Construct the quadrilateral $ ABCD $ given the lengths of the sides $ AB $ and $ CD $ and the angles of the quadrilateral.

Let $ n\geq 2$ be a positive integer. Find the [u]maximum[/u] number of segments with lenghts greater than $ 1,$ determined by $ n$ points which lie on a closed disc with radius $ 1.$

If $\sin\alpha>0,\cos\alpha<0,\sin\frac{\alpha}{3}>\cos\frac{\alpha}{3}$, then the range value of $\frac{\alpha}{3}$ is $\text{(A)}\left(2k\pi+\frac{\pi}{6},2k\pi+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(B)}\left(\frac{2k\pi}{3}+\frac{\pi}{6},\frac{2k\pi}{3}+\frac{\pi}{3}\right),k\in\mathbb{Z}$ $\text{(C)}\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$ $\text{(D)}\left(2k\pi+\frac{\pi}{4},2k\pi+\frac{\pi}{3}\right)\cup\left(2k\pi+\frac{5\pi}{6},2k\pi+\pi\right),k\in\mathbb{Z}$

Define a sequence $\left\{a_{i}\right\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \geq 1.$ Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i} ?$

Consider a grid board of $n \times n$, with $n \ge 5$. Two unit squares are said to be far [i]apart [/i] if they are neither on the same row nor on consecutive rows and neither in the same column nor in consecutive columns. Take $3$ rectangles with vertices and sides on the points and lines of board so that if two unit squares belong to different rectangles, then they are [i]apart [/i]. In how many ways is it possible to do this?

Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside the grid. In how many ways can the squares be chosen so that the final $1 \times 1$ square is the center of the original $5 \times 5$ grid? [i]Proposed by Nancy Kuang[/i]

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact. [i]Proposed by Mahyar Sefidgaran[/i]

For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$. Prove that the length of segment $NM$ doesn’t depend on point $C$.

Let $x$ and $y$ be distinct positive integers below $15$. For any two distinct numbers $a, b$ from the set $\{2, x,y\}$, $ab + 1$ is always a positive square. Find all possible values of the square $xy + 1$.

Let $S(n)$ be the sum of digits for any positive integer n (in decimal notation). Let $N=\displaystyle\sum_{k=10^{2003}}^{10{^{2004}-1}} S(k)$. Determine $S(N)$.

Let $\Gamma_1$ be a circle of radius $6$, and let $\Gamma_2$ be a circle of radius $1$. Next, let the circles be internally tangent at point $P$, and let $AP$ be a diameter of circle $\Gamma_1$. Finally, let $Y$ be a point on $\Gamma_2$ such that $AY$ is tangent to it. Compute the length of $PY$.

A positive integer $n$ is given. Consider all polynomials $P(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_0$, whose coefficients are nonnegative integers, not exceeding $100$. Call $P$ [i]reducible[/i] if it can be factored into two non-constant polynomials with nonnegative integer coeffiecients, and [i]irreducible[/i] otherwise. Prove that the number of [i]irreducible[/i] polynomials is at least twice as big as the number of [i]reducible[/i] polynomials. [i]D. Zmiaikou[/i]

There are $n$ coins aligned in a row. In each step, it is allowed to choose a coin with the tail up (but not one of the outermost markers), remove it and reverse the closest coin to the left and the closest coin to the right of it. Initially, all the coins have tails up. Prove that one can achieve the state with only two coins remaining if and only if $n-1$ is not divisible by $3$.

The seven dwarves are at work on day when they find a large pile of diamonds. They want to split the diamonds evenly among them, but find that they would need to take away one diamond to split into seven equal piles. They are still arguing about this when they get home, so Snow White sends them to bed without supper. In the middle of the night, Sneezy wakes up and decides that he should get the extra diamond. So he puts one diamond aside, splits the remaining ones in to seven equal piles, and takes his pile along with the extra diamond. Then, he runs off with the diamonds. His sneeze wakes up Grumpy, who, thinking along the same lines, removes one diamond, divides the remainder into seven equal piles, and runs off. Finally, Sleepy, for the first time in his life, wakes up before sunrise and performs the same operation. When the remaining four dwarves arise, they find that the remaining diamonds can be split into $5$ equal piles. Doc suggests that Snow White should get a share, so they have no problem splitting the remaining diamonds. Happy, Dopey, Bashful, Doc, and Snow White live happily ever after. What’s the smallest possible number of diamonds that the dwarves could have started out with?

Let $A$ be a set of real numbers satisfying the following: (a) $\sqrt(n^2+1) \in A$ for all positive integers $n$, (b) if $x \in A$ and $y \in A$, then $x-y \in A$. Prove that every integer can be written as a product of two different elements in $A$.

Let $D$ be a point on the side $AC$ of triangle $ABC$. Let $E$ and $F$ be points on the segments $BD$ and $BC$ respectively, such that $\angle BAE = \angle CAF$. Let $P$ and $Q$ be points on the segments $BC$ and $BD$ respectively, such that $EP \parallel CD$ and $FQ \parallel CD$. Prove that $\angle BAP = \angle CAQ$.

Three points $A\left(0,\frac{4}{3}\right),B(-1,0),C(1,0)$ are given. The distance from $P$ to line $BC$ is the geometric mean of that from $P$ to lines $AB$ and $AC$. [b](a)[/b] Find the path equation of point $P$. [b](b)[/b] If line $L$ passes $D$ ($D$ is the incenter of $\triangle ABC$ ), and it has three common points with the path of $P$, find the range value of slope $k$ of line $L$.

Extension of angle bisector $BL$ of the triangle $ABC$ (where $AB < BC$) meets its circumcircle at $N$. Let $M$ be the midpoint of $BL$. Isosceles triangle $BDC$ with base $BC$ and angle equal to $ABC$ at $D$ is constructed outside the triangle $ABC$. Prove that $CM \perp DN$. [i]Proposed by А. Mardanov[/i]

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]

Find two disjoint sets $N_1$ and $N_2$ with $N_1\cup N_2=\mathbb N$, so that neither set contains an infinite arithmetic progression.