Prove that every selection of $1325$ integers from $M=\{1, 2, \cdots, 1987 \}$ must contain some three numbers $\{a, b, c\}$ which are pairwise relatively prime, but that it can be avoided if only $1324$ integers are selected.

Construct a convex quadrilateral given two opposite angles and sides.

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{2}=12, \; a_{3}=20, \; a_{n+3}= 2a_{n+2}+2a_{n+1}-a_{n}.\] Prove that $1+4a_{n}a_{n+1}$ is a square for all $n \in \mathbb{N}$.

Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that \[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\] holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.

Let $n \ge 2$ be an even integer. Find the maximum integer $k$ (in terms of $n$) such that $2^k$ divides $\binom{n}{m}$ for some $0 \le m \le n$.

Let scalene triangle $ABC$ have altitudes $AD, BE, CF$ and circumcenter $O$. The circumcircles of $\triangle ABC$ and $\triangle ADO$ meet at $P \ne A$. The circumcircle of $\triangle ABC$ meets lines $PE$ at $X \ne P$ and $PF$ at $Y \ne P$. Prove that $XY \parallel BC$. [i]Proposed by Daniel Hu[/i]

Positive integer $l$ and positive real numbers $a_1, a_2, ..., a_l$ are given. For every positive integer $n$ we define $$c_n=\sum_{k_1+k_2+...+k_l=n}\frac{(2n)!}{(2k_1)!(2k_2)!...(2k_l)!}a_1^{k_1}a_2^{k_2}...a_l^{k_l}.$$ Prove that for every positive integer $n$ the inequality $\sqrt[n]{c_n}\leq \sqrt[n+1]{c_{n+1}}$ holds.

The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.

Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2=\sqrt{11}$. Find $\sum^4_{k=1}(CE_k)^2$.

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. [i]A. Skopenkov[/i]

Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$, \[\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}\] Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$.

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

Is it possible to have numbers $1, 2,..., 10$ put in a row in some order so that each of them, starting from the second, differs from the previous one by a whole number of percent?

Pasha placed numbers from $1$ to $100$ in the cells of the square $10$ × $10$, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green? [i]Bragin Vladimir[/i]

Find all pairs of real numbers $(x,y)$ which satisfy the system $$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

Hexagon $ABCDEF$ is inscribed in a circle. If $\measuredangle ACE = 35^{\circ}$ and $\measuredangle CEA = 55^{\circ}$, then compute the sum of the degree measures of $\angle ABC$ and $\angle EFA$. [i]Proposed by Isabella Grabski[/i]

Let $S$ be the set of all $10-$term arithmetic progressions that include the numbers $4$ and $10.$ For example, $(-2,1, 4,7,10,13, 16,19,22,25)$ and $(10,8\tfrac{1}{2}, 7,5\tfrac{1}{2}, 4, 2\tfrac{1}{2},1,\tfrac{1}{2},-2, -3\tfrac{1}{2})$ are both members of $S.$ Find the sum of all values of $a_{10}$ for each $(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}) \in S,$ that is, $\sum_{a_1, a_2, a_3, ... , a_{10} \in S} a_{10}.$

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2t^2}$ chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?

Let $a$ and $b$ be positive numbers. Prove the inequality $$\sqrt[3]{\frac ab}+\sqrt[3]{\frac ba}\le\sqrt[3]{2(a+b)\left(\frac1a+\frac1b\right)}.$$

The triangle $ABC$ is given. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, $Q_1$ and $Q_2$ are points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at the point $R{}$, and the circles $P_1P_2R$ and $Q_1Q_2R$ intersect a second time at the point $S{}$ lying inside the triangle $P_1Q_1R$. Let $M{}$ be the midpoint of the segment $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

For real numbers $ x_{i}>1,1\leq i\leq n,n\geq 2,$ such that: $ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$ for all $ i\equal{}1,2\dots, n$ find, with proof, $ \sup S.$

You live in an economy where all coins are of value $1/k$ for some positive integer $k$ (i.e. $1, 1/2, 1/3, \dots$). You just recently bought a coin exchanging machine, called the [i] Cape Town Machine [/i]. For any integer $n > 1$, this machine can take in $n$ of your coins of the same value, and return a coin of value equal to the sum of values of those coins (provided the coin returned is part of the economy). Given that the product of coins values that you have is $2015^{-1000}$, what is the maximum numbers of times you can use the machine over all possible starting sets of coins? [i] Proposed by Yang Liu [/i]

Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows: -for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$, -he erases all integers on the board, -he writes on the board the numbers $a_1, a_2,\ldots , a_n$. For instance, if $n = 5$ and the numbers initially on the board are $0, 7, 2, 6, 2$, after the first move the numbers on the board will be $1, 3, 3, 3, 3$, after the second they will be $1, 1, 5, 5, 5$, and so on. (a) Show that, whatever $n$ and whatever the initial configuration, the numbers on the board will eventually not change any more. (b) As a function of $n$, determine the minimum integer $k$ such that, whatever the initial configuration, moves from the $k$-th onwards will not change the numbers written on the board.