ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A robot is standing on the bottom left vertex $(0,0)$ of a $5 \times 5$ grid and wants to go to $(5,5)$, only moving to the right $(a,b) \mapsto (a+1,b)$ or upward $(a,b) \mapsto (a,b+1)$. However, this robot is not programmed perfectly and sometimes takes the upper-left diagonal path $(a,b) \mapsto (a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0 \leq a,b \leq 5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
Answer: 1650
Problem: Let $a, b, c,$ and $d$ be real numbers such that:
\[
a + b = c + d + 12
\]
and
\[
ab + cd - 28 = bc + ad.
\]
Find the minimum possible value of $a^4 + b^4 + c^4 + d^4$.
Answer: 772
Problem: About the pentagon $ABCDE$, we know that $AB = BC = CD = DE$, $\angle C = \angle D = 108^\circ$, $\angle B = 96^\circ$. Find the value in degrees of $\angle E$.
Answer: 102
Problem: 2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it. It is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right. Suppose the new numbers on the ducklings sum to 1000. What is the maximum possible sum of the original numbers on all 2018 slips?
Answer: 2000
Problem: A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.
Answer: 12
Problem: Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times 30 \times 30$ cube. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
Answer: 4500
Problem: How many solutions of the equation $\tan x = \tan(\tan x)$ are on the interval $0 \le x \le \tan^{-1}(942)$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written as $\arctan$.)
Answer: 300
Problem: For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that a **good** number. Conversely, if the number of $3$'s in its digits is greater than the number of $2$'s, we call that a **bad** number. For example, the number $2023$ has two $2$'s and one $3$, so $2023$ is a good number. However, in the number $123$, both the number of $2$'s and $3$'s are one, so $123$ is neither a good number nor a bad number.  

Find the difference between the number of good numbers and bad numbers among the positive integers not greater than $2023$.
Answer: 22
Problem: Find the number of positive integers $n \le 2014$ such that there exists an integer $x$ satisfying the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
Answer: 503
Problem: Let $E$ and $F$ be the respective midpoints of $BC$ and $CD$ in a convex quadrilateral $ABCD$. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.
Answer: 6
Problem: Let $ABCDA'B'C'D'$ be a rectangular parallelepiped, where $ABCD$ is the lower face and $A, B, C$, and $D$ are below $A', B', C'$, and $D'$, respectively. The parallelepiped is divided into eight parts by three planes parallel to its faces. For each vertex $P$, let $V_P$ denote the volume of the part containing $P$. Given that $V_A = 40$, $V_C = 300$, $V_{B'} = 360$, and $V_{C'} = 90$, find the volume of $ABCDA'B'C'D'$.
Answer: 2015
Problem: Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1 \leq n \leq 100$, find the maximum value of $\phi(n^2+2n) - \phi(n^2)$.
Answer: 72
Problem: How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$?
Answer: 2002
Problem: The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:
Answer: 11
Problem: Given that the expected amount of $1$s in a randomly selected $2021$-digit number is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Answer: 1828
Problem: Find all positive integers $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, such that $a^2 + b + 3$ is not divisible by the cube of any prime, and 
$$ n = \frac{ab + 3b + 8}{a^2 + b + 3}. $$
Answer: 2
Problem: Let $ABCD$ be a rectangle where $AB + BC + CD = 20$ and $AE = 9$, with $E$ being the midpoint of the side $BC$. Find the area of the rectangle.
Answer: 19
Problem: How many integers between 123 and 321 inclusive have exactly two digits that are 2?
Answer: 18
Problem: For reals $x \ge 3$, let $f(x)$ denote the function \[ f(x) = \frac{-x + x\sqrt{4x-3}}{2}. \]
Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[ a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3}{a_i^2 + a_ia_{i+1} + a_{i+1}^2}. \]
Answer: 4025
Problem: In a forest, there are five trees labeled $A$, $B$, $C$, $D$, and $E$ arranged in that order on a straight line. At the midpoint of $AB$, there is a daisy; at the midpoint of $BC$, there is a rose bush; at the midpoint of $CD$, there is a jasmine; and at the midpoint of $DE$, there is a carnation. The distance between $A$ and $E$ is $28$ meters, and the distance between the daisy and the carnation is $20$ meters. Calculate the distance between the rose bush and the jasmine.
Answer: 6
Problem: Circle $S_1$ has a radius of $5$. Circle $S_2$ has a radius of $7$ and its center lies on $S_1$. Circle $S_3$ has an integer radius and its center lies on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
Answer: 11
Problem: Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?
Answer: 3
Problem: Find the sum of all the prime numbers less than $100$ which are one more than a multiple of 6.
Answer: 527
Problem: Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0, 1, 2, \ldots, 12$ such that the value of the resulting expression is divisible by 5.
Answer: 816
Problem: Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 59
Problem: Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.
Answer: 17
Problem: Find the number of positive integers less than 100 that are divisors of 300.
Answer: 15
Problem: Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
Answer: 2019
Problem: A positive integer is said to be a "palindrome" if it reads the same from left to right as from right to left. For example, 2002 is a palindrome. Find the sum of all 4-digit palindromes.
Answer: 495000
Problem: We color some unit squares in a $99 \times 99$ square grid with one of $5$ given distinct colors, such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
Answer: 1900
Problem: Find \( \sum_{k=1}^\infty \frac{k^2-2}{(k+2)!} \).
Answer: 0
Problem: The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $\frac{i}{p}$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$. How many pages did you turn in?
Answer: 4028
Problem: Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$.
Answer: 16
Problem: Determine the smallest positive integer $n$ whose prime factors are all greater than 18, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.
Answer: 1843
Problem: How many positive integers $n < 2024$ are divisible by $\lfloor \sqrt{n} \rfloor - 1$?
Answer: 91
Problem: For $k \in \{0, 1, \ldots, 9\}$, let $\epsilon_k \in \{-1, 1\}$. If the minimum possible value of $\sum_{i=1}^9 \sum_{j=0}^{i-1} \epsilon_i \epsilon_j 2^{i+j}$ is $m$, find $|m|$.
Answer: 174762
Problem: A set $S$ consisting of $2019$ different positive integers has the following property: the product of every $100$ elements of $S$ is a divisor of the product of the remaining $1919$ elements. What is the maximum number of prime numbers that $S$ can contain?
Answer: 1820
Problem: Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA = 13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB + AC$ given that $BC = 7$.
Answer: 17
Problem: Triangle $ABC$ has a right angle at $A$, $AB=20$, and $AC=21$. Circles $\omega_A$, $\omega_B$, and $\omega_C$ are centered at $A$, $B$, and $C$ respectively and pass through the midpoint $M$ of $\overline{BC}$. $\omega_A$ and $\omega_B$ intersect at $X\neq M$, and $\omega_A$ and $\omega_C$ intersect at $Y\neq M$. Find $XY$.
Answer: 29
Problem: Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $b$, $c$, $d$ are non-negative integers.
Answer: 11
Problem: There is a polynomial $P$ such that for every real number $x$,  
\[  
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).  
\]  
When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
Answer: 341
Problem: When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number?
Answer: 27
Problem: When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ cm$^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ cm$^3$. What is the length (in cm) of the hypotenuse of the triangle?
Answer: 26
Problem: In $\triangle PQR$, $\angle Q + 10^\circ = \angle R$. Let $M$ be the midpoint of $\overline{QR}$. If $m\angle PMQ = 100^\circ$, then find the measure of $\angle Q$ in degrees.
Answer: 45
Problem: Let $a \diamond b = ab - 4(a + b) + 20$. Evaluate 
\[ 1 \diamond (2 \diamond (3 \diamond (\cdots (99 \diamond 100) \cdots))). \]
Answer: 4
Problem: Let \[ \frac{p}{q} = \frac{2017}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\cdots}}}}}}}}}}\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Answer: 10091
Problem: Let $n$ be an integer. Dayang is given $n$ sticks of lengths $1, 2, 3, \ldots, n$. She may connect the sticks at their ends to form longer sticks, but cannot cut them. She wants to use all these sticks to form a square.

For example, for $n = 8$, she can make a square of side length $9$ using these connected sticks: $1 + 8$, $2 + 7$, $3 + 6$, and $4 + 5$. 

How many values of $n$, with $1 \le n \le 2018$, allow her to do this?
Answer: 504
Problem: Find the number of positive integers less than or equal to $2020$ that are relatively prime to $588$.
Answer: 577
Problem: Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.
Answer: 109
Problem: Determine the number of ten-digit positive integers with the following properties:

- Each of the digits $0, 1, 2, \ldots, 8, 9$ is contained exactly once.
- Each digit, except $9$, has a neighboring digit that is larger than it.

(Note: For example, in the number $1230$, the digits $1$ and $3$ are the neighboring digits of $2$, while $2$ and $0$ are the neighboring digits of $3$. The digits $1$ and $0$ have only one neighboring digit.)
Answer: 256
Problem: Find the area enclosed by the graph of $|x| + |2y| = 12$.
Answer: 144
Problem: Determine the largest integer $N$ for which there exists a $6 \times N$ table $T$ that has the following properties:

- Every column contains the numbers $1, 2, \ldots, 6$ in some ordering.
- For any two columns $i \ne j$, there exists a row $r$ such that $T(r,i) = T(r,j)$.
- For any two columns $i \ne j$, there exists a row $s$ such that $T(s,i) \ne T(s,j)$.
Answer: 120
Problem: The integers $a$, $b$, and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?
Answer: 43
Problem: Find the maximal possible number of integers you can choose from the set \( \{1, 2, \ldots, 100\} \) such that no product of any non-empty subset of these numbers is a perfect square.
Answer: 25
Problem: The smallest three positive proper divisors of an integer $n$ are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.
Answer: 42
Problem: Solve for $x$ in the congruence: $$ x \equiv \left(\sum_{k=1}^{2007} k \right) \mod{2016}, $$ where $0 \leq x \leq 2015$.
Answer: 1044
Problem: Two boys are given a bag of potatoes, each bag containing $150$ tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero number of tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he had in his bag before any of his previous moves (if there were such moves). So, with his first move, a boy can move any non-zero quantity, and with his fifth move, a boy can move $200$ tubers, if before his first, second, third, and fourth move, the numbers of tubers in his bag were less than $200$. What is the maximal total number of moves the two boys can do?
Answer: 9
Problem: Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isosceles triangle with $a$, $b$, and $c$ as the lengths of sides. Determine how many possible three-digit numbers $N$ there are. 

($N=\overline{abc}$ means that $a$, $b$, and $c$ are digits of $N$, and not $N=a \times b \times c$.)
Answer: 165
Problem: Find the least positive integer $n$ such that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors.
Answer: 135
Problem: A rectangle $ABCD$ is divided into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints lie on the sides of $ABCD$. Given that the smallest three rectangles have areas of $48$, $18$, and $12$, determine the area of rectangle $ABCD$.
Answer: 150
Problem: Let $p(n)$ denote the product of the decimal digits of a positive integer $n$. Compute the sum $p(1) + p(2) + \ldots + p(2001)$.
Answer: 184320
Problem: Compute the last two digits of $9^{2020} + 9^{2020^2} + \cdots + 9^{2020^{2020}}$.
Answer: 20
Problem: How many subsets of \( \{1, 2, 3, 4, \ldots, 12\} \) contain exactly one prime number?
Answer: 640
Problem: Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided into exactly 2007 smaller cubes, each with integer side lengths.
Answer: 13
Problem: Suppose that a polynomial of the form $p(x) = x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?
Answer: 1005
Problem: Define the polynomials $P_0, P_1, P_2, \ldots$ by:
\[ P_0(x) = x^3 + 213x^2 - 67x - 2000 \]
\[ P_n(x) = P_{n-1}(x-n), \quad n \in \mathbb{N} \]
Find the coefficient of $x$ in $P_{21}(x)$.
Answer: 61610
Problem: To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate, and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed?
Answer: 1092
Problem: David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?
Answer: 5
Problem: Define $f(x) = -\frac{2x}{4x+3}$ and $g(x) = \frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x) = g(f(h^n(x)))$, where $h^1(x) = g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k \cdot h^{100}(k)$ can be written in the form $a b^c$, for some integers $a, b, c$ where $c$ is as maximal as possible and $b \ne 1$, find $a + b + c$.
Answer: 128
Problem: Let $r$, $s$, $t$ be the roots of the polynomial $x^3 + 2x^2 + x - 7$. Then, \[ \left(1 + \frac{1}{(r+2)^2}\right)\left(1 + \frac{1}{(s+2)^2}\right)\left(1 + \frac{1}{(t+2)^2}\right) = \frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Answer: 4181
Problem: A natural number $n$ is considered \textit{good} if it can be expressed as the sum of $r$ consecutive positive integers, where $r \geq 2$. Determine the number of good numbers within the set $\{1, 2, \dots, 100\}$.
Answer: 93
Problem: In the quadrilateral $ABCD$, the points $E$, $F$, and $K$ are midpoints of the sides $AB$, $BC$, and $AD$, respectively. It is known that $KE \perp AB$, $KF \perp BC$, and the angle $\angle ABC = 118^\circ$. Find $\angle ACD$ (in degrees).
Answer: 62
Problem: How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
Answer: 233
Problem: Compute the number of positive integer divisors of $100000$ that do not contain the digit $0$.
Answer: 11
Problem: Find the sum of all positive integers $n$ such that $$ \frac{n+11}{\sqrt{n-1}} $$ is an integer.
Answer: 216
Problem: Suppose $x, y, z$, and $w$ are positive reals such that:
\[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \]
\[ xz + yw = 30. \]
Find the largest possible value of $(xy + wz)^2$.
Answer: 960
Problem: For some integers $b$ and $c$, neither of the equations below have real solutions:
\[
2x^2+bx+c=0
\]
\[
2x^2+cx+b=0.
\]
What is the largest possible value of $b+c$?
Answer: 14
Problem: Suppose $x$ is a real number satisfying the equation:  
\[ x^2 - 990x + 1 = (x + 1)\sqrt{x} \]  
Find the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \).
Answer: 32
Problem: What is the smallest positive integer that is divisible by $225$ and has only the digits $1$ and $0$?
Answer: 11111111100
Problem: A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border forms a rectangle that is twice as long as it is wide. How many inches long is that rectangle?
Answer: 24
Problem: In $\Delta ABC$, $\angle ABC = 120^\circ$. The internal bisector of $\angle B$ meets $AC$ at $D$. If $BD = 1$, find the smallest possible value of $4BC + AB$.
Answer: 9
Problem: When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer?
Answer: 14
Problem: The six sides of a convex hexagon $A_1 A_2 A_3 A_4 A_5 A_6$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If $N$ is the number of colorings such that every triangle $A_i A_j A_k$, where $1 \leq i<j<k \leq 6$, has at least one red side, find the sum of the squares of the digits of $N$.
Answer: 94
Problem: Consider the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \in \{-1, 0, 1\}$ and $(x,y) \neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints?
Answer: 14
Problem: A three-digit number $x$ in base 10 has a units-digit of 6. When $x$ is written in base 9, the second digit of the number is 4, and the first and third digits are equal in value. Compute $x$ in base 10.
Answer: 446
Problem: What is the largest positive integer \( n \) satisfying the following inequality:
\[ n^{2006} < 7^{2007} \]
Answer: 7
Problem: Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value of $mn$.
Answer: 2350
Problem: We will say that a rearrangement of the letters of a word has no *fixed letters* if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance, $HBRATA$ is a rearrangement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed letters does $BHARAT$ have? (The two $A$s are considered identical.)
Answer: 84
Problem: How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
Answer: 19
Problem: You have a $1 \times 2024$ grid of squares in a column, vertices labelled with coordinates $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. When a weed is attempting to be placed at coordinates $(x,y)$, it will be placed with a $50\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it, otherwise the attempt will fail with probability $1$. The placement attempts are made in the following order: For each vertex with $x$ coordinate $0$, attempt a placement for each vertex starting from $y$ coordinate $0$, incrementing by $1$ until $2024$. Then, attempts will be made on the vertices with $x$ coordinate $1$ in the same fashion. Each placement attempt is made exactly once.

The probability that a weed appears on $(1,2024)$ after placing the weed at $(0,0)$ and attempting to place weeds on every vertex is $p$. Estimate $9p \cdot 2^{2025}$ to the nearest integer.
Answer: 18
Problem: Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$.
Answer: 3162
Problem: A five-digit positive integer $abcde_{10}$ ($a \neq 0$) is said to be a *range* if its digits satisfy the inequalities $a < b > c < d > e$. For example, $37452$ is a range. How many ranges are there?
Answer: 8622
Problem: Evaluate the sum: 
\[
\frac{3!+4!}{2(1!+2!)} + \frac{4!+5!}{3(2!+3!)} + \cdots + \frac{12!+13!}{11(10!+11!)}
\]
Answer: 95
Problem: Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$.
Answer: 24
Problem: Let $\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \ge 2$ and all $n \ge 3$, we have that \[\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.\] Find the largest possible value of $m$.
Answer: 0