sungyub/dapo-math-17k-verl huggingface dataset:
user: The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2) = 0$, $f(3) > 0$, and $f(9999) = 3333$. For all positive integers $m$ and $n$, the following condition holds:
\[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \]
Determine the value of $f(1982)$.

Answer: 660
user: The \textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules:
\begin{itemize}
 \item $D(1) = 0$;
 \item $D(p) = 1$ for all primes $p$;
 \item $D(ab) = D(a)b + aD(b)$ for all positive integers $a$ and $b$.
\end{itemize}
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n) = n$.

Answer: 31
user: Let $f(n)$ denote the sum of the distinct positive integer divisors of $n$. Evaluate:
\[
f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9).
\]

Answer: 69
user: Let $ABC$ be a triangle with $AB = 20$ and $AC = 18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE = AF = 8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC = m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m + n$.

Answer: 305
user: After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.

Answer: 60
user: There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What is the largest possible value of $n$ under such conditions?

Answer: 6
user: If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?

Answer: 3
user: Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^{\circ}$. Find the measure of $\angle ABC$.

Answer: 30
user: What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101?

Answer: 22
user: Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle XAC=32^\circ$, and $\angle XCA=35^\circ$. Compute $\angle BAC$ in degrees.

Answer: 81
user: Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board:
$n$ is larger than any integer currently on the board.
$n$ cannot be written as the sum of two distinct integers already on the board.

Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.

Answer: 388
user: Compute the number of nonnegative integer triples \((x, y, z)\) which satisfy the inequality:
\[ 4x + 2y + z \leq 36 \]

Answer: 1330
user: How many pairs of positive integers $(a, b)$ satisfy $\frac{1}{a} + \frac{1}{b} = \frac{1}{2004}$?

Answer: 45
user: In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$.

Answer: 40
user: Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.

Answer: 10004
user: Find the three-digit number that has the greatest number of different divisors.

Answer: 840
user: The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off each of the mirrors once, and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\frac{AB}{AX} = 64$, and $\frac{CA}{AY} = \frac{p}{q}$ for coprime positive integers $p$ and $q$, find $p + q$.

Answer: 131
user: Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]

Answer: 30
user: What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \]

Answer: 972
user: Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

Answer: 311
user: In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$.

Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise, he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\frac{m}{1024}$, find $m$.

Answer: 147
user: The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.

Answer: 24
user: The function $f : \mathbb{R} \to \mathbb{R}$ satisfies the equation \( f(x^2) f''(x) = f'(x) f'(x^2) \) for all real numbers \( x \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), determine the value of \( f'(1) + f''(1) \).

Answer: 6
user: The value of $21!$ is $51,090,942,171,abc,440,000$, where $a$, $b$, and $c$ are digits. What is the value of $100a + 10b + c$?

Answer: 709
user: A cube $ABCDA'B'C'D'$ is given with an edge of length $2$, with vertices marked as in the figure. The point $K$ is the center of the edge $AB$. The plane containing the points $B', D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.

Answer: 1
user: Let $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression:
\[
(r^2 + s + t)(s^2 + t + r)(t^2 + r + s)
\]

Answer: 1100
user: How many zeros does $101^{100} - 1$ end with?

Answer: 4
user: Find the number of integer solutions to $||x| - 2020| < 5$.

Answer: 18
user: Find the last eight digits of the binary representation of $27^{1986}$.

Answer: 11011001
user: The roots of the equation $x^5 - 180x^4 + Ax^3 + Bx^2 + Cx + D = 0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

Answer: 243
user: The figure on the right shows the ellipse \( \frac{(x-19)^2}{19} + \frac{(y-98)^2}{98} = 1998 \).

Let \( R_1, R_2, R_3, \) and \( R_4 \) denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of \( R_1 - R_2 + R_3 - R_4 \).

Answer: 7448
user: Let $x \neq y$ be positive reals satisfying $x^3 + 2013y = y^3 + 2013x$, and let $M = (\sqrt{3} + 1)x + 2y$. Determine the maximum possible value of $M^2$.

Answer: 16104
user: Let $\triangle ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 6$. The internal angle bisector of $\angle C$ intersects the side $AB$ at point $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM \parallel AC$ and $DN \parallel BC$. If $(MN)^2 = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, what is the sum of the digits of $|p - q|$?

Answer: 2
user: In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C$?

Answer: 20
user: Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$?

Answer: 2
user: Find $ \lim_{x\rightarrow 0} \frac{1}{x} \int_{-x}^x (t\sin 2006 t + 2007t + 1004)\ dt$.

Answer: 2008
user: Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppose $AB = 6$ and $CD = 8$. Further, suppose that the area of the part of the circle lying between the chords $AB$ and $CD$ is given by \( \frac{m\pi + n}{k} \), where $m$, $n$, and $k$ are positive integers with \( \gcd(m, n, k) = 1 \). What is the value of $m + n + k$?

Answer: 75
user: Find the largest prime that divides $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 44 \cdot 45 \cdot 46$.

Answer: 47
user: For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?

Answer: 18
user: Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number.

Answer: 4
user: Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.

Answer: 50
user: Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2} = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?

Answer: 462
user: Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers.

Answer: 400185
user: Find the remainder when $2^{5^9} + 5^{9^2} + 9^{2^5}$ is divided by $11$.

Answer: 8
user: The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence.

Answer: 16384
user: What is the last three digits of the sum \(11! + 12! + 13! + \ldots + 2006!\)?

Answer: 400
user: What is the smallest number $k$ greater than $6$ for which:
\[1 + 2 + \ldots + k = k + (k+1) + \ldots + n\]
where $n$ is an integer greater than $k$?

Answer: 35
user: Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

Answer: 13
user: Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_A$ through $O$ and $A$, and another circle $\omega_B$ through $O$ and $B$; the circles $\omega_A$ and $\omega_B$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega$, $\omega_A$, $\omega_B$ are congruent. If $CO = \sqrt{3}$, what is the perimeter of $\triangle ABC$?

Answer: 6
user: A company that sells keychains incurs a daily maintenance cost of $\$500$. Additionally, each worker is paid $\$15$ per hour. Each worker produces 5 keychains per hour, and each keychain is sold for $\$3.10$. What is the minimum number of workers the company must hire to make a profit during an 8-hour workday?

Answer: 126
user: Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make?

Answer: 112
user: How far is it from the point \((9, 17)\) to its reflection across the line \(3x + 4y = 15\)?

Answer: 32
user: Find the numerator of \[ \frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} \] when reduced.

Answer: 9091
user: The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x) \cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$.

Answer: 100
user: Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

Answer: 702
user: Let $a$, $b$, and $c$ be integers simultaneously satisfying the equations:
\[
4abc + a + b + c = 2018
\]
and
\[
ab + bc + ca = -507
\]
Find \(|a| + |b| + |c|\).

Answer: 46
user: \( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What is the area of \( \triangle IJK \)?

Answer: 21
user: For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?

Answer: 1680
user: An integer $n \ge 2$ is called *delicious* if there exist $n$ positive integers that sum up to 2014 and have distinct remainders when divided by $n$. Determine the smallest delicious integer.

Answer: 4
user: A magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his choice. What is the smallest value of $k$ such that the revealed cards can determine the exact positions of all other cards?

Answer: 2
user: A building contractor needs to pay his $108$ workers \(\$200\) each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two \(\$100\) bills. Find the number of workers who get paid with four \(\$50\) bills.

Answer: 31
user: Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

Answer: 1847
user: A courtyard has the shape of a parallelogram $ABCD$. At the corners of the courtyard, there stand poles $AA'$, $BB'$, $CC'$, and $DD'$, each of which is perpendicular to the ground. The heights of these poles are $AA' = 68$ centimeters, $BB' = 75$ centimeters, $CC' = 112$ centimeters, and $DD' = 133$ centimeters. Find the distance in centimeters between the midpoints of $A'C'$ and $B'D'$.

Answer: 14
user: Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. Find the minimum possible value of $|b|$.

Answer: 4
user: Determine the number of ordered pairs of positive integers $(x, y)$ with $y < x \le 100$ such that $x^2 - y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are relatively prime if they have no common factor other than $1$.)

Answer: 99
user: Find all positive integers $n$ such that both $837 + n$ and $837 - n$ are cubes of positive integers.

Answer: 494
user: A positive integer is written on each of the six faces of a cube. For each vertex of the cube, we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?

Answer: 31
user: Find the sum of all positive integers $n$ such that \( \frac{2020}{n^3 + n} \) is an integer.

Answer: 13
user: What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

Answer: 672
user: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

Answer: 7
user: Compute $ \left\lfloor \frac{2005^3}{2003 \cdot 2004} - \frac{2003^3}{2004 \cdot 2005} \right\rfloor $

Answer: 8
user: Let $n$ be a positive integer. Define $f(n)$ as the probability that if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the smallest possible value of $s(n)$.

Answer: 58
user: Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

$f(1) = 2$
$f(a) + f(b) \leq 2\sqrt{f(a)}$
For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.

Answer: 10201
user: Let $a, b$ be positive real numbers with $a > b$. Compute the minimum possible value of the expression

\[
\frac{a^2b - ab^2 + 8}{ab - b^2}.
\]

Answer: 6
user: How many ordered triples \((a,b,c)\) of integers with \(1 \leq a \leq b \leq c \leq 60\) satisfy \(a \cdot b = c\)?

Answer: 134
user: An alarm clock runs 4 minutes slow every hour. It was set right $3 \frac{1}{2}$ hours ago. Now another clock, which is correct, shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?

Answer: 15
user: Find the value of \((2^{40} + 12^{41} + 23^{42} + 67^{43} + 87^{44})^{45! + 46} \mod 11\).

Answer: 5
user: Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.

Answer: 29
user: Let $P(x)$ be the product of all linear polynomials $ax+b$, where $a,b \in \{0,\ldots,2016\}$ and $(a,b) \neq (0,0)$. Let $R(x)$ be the remainder when $P(x)$ is divided by $x^5-1$. Determine the remainder when $R(5)$ is divided by $2017$.

Answer: 121
user: Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$.

Answer: 2009
user: Let triangle $\triangle ABC$ have side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $M$ be the midpoint of $\overline{BC}$, and $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle AEM = m\angle AFM = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle AED$ and $\angle AFD$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.

Answer: 28
user: A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?

Answer: 2004
user: The number $27,000,001$ has exactly four prime factors. Find the sum of these prime factors.

Answer: 652
user: In triangle $ABC$, where $\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\angle ABD$ is a right angle. Point $E$ is on side $AC$ between points $A$ and $D$, such that line $BD$ bisects $\angle EBC$. Given that $AC = 35$, $BC = 7$, and $BE = 5$, find the length of $CE$.

Answer: 10
user: What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?

Answer: 17
user: Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
All other pairs of the five integers are relatively prime.

Answer: 47
user: A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?

Answer: 962
user: The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots and form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square.

Answer: 180
user: Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$.

Answer: 24
user: The four-digit base ten number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ has all nonzero digits and is a multiple of $99$. Additionally, the two-digit base ten number $\underline{a}\;\underline{b}$ is a divisor of $150$, and the two-digit base ten number $\underline{c}\;\underline{d}$ is a divisor of $168$. Find the remainder when the sum of all possible values of the number $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$ is divided by $1000$.

Answer: 108
user: We define the ridiculous numbers recursively as follows:
\begin{itemize}
\item 1 is a ridiculous number.
\item If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
\end{itemize}

A closed interval $I$ is "boring" if:
\begin{itemize}
\item $I$ contains no ridiculous numbers, and
\item There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
\end{itemize}

The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?

Answer: 9
user: Find all positive integers $n$ such that \( \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \), where \( \lfloor x \rfloor \) represents the largest integer less than the real number $x$.

Answer: 24
user: How many ordered pairs of integers \((m,n)\) are there such that \(m\) and \(n\) are the legs of a right triangle with an area equal to a prime number not exceeding 80?

Answer: 87
user: Let $a_1, a_2, \ldots$ be a sequence defined by $a_1 = 1$ and for $n \ge 1$, $a_{n+1} = \sqrt{a_n^2 - 2a_n + 3} + 1$. Find $a_{513}$.

Answer: 33