Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. ([I]Two colorings are different as long as they differ at some vertices. [/i])

Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that (a) $xy+yz+xz\ge 9xyz$; (b) $xy+yz+xz<\frac{1}{4}+3xyz$;

Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \] for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \] for every $n \in \mathbb{N}$.

Is it possible to divide a circle by three chords, different from diameters, into several equal parts?

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

[b]p1.[/b] Calculate $\left( \frac12 + \frac13 + \frac14 \right)^2$. [b]p2.[/b] Find the $2010^{th}$ digit after the decimal point in the expansion of $\frac17$. [b]p3.[/b] If you add $1$ liter of water to a solution consisting of acid and water, the new solutions will contain of $30\%$ water. If you add another $5$ liters of water to the new solution, it will contain $36\frac{4}{11}\%$ water. Find the number of liters of acid in the original solution. [b]p4.[/b] John places $5$ indistinguishable blue marbles and $5$ indistinguishable red marbles into two distinguishable buckets such that each bucket has at least one blue marble and one red marble. How many distinguishable marble distributions are possible after the process is completed? [b]p5.[/b] In quadrilateral $PEAR$, $PE = 21$, $EA = 20$, $AR = 15$, $RE = 25$, and $AP = 29$. Find the area of the quadrilateral. [b]p6.[/b] Four congruent semicircles are drawn within the boundary of a square with side length $1$. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$, where $a$ and $b$ are positive rational numbers. Compute $a + b$. [b]p7.[/b] Let $x$ and $y$ be two numbers satisfying the relations $x\ge 0$, $y\ge 0$, and $3x + 5y = 7$. What is the maximum possible value of $9x^2 + 25y^2$? [b]p8.[/b] In the Senate office in Exie-land, there are $6$ distinguishable senators and $6$ distinguishable interns. Some senators and an equal number of interns will attend a convention. If at least one senator must attend, how many combinations of senators and interns can attend the convention? [b]p9.[/b] Evaluate $(1^2 - 3^2 + 5^2 - 7^2 + 9^2 - ... + 2009^2) -(2^2 - 4^2 + 6^2 - 8^2 + 10^2- ... + 2010^2)$. [b]p10.[/b] Segment $EA$ has length $1$. Region $R$ consists of points $P$ in the plane such that $\angle PEA \ge 120^o$ and $PE <\sqrt3$. If point $X$ is picked randomly from the region$ R$, the probability that $AX <\sqrt3$ can be written in the form $a - \frac{\sqrt{b}}{c\pi}$ , where $a$ is a rational number, $b$ and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find the ordered triple $(a, b, c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets $A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.

Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$ in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent. (D.Shvetsov, A.Zaslavsky)

Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?

Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

The letters $AAABBCC$ are arranged in random order. The probability no two adjacent letters will be the same is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that \[ \sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}. \] [i]Note:[/i] $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

Let $a, b, c$ be positive real numbers. Prove the inequality $(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$. [i]Proposed by Tajikistan[/i]

The Serpent Gorynych has $1976$ heads. The fabulous hero can cut down $33, 21, 17$ or $1$ head with one blow of the sword, but at the same time, the Serpent grows, respectively, $48, 0, 14$ or $349$ heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?

In a handball tournament with $n$ teams, each team played against other teams exactly once. In each game, the winner got $2$ points, the loser got $0$ point, and each team got $1$ point if there was a tie. After the tournament ended, each team had different score than the others, and the last team defeated the first three teams. What is the least possible value of $n$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of above} $

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

Determine all positive integers expressible, for every integer $ n \geq 3 $, in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where $ a_1, a_2, \ldots, a_n $ are pairwise distinct positive integers.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3,\cdots, n$. For how many integers $n$ with $1 \le n \le 22$ is $k_n<L_n$? $\textbf{(A)} ~0 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~10 $

On the plane we consider $70$ points $A_1,A_2,...,A_{70}$ with integer coodinates. Suppose each pooints has weight $1$ and the centers of gravity of the triangles $ A_1A_2A_3$, $A_2A_3A_4$, $..$., $A_{68}A_{69}A_{70}$, $A_{69}A_{70}A_{1}$, $A_{70}A_{1}A_{2}$ have integer coodinates. Prove that the centers of gravity of any triple $A_i,A_j,...,A_{k}$ has integer coodinates.

Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below: [img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img] Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses. (a) Show that, even if Chris and Michael play cooperatively, the game will come to an end. (b) Which player has a winning strategy?

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that \[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]