Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

On the side $AC$ of triangle $ABC$ point $D$ is chosen. Let $I_1, I_2, I$ be the incenters of triangles $ABD, BCD, ABC$ respectively. It turned out that $I$ is the orthocentre of triangle $I_1I_2B$. Prove that $BD$ is an altitude of triangle $ABC$.

One country consists of islands $A_1,A_2,\cdots,A_N$,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$. We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge). 1-Prove that $a_3 = 3$ 2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square 3-Compute $a_4$ 4-Compute $a_6$ 5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$

Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.

Find all positive integers N with at most 4 digits such that the number obtained by reversing the order of digits of N is divisible by N and differs from N.

In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.

Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{None}$

Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

If 1 pint of paint is needed to paint a statue 6 ft. high, then the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high is $ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 15$

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$? $\textbf{(A) } \frac{5}{512} \qquad \textbf{(B) } \frac{45}{1024} \qquad \textbf{(C) } \frac{127}{1024} \qquad \textbf{(D) } \frac{511}{1024} \qquad \textbf{(E) } \frac{1}{2}$

Regular hexagon $ABCDEF$ has side length $2$. Line segment $BD$ is drawn, and circle $O$ is inscribed inside the pentagon $ABDEF$ such that $O$ is tangent to $AF$, $BD$, and $EF$. Compute the radius of $O$.

How many ordered pairs of integers $ (x, y)$ are there such that \[ 0 < \left\vert xy \right\vert < 36?\]

Let $ABCD$ be a non-isosceles trapezoid with $AB \parallel CD$. A circle through $A$ and $B$ meets $AD$, $BC$ at $E, F$. The segments $AF, BE$ meet at $G$. The circumcircles of $\triangle ADG$ and $\triangle BCG$ meet at $H$. Show that if $GD=GC$, $H$ is the orthocenter of $\triangle ABG$.

For positive integers $a,n$, define $F_n(a)=q+r$, where $a=qn+r$ ($q,r$ are nonnegative integers, $0\leq q<n$). Find the largest integer $A$, there are positive integers $n_1,n_2,n_3,n_4,n_5,n_6$, for all positive integer $a\leq A$, $F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1$.

For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.

Call a set $S$ [i]product-free[/i] if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.

Point $M$ is the midpoint of the side $CD$ of the trapezoid $ABCD$, point $K$ is the foot of the perpendicular drawn from point $M$ to the side $AB$. Give that $3BK \le AK$. Prove that $BC + AD\ge 2BM$.

A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

Let \[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\] Compute $\Big|\lfloor 2008 X\rfloor\Big|$.

Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions: (i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$ (ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.) Find the locus of the center of circle $\Gamma$, and explain how to construct the locus.

Let $ABCD$ be a regular tetrahedron and $\mathbf{Z}$ an isometry mapping $A,B,C,D$ into $B,C,D,A$, respectively. Find the set $M$ of all points $X$ of the face $ABC$ whose distance from $\mathbf{Z}(X)$ is equal to a given number $t$. Find necessary and sufficient conditions for the set $M$ to be nonempty.

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?