Professor David proposed to his wife to calculate the steps of an escalator that worked in a shopping mall, asking him to walk up counting the steps that rise from the bottom to the end. The teacher, in turn, left with his wife, but walking the twice as fast, so that the woman advanced one step each time her husband advanced $2$. When The lady arrived at the top reported that she had counted $21$ steps, while the teacher counted $28$ of them. How many steps are there in sight on the ladder at any given time? [hide=original wording]El profesor David propuso a su senora calcular los escalones de una escalera mecanica que funcionaba en un centro comercial, pidiendole que caminara hacia arriba contando los escalones que subiera desde la base hasta el final. El profesor a su vez, partio junto a su senora, pero caminando el doble de rapido, de modo que la senora avanzaba un escalon cada vez que su marido avanzaba 2. Cuando la senora llego arriba informo que habıa contado 21 escalones, mientras que el profesor conto 28 de ellos, ¿Cuantos escalones hay a la vista en la escalera en un instante cualquiera?[/hide]

S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

Find all functions $f: \mathbb N \to \mathbb N$ for which \[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\] holds for all positive integers $n$.

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x.$ [asy] size(100);defaultpen(linewidth(0.7)); int i; for(i=0; i<4; i=i+1) { draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6)); } label("$x$", (1,5)); label("$1$", (1,3)); label("$19$", (3,5)); label("$96$", (5,5));[/asy]

Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$ [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate? b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .

Find the values of the real numbers $x,y,z$ that correspond to the system of equations. $$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$ $$xy + yz + zx=1$$ [i](Heir of Ramanujan)[/i]

We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q. a) if P,Q are similar, then $P(2007)-Q(2007)$ is even b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?

Let $a, b, c$ be distinct real numbers. Prove that $$\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c} \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5$$

The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

Let $a>1$ be a positive integer and let $f(n)=n+[a\{n\sqrt{2}\}]$. Show that there exists a positive integer $n$, such that $f(f(n))=f(n)$, but $f(n) \neq n$.

Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have? [i]Mihail Bălună[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ [i]Based on a problem of Romanian Masters of Mathematics[/i]

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.