Let $A_1A_2...A_n$ ba a polygon. Prove that there is a convex polygon $B_1B_2...B_n$ such that $B_iB_{i + 1} = A_iA_{i + 1}$ for $i \in \{1, 2,...,n-1\}$ and $B_nB_1 = A_nA_1$ (some of the successive vertices of the polygon $B_1B_2...B_n$ can be colinear).

The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.

Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear. [i]Proposed by Mads Christensen, Denmark[/i]

Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$.

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

Let $n$ and $k$ be positive integers. A simple graph $G$ does not contain any cycle whose length be an odd number greater than $1$ and less than $ 2k + 1$. If $G$ has at most $n + \frac{(k-1) (n-1) (n+2)}{2}$ vertices, prove that the vertices of $G$ can be painted with $n$ colors in such a way that any edge of $G$ has its ends of different colors.

Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$ Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$

The sequence $\{a_n\}$ satisfies $a_1 = \frac{1}{2}$, $a_2 = \frac{3}{8}$, and $a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) (n \in \mathbb{N^*})$. $(1)$ Determine the general formula of the sequence $\{a_n\}$; $(2)$ Prove that for any positive integer $n$, there is $0 < a_n < \frac{1}{\sqrt{2n + 1}}$.

Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this? [asy] unitsize(12); draw((0,9)--(0,0)--(9,0)); label("time",(4.5,0),S); label("s",(0,7),W); label("p",(0,6),W); label("e",(0,5),W); label("e",(0,4),W); label("d",(0,3),W); label("(A)",(-1,9),NW); draw((0,4)--(4,4),dashed); label("M",(4,4),E); draw((0,8)--(4,8),linewidth(1)); label("N",(4,8),E); draw((15,9)--(15,0)--(24,0)); label("time",(19.5,0),S); label("s",(15,7),W); label("p",(15,6),W); label("e",(15,5),W); label("e",(15,4),W); label("d",(15,3),W); label("(B)",(14,9),NW); draw((15,4)--(19,4),dashed); label("M",(19,4),E); draw((15,8)--(23,8),linewidth(1)); label("N",(23,8),E); draw((30,9)--(30,0)--(39,0)); label("time",(34.5,0),S); label("s",(30,7),W); label("p",(30,6),W); label("e",(30,5),W); label("e",(30,4),W); label("d",(30,3),W); label("(C)",(29,9),NW); draw((30,4)--(34,4),dashed); label("M",(34,4),E); draw((30,2)--(34,2),linewidth(1)); label("N",(34,2),E); draw((0,-6)--(0,-15)--(9,-15)); label("time",(4.5,-15),S); label("s",(0,-8),W); label("p",(0,-9),W); label("e",(0,-10),W); label("e",(0,-11),W); label("d",(0,-12),W); label("(D)",(-1,-6),NW); draw((0,-11)--(4,-11),dashed); label("M",(4,-11),E); draw((0,-7)--(2,-7),linewidth(1)); label("N",(2,-7),E); draw((15,-6)--(15,-15)--(24,-15)); label("time",(19.5,-15),S); label("s",(15,-8),W); label("p",(15,-9),W); label("e",(15,-10),W); label("e",(15,-11),W); label("d",(15,-12),W); label("(E)",(14,-6),NW); draw((15,-11)--(19,-11),dashed); label("M",(19,-11),E); draw((15,-13)--(23,-13),linewidth(1)); label("N",(23,-13),E);[/asy]

Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.

Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that (a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and (a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.

Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that \begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}

Don't have original wording: In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$ [asy] import geometry; unitsize(2cm); real arg(pair p) { return atan2(p.y, p.x) * 180/pi; } pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C; pair t(pair p) { return rotate(-arg(dir(B--C)))*p; } path t(path p) { return rotate(-arg(dir(B--C)))*p; } void d(path p, pen q = black+linewidth(1.5)) { draw(t(p),q); } void o(pair p, pen q = 5+black) { dot(t(p),q); } void l(string s, pair p, pair d) { label(s, t(p),d); } d(A--B--C--cycle); d(A--D); d(B--E); o(A); o(B); o(C); o(D); o(E); o(G); l("$A$",A,N); l("$B$",B,SW); l("$C$",C,SE); l("$D$",D,S); l("$E$",E,NE); l("$G$",G,NW); [/asy] $\textbf{(A)}44~\textbf{(B)}48~\textbf{(C)}52~\textbf{(D)}56~\textbf{(E)}60$

How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.) $ \mathrm{(A) \ } 16 \qquad \mathrm{(B) \ } 18 \qquad \mathrm {(C) \ } 64 \qquad \mathrm{(D) \ } 100 \qquad \mathrm{(E) \ } 125$

Let \( p_1, p_2, \cdots, p_{2025} \) be real numbers. For \( 1 \leq i \leq 2025 \), let \[\{a_n^{(i)}\}_{n \geq 0}\] be an infinite real sequence satisfying \[a_0^{(i)} = 0.\] It is known that: (1) \[a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(2025)}\] are not all zero. (2) For any integer \( n \geq 0 \) and any \( 1 \leq i \leq 2025 \), the following holds: \[p_i \cdot a_n^{(i+1)} = a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)},\] where the sequence \[\{a_n^{(2026)}\}\] satisfies \[a_n^{(2026)} = a_n^{(1)}, \, n = 0, 1, 2, \cdots.\] Prove that there exists a positive real number \( r \) such that for infinitely many positive integers \( n \), \[\max \left\{ |a_n^{(1)}|, |a_n^{(2)}|, \cdots, |a_n^{(2025)}|\right\} \geq r.\]

Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally? $ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$

[u]Round 1[/u] [b]p1.[/b] Tom and Jerry stole a chain of $7$ sausages and are now trying to divide the bounty. They take turns biting the sausages at one of the connections. When one of them breaks a connection, he may eat any single sausages that may fall out. Tom takes the first bite. Each of them is trying his best to eat more sausages than his opponent. Who will succeed? [b]p2. [/b]The King of the Mountain Dwarves wants to light his underground throne room by placing several torches so that the whole room is lit. The king, being very miserly, wants to use as few torches as possible. What is the least number of torches he could use? (You should show why he can't do it with a smaller number of torches.) This is the shape of the throne room: [img]https://cdn.artofproblemsolving.com/attachments/b/2/719daafd91fc9a11b8e147bb24cb66b7a684e9.png[/img] Also, the walls in all rooms are lined with velvet and do not reflect the light. For example, the picture on the right shows how another room in the castle is partially lit. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0f6971274e8c2ff3f2d0fa484b567ff3d631fb.png[/img] [b]p3.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p4.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p5.[/b] There are $40$ piles of stones with an equal number of stones in each. Two players, Ann and Bob, can select any two piles of stones and combine them into one bigger pile, as long as this pile would not contain more than half of all the stones on the table. A player who can’t make a move loses. Ann goes first. Who wins? [u]Round 2[/u] [b]p6.[/b] In a galaxy far, far away, there is a United Galactic Senate with $100$ Senators. Each Senator has no more than three enemies. Tired of their arguments, the Senators want to split into two parties so that each Senator has no more than one enemy in his own party. Prove that they can do this. (Note: If $A$ is an enemy of $B$, then $B$ is an enemy of $A$.) [b]p7.[/b] Harry has a $2012$ by $2012$ chessboard and checkers numbered from $1$ to $2012 \times 2012$. Can he place all the checkers on the chessboard in such a way that whatever row and column Professor Snape picks, Harry will be able to choose three checkers from this row and this column such that the product of the numbers on two of the checkers will be equal to the number on the third? [img]https://cdn.artofproblemsolving.com/attachments/b/3/a87d559b340ceefee485f41c8fe44ae9a59113.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.

Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo $n$. If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Yang Liu[/i]

Find the sum of the $2007$ roots of \[(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007).\]

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities \[ 2\le a^2+b^2+c^2+d^2\le 4. \]

For every positive $a_1, a_2, \dots, a_6$, prove the inequality \[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region? [asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy] $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Let $p$ be an odd prime. A degree $d$ polynomial $f$ with non-negative integer coefficients less than $p$ is called $p-floppy$ if the coefficients of $f(x)f(-x) - f(x^2)$ are all divisible by $p$ and if exactly $d$ entries in the sequence $(f(0), f(1), f(2), \dots , f(p-1))$ are divisible by $p$. How many non-constant $61$-floppy polynomials are there?