Two cyclists start moving simultaneously in opposite directions on a circular circuit. The first cyclist maintains a constant speed $c_1$ meters per second, the second maintains $c_2$ meters per second. How many times did they meet when the first cyclist completed $n$ laps? Compute for $c_1=10,c_2=7,n=11$.

Find all monotonic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[f(f(x) - y) + f(x+y) = 0,\] for every real $x, y$. (Note that monotonic means that function is not increasing or not decreasing)

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

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.