For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that \[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\] for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.

Compute: $ \frac{777^2 \minus{} 66^2}{777\plus{}66}$

Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that: \[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\] [i]Proposed by Juhan Aru, Estonia[/i]

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$. $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that the triangle $TRS$ is equilateral.

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 $A$.