the area of $\triangle ABC$.

Answer: 150
user: Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

$f(1) = 1.$

$\frac{(x + y)}{2} < f(x + y) \le f(x) + f(y)$ for all $x, y \in \mathbb{N}$.

$f(4n + 1) < 2f(2n + 1)$ for all $n \ge 0.$

$f(4n + 3) \le 2f(2n + 1)$ for all $n \ge 0.$

Find the sum of all possible values of $f(2023)$.

Answer: 1012
user: Positive integers $a$, $b$, and $c$ satisfy the equation:
\[
\frac{ab}{a - b} = c
\]
What is the largest possible value of $a + b + c$ that does not exceed 99?

Answer: 99
user: Find the number of 5-digit numbers $n$, such that every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$.

Answer: 40
user: The function $g(x)$ is defined as $\sqrt{\frac{x}{2}}$ for all positive $x$. 

Calculate the value of the expression:
$$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$

This expression can be expressed as $\cos(b)$ in degrees, where $0^\circ < b < 90^\circ$ and $b = \frac{p}{q}$ for some relatively prime positive integers $p$ and $q$. Find $p+q$.

Answer: 19
user: Sokal da is trying to find the largest positive integer $n$ such that when $n$ is transformed to base-7, it appears as twice its value in base-10. For example, $156$ is such a number because $156_{10} = 312_7$ and $312 = 2 \times 156$. Determine Sokal da's number.

Answer: 315
user: Point $P$ is taken inside the square $ABCD$ such that $BP + DP = 25$, $CP - AP = 15$, and $\angle ABP = \angle ADP$. What is the radius of the circumcircle of $ABCD$?

Answer: 10
user: Call an $8$-digit number \textit{cute} if its digits are a permutation of $1,2,\ldots,8$. For example, $23615478$ is \textit{cute} but $31234587$ is not. Find the number of $8$-digit \textit{cute} numbers that are divisible by $11$.

Answer: 4608
user: Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776, -1765, -1754, \ldots$. Joshua computes the (positive) difference between the $1980^\text{th}$ term in the sequence and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?

Answer: 33
user: Let $ABCD$ be a square with side length $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, $CD$, and $DA$. A third square $A_2B_2C_2D_2$ is formed in the same way from $A_1B_1C_1D_1$, and a fourth square $A_3B_3C_3D_3$ from $A_2B_2C_2D_2$. Find the sum of the areas of these four squares.

Answer: 120
user: Let $\alpha$ and $\beta$ be positive integers such that:
$$
\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .
$$
Find the smallest possible value of $\beta$.

Answer: 23
user: Colour a $20000 \times 20000$ square grid using 2000 different colours, with one colour per square. Two squares are considered neighbours if they share a vertex. Define a path as a sequence of squares where each pair of successive squares are neighbours.

Mark $k$ squares on the grid. For each unmarked square $x$, there is exactly one marked square $y$ of the same colour such that $x$ and $y$ are connected by a path consisting of squares of the same colour.

Furthermore, for any two marked squares of the same colour, any path connecting them must pass through squares of all the different colours.

Find the maximum value of $k$.

Answer: 2000
user: Among the numbers $11n + 10^{10}$, where $1 \le n \le 10^{10}$ is an integer, how many are squares?

Answer: 44801
user: Within a group of $2009$ people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends.

Answer: 2006
user: Positive integers $x, y, z$ satisfy the equation $x^3 + xy + x^2 + xz + y + z = 301$. Compute $y + z - x$.

Answer: 1
user: Find the least positive integer $m$ such that $\text{lcm}(15, m) = \text{lcm}(42, m)$. Here $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$.

Answer: 70
user: Consider digits $A, B, C, D$, with $A \neq 0$, such that $ABCD = (CD)^2 - (AB)^2$. Compute the sum of all distinct possible values of $A + B + C + D$.

Answer: 21
user: Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that:
\[
Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) 
\]
for some complex number $w$ with $||w||=\frac{\sqrt{3}}{2}$, $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

Answer: 68
user: Find the sum of the prime factors of $67208001$, given that $23$ is one.

Answer: 781
user: Given that the polynomial $P(x) = x^5 - x^2 + 1$ has 5 roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.

Answer: 5
user: Find the sum of all integers $n$ with $2 \le n \le 999$ that have the following property: if $x$ and $y$ are randomly selected without replacement from the set $\{ 1, 2, \dots, n \}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.

Answer: 600
user: Find the smallest positive integer $k$ such that 
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] 
for some ordered triple of positive integers $(a,b,c)$.

Answer: 1297
user: Let $f$ be a polynomial such that, for all real numbers $x$, \( f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019 \).
Compute \( f(2018) \).

Answer: -2019
user: The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 < m < 6.5 \cdot 10^9$. Find $m$.

Answer: 6490716149
user: In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?

Answer: 2
user: Find the last digit of the number $$\frac{400!}{(200!)(2^{200})}$$

Answer: 5
user: You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same direction, meaning their longer sides must be parallel to each other. How many such coverings are possible?

Answer: 257
user: Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM = MB = BP = 15$ and $AN = NC = CP = 25$.

Answer: 150
user: An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$. 

Let $n!$ denote the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, $24$ is a factorial number since $24 = 4! = 1 \cdot 2 \cdot 3 \cdot 4$. 

Find all perfect numbers greater than $1$ that are also factorials.

Answer: 6
user: How many positive integers have a square less than $10^7$?

Answer: 3162
user: A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the least number of dominoes that are entirely inside some $2 \times 2$ square?

Answer: 100
user: A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube.

Answer: 125
user: Let $P(X) = X^5 + 3X^4 - 4X^3 - X^2 - 3X + 4$. Determine the number of monic polynomials $Q(x)$ with integer coefficients such that $\frac{P(X)}{Q(X)}$ is a polynomial with integer coefficients. Note: A monic polynomial is one with leading coefficient $1$ (e.g., $x^3 - 4x + 5$ is monic, but $5x^3 - 4x^2 + 1$ or $x^2 + 3x^3$ are not).

Answer: 12
user: In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points.

Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points than any other player. Determine all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.

Answer: 50
user: Let $a_1$, $a_2$, \ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence?

Answer: 8
user: Is there a whole number that becomes exactly $57$ times less than itself when its first digit is crossed out?

Answer: 7125
user: What is the largest prime factor of 8091?

Answer: 31
user: Let $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

Answer: 4544
user: Compute the remainder when $98!$ is divided by $101$.

Answer: 50
user: Find $\sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z}, m,n \geq 2 \}$.

Answer: 1
user: Find the remainder when $7^{7^7}$ is divided by $1000$.

Answer: 343
user: Let $a_1, a_2, a_3, \ldots$ be a non-decreasing sequence of positive integers. For $m \ge 1$, define $b_m = \min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n \ge m$. If $a_{19} = 85$, determine the maximum value of $a_1 + a_2 + \cdots + a_{19} + b_1 + b_2 + \cdots + b_{85}$.

Answer: 1700
user: How many ordered pairs of integers $(x, y)$ satisfy $2 \cdot |y| \le x \le 40$?

Answer: 841
user: For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23} \cdot 23^{20}$, compute
\[\frac{f(N)}{f(f(f(N)))}.\]

Answer: 25
user: Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(T_k - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 451
user: Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

Answer: 84