The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8,15,8,8,8,6,8,$ and $29$ as shown. All of its angles are right angles. Turn this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$. [asy] size(200); defaultpen(linewidth(2)); pen qq=font("phvb"); pair rectangle[] = {origin,(0,-8),(15,-8),(15,-16),(23,-16),(23,-8),(29,-8),(29,0)}; string point[] = {"A","B","C","D","E","F","G","H"}; int dirlbl[] = {135,225,225,225,315,315,315,45}; string value[] = {"8","15","8","8","8","6","8","29"}; int direction[] = {0,90,0,90,180,90,180,270}; for(int i=0;i<=7;i=i+1) { draw(rectangle[i]--rectangle[(i+1) % 8]); label(point[i],rectangle[i],dir(dirlbl[i]),qq); label(value[i],(rectangle[i]+rectangle[(i+1) % 8])/2,dir(direction[i]),qq); } [/asy]

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.

Vaggelis has a box that contains $2015$ white and $2015$ black balls. In every step, he follows the procedure below: He choses randomly two balls from the box. If they are both blacks, he paints one white and he keeps it in the box, and throw the other one out of the box. If they are both white, he keeps one in the box and throws the other out. If they are one white and one black, he throws the white out, and keeps the black in the box. He continues this procedure, until three balls remain in the box. He then looks inside and he sees that there are balls of both colors. How many white balls does he see then, and how many black?

$P$ is inside rectangle $ABCD$. $PA = 2$, $PB = 3$, and $PC = 10$. Find $PD$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N_{21}$ be the answer to question 21. Suppose a jar has $3N_{21}$ colored balls in it: $N_{21}$ red, $N_{21}$ green, and $N_{21}$ blue balls. Jonathan takes one ball at a time out of the jar uniformly at random without replacement until all the balls left in the jar are the same color. Compute the expected number of balls left in the jar after all balls are the same color. [b]p20.[/b] Let $N_{19}$ be the answer to question 19. For every non-negative integer $k$, define $$f_k(x) = x(x - 1) + (x + 1)(x - 2) + ...+ (x + k)(x - k - 1),$$ and let $r_k$ and $s_k$ be the two roots of $f_k(x)$. Compute the smallest positive integer $m$ such that $|r_m - s_m| > 10N_{19}$. [b]p21.[/b] Let $N_{20}$ be the answer to question 20. In isosceles trapezoid $ABCD$ (where $\overline{BC}$ and $\overline{AD}$ are parallel to each other), the angle bisectors of $A$ and $D$ intersect at $F$, and the angle bisectors of points $B$ and $C$ intersect at $H$. Let $\overline{BH}$ and $\overline{AF}$ intersect at $E$, and let $\overline{CH}$ and $\overline{DF}$ intersect at $G$. If $CG = 3$, $AE = 15$, and $EG = N_{20}$, compute the area of the quadrilateral formed by the four tangency points of the largest circle that can fit inside quadrilateral $EFGH$.

Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$): \[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

On the board, we write the integers $1, 2, 3, \dots, 2019$. At each minute, we pick two numbers on the board $a$ and $b$, delete them, and write down the number $s(a + b)$ instead, where $s(n)$ denotes the sum of the digits of the integer $n$. Let $N$ be the last number on the board at the end. [list=a] [*] Is it possible to get $N = 19$? [*] Is it possible to get $N = 15$? [/list]

We divide the plane in squares shaped of side 1, tracing straight lines parallel bars to the coordinate axles. Each square is painted of black white or. To each as, we recolor all simultaneously squares, in accordance with the following rule: each square $Q$ adopts the color that more appears in the configuration of five squares indicated in the figure. The recoloration process is repeated indefinitely. Determine if exists an initial coloration with black a finite amount of squares such that always has at least one black square, not mattering how many seconds if had passed since the beginning of the process.

Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?

Oh no! While playing Mario Party, Theo has landed inside the Bowser Zone. If his next roll is between $1$ and $5$ inclusive, Bowser will shoot his ``Zero Flame" that sets a player's coin and star counts to zero. Fortunately, Theo has a double dice block, which lets him roll two fair $10$-sided dice labeled $1$-$10$ and take the sum of the rolls as his "roll". If he uses his double dice block, what is the probability he escapes the Bowser zone without losing his coins and stars? [i]Proposed by Connor Gordon[/i]

How many ways are there to cut a $1$ by $1$ square into $8$ congruent polygonal pieces such that all of the interior angles for each piece are either $45$ or $90$ degrees? Two ways are considered distinct if they require cutting the square in different locations. In particular, rotations and reflections are considered distinct.

Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to offer. His many hobbies include playing soccer, volleyball, and the many forms of chess. We hope that he likes the questions that we’ve dedicated to him! [b]p1.[/b] Ephram is scared of freshmen boys. How many ways can Ephram and $4$ distinguishable freshmen boys sit together in a row of $5$ chairs if Ephram does not want to sit between $2$ freshmen boys? [b]p2.[/b] Ephram, who is a chess enthusiast, is trading chess pieces on the black market. Pawns are worth $\$100$, knights are worth $\$515$, and bishops are worth $\$396$. Thirty-four minutes ago, Ephrammade a fair trade: $5$ knights, $3$ bishops, and $9$ rooks for $8$ pawns, $2$ rooks, and $11$ bishops. Find the value of a rook, in dollars. [b]p3.[/b] Ephramis kicking a volleyball. The height of Ephram’s kick, in feet, is determined by $$h(t) = - \frac{p}{12}t^2 +\frac{p}{3}t ,$$ where $p$ is his kicking power and $t$ is the time in seconds. In order to reach the height of $8$ feet between $1$ and $2$ seconds, Ephram’s kicking power must be between reals $a$ and $b$. Find is $100a +b$. [b]p4.[/b] Disclaimer: No freshmen were harmed in the writing of this problem. Ephram has superhuman hearing: He can hear sounds up to $8$ miles away. Ephramstands in the middle of a $8$ mile by $24$ mile rectangular grass field. A freshman falls from the sky above a point chosen uniformly and randomly on the grass field. The probability Ephram hears the freshman bounce off the ground is $P\%$. Find $P$ rounded to the nearest integer. [img]https://cdn.artofproblemsolving.com/attachments/4/4/29f7a5a709523cd563f48176483536a2ae6562.png[/img] [b]p5.[/b] Ephram and Brandon are playing a version of chess, sitting on opposite sides of a $6\times 6$ board. Ephram has $6$ white pawns on the row closest to himself, and Brandon has $6$ black pawns on the row closest to himself. During each player’s turn, their only legal move is to move one pawn one square forward towards the opposing player. Pawns cannot move onto a space occupied by another pawn. Players alternate turns, and Ephram goes first (of course). Players take turns until there are no more legal moves for the active player, at which point the game ends. Find the number of possible positions the game can end in. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was $\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$

Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right)$ is an integer. (a) Prove that $ \sqrt{k}$ is rational. (b) Prove that $ \sqrt{k}$ is a positive integer.

It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?

Written in mm/dd format, a date is called [i]cute[/i] if the month is divisible by the day. For example, the date [i]cute[/i] is a [i]cute[/i] date because $8$ is divisible by $2$. Find the number of [i]cute[/i] dates in a year. [i]Proposed by Andy Xu[/i]

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true. [i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i]