Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality \[ \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\] does holds.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$

Suppose that $x,y$ and $z$ are positive numbers such that $$1=2xyz+xy+yz+zx$$ Prove that (i) $$\frac{3}{4}\le xy+yz+zx<1$$ (ii) $$xyz\le \frac{1}{8}$$ Using (i) or otherwise, deduce that $$x+y+z\ge \frac{3}{2}$$ and derive the case of equality.

Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2+(\log y)^2=\log(x^2)+\log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$.

Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\] in which $a,~ b,~ c$, and $d$ vary over the set of positive integers. (Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string) and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that: a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$ b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition: For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.

Let a.b.c be positive reals such that their sum is 1. Prove that $ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that \[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\] for all $x,y\in\mathbb{Q}$