Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.

Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied: $(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. $(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even . [i]Proposed by Poland.[/i]

Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|BC|+|AD| = 7$, $|AB| = 9$ and $|BC| = 14$. What is the ratio of the area of the triangle formed by $CD$, angle bisector of $\widehat{BCD}$ and angle bisector of $\widehat{CDA}$ over the area of the trapezoid? $ \textbf{a)}\ \dfrac{9}{14} \qquad\textbf{b)}\ \dfrac{5}{7} \qquad\textbf{c)}\ \sqrt 2 \qquad\textbf{d)}\ \dfrac{49}{69} \qquad\textbf{e)}\ \dfrac 13 $

Does there exist an integer $ n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $ n$ nonempty subsets, such that an arbitrary sum of $ n \minus{} 1$ integers, one taken from each of any $ n \minus{} 1$ of the subsets, lies in the remaining subset.

[b]Problem 3[/b] Determine the set of real values of $x$ for which the following series converges, and find its sum: $$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

Let $ABC$ be an obtuse-angled triangle with $ \angle ABC>90^{\circ}$, and let $(c)$ be its circumcircle. The internal angle bisector of $\angle BAC$ meets again the circle $(c)$ at the point $E$, and the line $BC$ at the point $D$. The circle of diameter $DE$ meets the circle $(c)$ at the point $H$. If the line $HE$ meets the line $BC$ at the point $K$, prove that: (a) the points $K, H, D$ and $A$ are concyclic (b) the line $AH$ passes through the point of intersection of the tangents to the circle $(c)$ at the points $B$ and $C$.

Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true: $(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.

In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$. [i]Proposed by Lewis Chen[/i]

Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals $\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6$

Show that there are no integers $x, y, z$, and $t$ such that $$\sqrt[3]{x^5 + y^5 + z^5 + t^5} = 2016.$$

$b,c$ are naturals. $b|c+1$ Prove that exists such natural $x,y,z$ that $x+y=bz,xy=cz$

Let $ABC$ be an equilateral triangle with side length $10$ and $P$ be a point inside the triangle such that $|PA|^2+ |PB|^2 + |PC|^2 = 128$. What is the area of a triangle with side lengths $|PA|,|PB|,|PC|$? $ \textbf{(A)}\ 6\sqrt 3 \qquad\textbf{(B)}\ 7 \sqrt 3 \qquad\textbf{(C)}\ 8 \sqrt 3 \qquad\textbf{(D)}\ 9 \sqrt 3 \qquad\textbf{(E)}\ 10 \sqrt 3 $

Find the linking coefficient of the phase trajectories of the equation of small oscillations $\ddot{x}=-4x$, $\ddot{y}=-9y$ on a level surface of the total energy.

In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken more that by the half of the participants. What is the least value of $n$?

When the repeating decimal $ 0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 114 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 150$

Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that $$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.

The equation $$4^x -5 \cdot 2^{x+1} +16 = 0$$ has two integer solutions for $x.$ Find their sum.

An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by [b]HrishiP[/b][/i]

In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_0, P_1, P_2, \ldots, \ldots, P_{2n}$ such that $P_0$ and $P_{2n}$ are both the origin and $P_{i}P_{i+1}=1$ for each $i.$ A closed lattice loop of length $2026$ is chosen uniformly at random from all such loops. Let $k$ be the maximum integer such that the line $\ell$ with equation $x+y=k$ passes through at least one point of the loop. Compute the expected number of indices $i$ such that $0 \le i \le 2025$ and $P_i$ lies on $\ell.$ (A lattice point is a point with integer coordinates.)

There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?

For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$