ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: 2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?
Answer: 1000
Problem: Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.
Answer: 45
Problem: Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
Answer: 8
Problem: Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 - x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
Answer: 3984
Problem: If $a$, $b$ are integers and $s = a^3 + b^3 - 60ab(a + b) \geq 2012$, find the least possible value of $s$.
Answer: 2015
Problem: Find the sum of the real roots of the polynomial:
\[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \]
Answer: 330
Problem: In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion indefinitely. The total length of his path, if $AB=5$ and $BC=12$, is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.
Answer: 793
Problem: Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line $AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD=16$, find the value of the product $r_1r_2$.
Answer: 28
Problem: Find all the real numbers $k$ such that for any non-zero real numbers $a$ and $b$, at least one of the numbers $a$, $b$, or $\frac{5}{a^2} + \frac{6}{b^3}$ is less than or equal to $k$.
Answer: 2
Problem: All vertices of a convex polyhedron are endpoints of exactly four edges. Find the minimal possible number of triangular faces of the polyhedron.
Answer: 8
Problem: Jerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what time will the clock show when he takes the last pill from the bottle?
Answer: 1
Problem: Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, and $252$, in no particular order. Determine the maximum possible value of $x + y$.
Answer: 804
Problem: Consider a $2008 \times 2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard such that the sides of every cell of the board are contained within the sides of one of the rectangles. Find the value of $M$. (For example, for a $2 \times 3$ chessboard, the value of $M$ is 3.)
Answer: 2009
Problem: In a rectangle $ABCD$, let $E$ be the midpoint of $AB$, and $F$ be a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ is perpendicular to $BD$. Suppose $BC = 8\sqrt{3}$. Find $AB$.
Answer: 24
Problem: Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.
Answer: 20
Problem: Given the following system of equations:  
\[ a+4b+9c+16d+25e+36f+49g=1 \]  
\[ 4a+9b+16c+25d+36e+49f+64g=12 \]  
\[ 9a+16b+25c+36d+49e+64f+81g=123 \]  
Determine the value of \( 16a+25b+36c+49d+64e+81f+100g \).
Answer: 334
Problem: Let $S$ be a randomly selected four-element subset of \( \{1, 2, 3, 4, 5, 6, 7, 8\} \). Let \( m \) and \( n \) be relatively prime positive integers so that the expected value of the maximum element in \( S \) is \( \frac{m}{n} \). Find \( m + n \).
Answer: 41
Problem: Call a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the digits of the year. For example, February $28$, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.
Answer: 36
Problem: Let $a, b,$ and $c$ be real numbers such that $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ and $abc=5$. The value of \(\left(a-\frac{1}{b}\right)^3+\left(b-\frac{1}{c}\right)^3+\left(c-\frac{1}{a}\right)^3\) can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 77
Problem: Find $n$ such that $20^{2009} = 10^{2000} \cdot 40^9 \cdot 2^n$.
Answer: 1991
Problem: Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Answer: 5
Problem: Let $[ABCD]$ be a convex quadrilateral with an area of $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have an area of $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.
Answer: 2013
Problem: Evaluate $ \sqrt{\pi \int_0^{2008} x|\sin \pi x|\, dx}$
Answer: 2008
Problem: The side lengths of a triangle are natural number multiples of $7$ and are less than $40$. How many such triangles satisfy these conditions?
Answer: 22
Problem: Find the smallest natural $k$ such that among any $k$ distinct and pairwise coprime naturals smaller than $2018$, a prime can be found.
Answer: 16
Problem: Compute the expected sum of elements in a subset of $\{1, 2, 3, \ldots, 2020\}$ (including the empty set) chosen uniformly at random.
Answer: 1020605
Problem: A circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.
Answer: 96
Problem: Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. Determine the smallest $N$ such that $a_{N} > 2017$.
Answer: 95
Problem: Determine the number of ordered pairs of positive integers $(m,n)$ with $1 \leq m \leq 100$ and $1 \leq n \leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]
Answer: 52
Problem: Determine the minimum value of \( n \) such that in a group of 1991 participants, where each participant knows at least \( n \) other participants (with mutual friendship), there exists a subset of 6 participants in which every pair of participants knows each other.
Answer: 1593
Problem: Let $a, b, c, x, y,$ and $z$ be complex numbers such that \[ a = \frac{b+c}{x-2}, \ b = \frac{c+a}{y-2}, \ c = \frac{a+b}{z-2}. \] If $xy + yz + xz = 67$ and $x + y + z = 2010$, find the value of $xyz$.
Answer: -5892
Problem: Given 2005 distinct numbers $a_1, a_2, \dots, a_{2005}$. By one question, we may take three different indices $1 \le i < j < k \le 2005$ and find out the set of numbers $\{a_i, a_j, a_k\}$ (unordered, of course). Find the minimal number of questions necessary to determine all numbers $a_i$.
Answer: 1003
Problem: Point $P_1$ is located 600 miles West of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives East at a speed of 50 mph. At 8:00 AM, another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?
Answer: 60
Problem: Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $$\{10a+b \mid 1 \leq a \leq 5, 1 \leq b \leq 5\}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?
Answer: 47
Problem: Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.
Answer: 917
Problem: Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$.  
Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$.
Answer: 194763
Problem: There is a sequence defined as follows:
- $a(2) = 0$
- $a(3) = 1$
- 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 $a(2014)$.

Note: $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floor function (largest integer $\leq \frac{n}{2}$) and the ceiling function (smallest integer $\geq \frac{n}{2}$), respectively.
Answer: 34
Problem: Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, \ldots, x_{10})$ of nonnegative integers such that:
\[ x_1^2 + x_2^2 + \ldots + x_{10}^2 = k. \]
Define $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Determine the value of:
\[ \sum_{i=1}^{2012} b_{4i}. \]
Answer: 1006
Problem: A right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its area to its perimeter is a perfect square. Find the minimum possible area of this triangle.
Answer: 30
Problem: Let \( x \) be the answer to this question. Find the value of \( 2017 - 2016x \).
Answer: 1
Problem: Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They agreed on the following rules of attendance:  

1. There should always be at least one person present on each day.  
2. On no two days should the same subset of politicians attend.  
3. The members present on day $N$ should include, for each $K<N$ (where $K \ge 1$), at least one member who was present on day $K$.  

**Question:** For how many days can the parliament sit before one of the rules is broken?
Answer: 128
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the integer part of the number \( \frac{2^1}{1!} + \frac{2^2}{2!} + \cdots + \frac{2^{2018}}{2018!} \).
Answer: 6
Problem: 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
Problem: What is the greatest three-digit divisor of $111777$?
Answer: 703
Problem: 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
Problem: For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
Answer: 205
Problem: Evaluate $\int_{1}^{4} \frac{x-2}{(x^2+4)\sqrt{x}}\,dx$
Answer: 0
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?
Answer: 250
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: A regular $n$-gon $P_1P_2\ldots P_n$ satisfies $\angle P_1P_7P_8 = 178^\circ$. Compute $n$.
Answer: 630
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the largest positive integer $n$ such that the number $(2n)!$ ends with 10 more zeroes than the number $n!$.
Answer: 42
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.
Answer: -7
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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