Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.

You have $9$ colors of socks and $5$ socks of each type of color. Pick two socks randomly. What is the probability that they are the same color?

Determine all integers $ n>1$ for which the inequality \[ x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$.

Let $x<0.1$ be a positive real number. Let the [i]foury series[/i] be $4+4x+4x^2+4x^3+\dots$, and let the [i]fourier series[/i] be $4+44x+444x^2+4444x^3+\dots$. Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$.

A triple of positive integers $(a,b,c)$ is [i]brazilian[/i] if $$a|bc+1$$ $$b|ac+1$$ $$c|ab+1$$ Determine all the brazilian triples.

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

Consider a nonconstant arithmetic progression $a_1, a_2,\cdots, a_n,\cdots$. Suppose there exist relatively prime positive integers $p>1$ and $q>1$ such that $a_1^2, a_{p+1}^2$ and $a_{q+1}^2$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.

A merchant bought some goods at a discount of $ 20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $ 20 \%$ of the marked price and still make a profit of $ 20\%$ of the selling price. The percent of the list price at which he should mark them is: $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 120$

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

Let $ABC$ be a given triangle. The circumcircle of the triangle has radius $R$, the incircle has radius $r$, the longest side of the triangle is $a$, while the shortest altitude is $h$. Show that: $\frac{R}{r} > \frac{a}{h}$.

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$: a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$ b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?

When $ x^9\minus{}x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: $ \textbf{(A)}\ \text{more than 5} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

Evaluate the sum \[ \dfrac {1}{2 \lfloor \sqrt {1} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {2} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {3} \rfloor + 1} + \cdots + \dfrac {1}{2 \lfloor \sqrt {100} \rfloor + 1} \]

A rook stands at a corner of an $m \times n$ squared board. Two players move the rook in turn (vertically or horizontally through any numbers of squares). As the rook moves, it paints the squares that it visits (stopping or passing through). The rook is not allowed to pass through or stop at the painted squares. The player who cannot move, loses. Who has a guaranteed win: the first player (who starts the game) or the other, and how should he/she play? (B Begun)

A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$

Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences. How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2\equal{}40 \cdot 2009$ are there?

Let $m, n$ be positive integers. There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$. Consider the following game: Step 1. Nadech picks sets $B_1, B_2, ... , B_n$, where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$, and gives them to Yaya. Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$. Step 3. For each $i = 1, 2, . . . , n$, Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements. Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$, Yaya can always pick $S$ so that she ends up with more gold coins than Nadech

Let $P$ be a convex polygon, and let $n \ge 3$ be a positive integer. On each side of $P$, erect a regular $n$-gon that shares that side of $P$, and is outside $P$. If none of the interiors of these regular n-gons overlap, we call P $n$-[i]good[/i]. (a) Find the largest value of $n$ such that every convex polygon is $n$-[i]good[/i]. (b) Find the smallest value of $n$ such that no convex polygon is $n$-[i]good[/i].

Find all pairs of positive integers $m, n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously.

Let $n$ be a positive integer. Show that if p is prime dividing $5^{4n}-5^{3n}+5^{2n}-5^{n}+1$, then $p\equiv 1 \;(\bmod\; 4)$.

An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is $ \textbf{(A)}\ 2\pi \qquad\textbf{(B)}\ 2\pi+2 \qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 $

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$.