Let $ABC$ be a triangle inscribed in a semicircle with center $O$ and diameter $BC.$ Two tangent lines to the semicircle at $A$ and $B$ intersect at $D.$ Prove that $DC$ goes through the midpoint of the altitude $AH$ of triangle $ABC.$

A gardener plants three maple trees, four oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m + n$.

Solve the equation in nonnegative integers:\\ $2^x=5^y+3$

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying: (i) $ f(1) \equal{} 1$ (ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.

An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines. Proposed by: M.Evdokimov

If we separate the numbers $1,2,3,4,\dots, 100$ in two lists with $$a_1<a_2<\cdots<a_{50}$$ and $$b_1>b_2>\cdots>b_{50}$$ Prove that, no matter how we do the separation, $$\vert a_1-b_1\vert +\vert a_2-b_2\vert+\cdots +\vert a_{50}-b_{50}\vert=2500$$

Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$. [i] Proposed by Justin Stevens [/i]

Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need?

Let $x, y, z$ be real numbers such that $x+y+z=-2$ and \[\begin{aligned} & (x^2+xy+y^2)(y^2+yz+z^2) \\ &+ (y^2+yz+z^2)(z^2+zx+x^2) \\ &+ (z^2+zx+x^2)(x^2+xy+y^2) \\ & = 625+ \tfrac34(xy+yz+zx)^2. \end{aligned}\] Compute $|xy+yz+zx|$. [i]Proposed by Michael Tang[/i]

We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.

$ABC$ is a triangle with $\angle A > 90^o$ . On the side $BC$, two distinct points $P$ and $Q$ are chosen such that $\angle BAP = \angle PAQ$ and $BP \cdot CQ = BC \cdot PQ$. Calculate the size of $\angle PAC$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

A squirrel has infinitely many nuts; one nut of each of the masses $1g, 2g, 3g, \ldots$. The squirrel took $100$ bags, in each put a finite number of nuts, after which wrote on each bag the total mass of the nuts inside it. Prove that it is possible to create bags of the same mass using no more than $500$ nuts.

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.

Prove that the polynomial $$P(x) = nx^{n+2} -(n + 2)x^{n+1} + (n + 2)x-n$$ is divisible by the polynomial $(x - 1)^3$.

Let $(k_i)$ be a sequence of unique nonzero integers such that $x^2- 5x + k_i$ has rational solutions. Find the minimum possible value of $$\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}$$

Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that : (1) $(a+1, \frac{a^p+1}{a+1}) = p$ (2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$. (3) $p!|(a +1) $.

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there? (A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.

Let $G$ be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.

Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.) Boris Frenkin