Let $x, y, z > 0$ be real numbers such that $xyz + xy + yz + zx = 4$. Prove that $x + y + z \ge 3$.

Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

Consider six points in the interior of a square of side length $3$. Prove that among the six points, there are two whose distance is less than $2$.

Let $ABCD$ be a quadrilateral with $\angle A + \angle C = 60^o$. If $AB \cdot CD = BC \cdot AD$, prove that $AB \cdot CD = AC \cdot BD$. Leonard Giugiuc

Let $D$ and $E$ be the midpoints of sides $[AB]$ and $[AC]$ of the triangle $ABC$. The circle of diameter $[AB]$ intersects the line $DE$ on the opposite side of $AB$ than $C$, in $X$. The circle of diameter $[AC]$ intersects $DE$ on the opposite side of $AC$ than $B$ in $Y$ . Let $T$ be the intersection of $BX$ and $CY$. Prove that the orthocenter of triangle $XY T$ lies on $BC$.

Find all positive integers $a$ and $b$ such that \[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \] are both integers.

Determine all real numbers $x, y, z \in (0, 1)$ that satisfy simultaneously the conditions: $(x^2 + y^2)\sqrt{1- z^2}\ge z$ $(y^2 + z^2)\sqrt{1- x^2}\ge x$ $(z^2 + x^2)\sqrt{1- y^2}\ge y$

Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.

Let $n \ge 6$ be an integer. We have at our disposal $n$ colors. We color each of the unit squares of an $n \times n$ board with one of the $n$ colors. a) Prove that, for any such coloring, there exists a path of a chess knight from the bottom-left to the upper-right corner, that does not use all the colors. b) Prove that, if we reduce the number of colors to $\lfloor 2n/3 \rfloor + 2$, then the statement from a) is true for infinitely many values of $n$ and it is false also for infinitely many values of $n$

Let $a, b, c, d$ be positive real numbers so that $abc+bcd+cda+dab = 4$. Prove that $a^2 + b^2 + c^2 + d^2 \ge 4$

Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .

Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that: for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$. Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .

In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$. a) Prove that $T$ is the incircle of triangle $XYZ$. b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$. c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.

We call a composite positive integer $n$ nice if it is possible to arrange its factors that are larger than $1$ on a circle such that two neighboring numbers are not coprime. How many of the elements of the set $\{1, 2, 3, ..., 100\}$ are nice?

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.

Let n be a positive integer and $x_1, x_2, ..., x_n > 0$ be real numbers so that $x_1 + x_2 +... + x_n =\frac{1}{x_1^2}+\frac{1}{x_2^2}+...+\frac{1}{x_n^2}$ Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, ..., x_n $ whose sum is at least $k$.

Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. Colombia 1997