ISL 1995

Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ where $c$ is an integer. Prove that there is a number $n$ and integers $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that

$$\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$

Let $n$ be an integer, $n \geq 3.$ Let $a_1, a_2, \ldots, a_n$ be real numbers such that $2 \leq a_i \leq 3$ for $i = 1, 2, \ldots, n.$ If $s = a_1 + a_2 + \ldots + a_n,$ prove that

$$\frac{a^2_1 + a^2_2 - a^2_3}{a_1 + a_2 - a_3} + \frac{a^2_2 + a^2_3 - a^2_4}{a_2 + a_3 - a_4} + \ldots + \frac{a^2_n + a^2_1 - a^2_2}{a_n + a_1 - a_2} \leq 2s - 2n.$$

Find all of the positive real numbers like $x,y,z,$ such that :

1.) $x + y + z = a + b + c$

2.) $4xyz = a^2x + b^2y + c^2z + abc$

Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

Let $\mathbb{R}$ be the set of real numbers. Does there exist a function $f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions?(a) There is a positive number $M$ such that $\forall x:$ $- M \leq f(x) \leq M.$ (b) The value of $f(1)$ is $1$ .(c) If $x \neq 0,$ then $$f \left(x + \frac {1}{x^2} \right) = f(x) + \left[ f \left(\frac {1}{x} \right) \right]^2$$

Let $n$ be an integer, $n \geq 3.$ Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_i < x_{i+1}$ for $1 \leq i \leq n - 1$ . Prove that

$$\frac{n(n-1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n-1}_{i=1} (n-i)\cdot x_i \right) \cdot \left(\sum^{n}_{j=2} (j-1) \cdot x_j \right)$$

The incircle of triangle $\triangle ABC$ touches the sides $BC$ , $CA$ , $AB$ at $D, E, F$ respectively. $X$ is a point inside triangle of $\triangle ABC$ such that the incircle of triangle $\triangle XBC$ touches $BC$ at $D$ , and touches $CX$ and $XB$ at $Y$ and $Z$ respectively. Show that $E, F, Z, Y$ are concyclic.

An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$ ), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$ ), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$ ) so that

$$\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.$$

The lines $AA_1,BB_1,$ and $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A'_1, A'_2, A'_3,$ and $A'_4$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1,GA_2,GA_3,$ and $GA_4,$ respectively. Prove that

$$GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4$$

and

$$\frac{1}{GA'_1} + \frac{1}{GA'_2} + \frac{1}{GA'_3} + \frac{1}{GA'_4} \leq \frac{1}{GA_1} + \frac{1}{GA_2} + \frac{1}{GA_3} + \frac{1}{GA_4}.$$

Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that $\sqrt {\left|AHOE\right|} + \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}$ , where $\left|P_1P_2...P_n\right|$ is an abbreviation for the non-directed area of an arbitrary polygon $P_1P_2...P_n$ .

Suppose that $ABCD$ is a cyclic quadrilateral. Let $E = AC\cap BD$ and $F = AB\cap CD$ . Denote by $H_{1}$ and $H_{2}$ the orthocenters of triangles $EAD$ and $EBC$ , respectively. Prove that the points $F$ , $H_{1}$ , $H_{2}$ are collinear.

Original formulation:

Let $ABC$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ again at $C'$ and $B',$ respectively. Prove that $BB'$ , $CC'$ and $HH'$ are concurrent, where $H$ and $H'$ are the orthocentres of triangles $ABC$ and $AB'C'$ respectively.

Let $k$ be a positive integer. Show that there are infinitely many perfect squares of the form $n \cdot 2^k - 7$ where $n$ is a positive integer.

Let $\mathbb{Z}$ denote the set of all integers. Prove that for any integers $A$ and $B,$ one can find an integer $C$ for which $M_1 = \{x^2 + Ax + B : x \in \mathbb{Z}\}$ and $M_2 = {2x^2 + 2x + C : x \in \mathbb{Z}}$ do not intersect.

Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$ .

At a meeting of $12k$ people, each person exchanges greetings with exactly $3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

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 $n - 1$ integers, one taken from each of any $n - 1$ of the subsets, lies in the remaining subset.

Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \leq y$ and $\sqrt{2p} - \sqrt{x} - \sqrt{y}$ is non-negative and as small as possible.