Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

$ a>0,\quad\lim_{n\to\infty }\sum_{i=1}^n \frac{1}{n+a^i} $

Let $n$ be a positive integer. Let $x_1, x_2, \dots, x_n > 1$ be real numbers whose product is $n+1$. Prove that \[\left(\frac{1}{1^2(x_1-1)}+1\right)\left(\frac{1}{2^2(x_2-1)}+1\right)\cdots\left(\frac{1}{n^2(x_n-1)}+1\right)\geq n+1\] and find for which values equality holds.

Decide whether there exists a largest positive integer $n$ such that the inequality \[\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}\] holds for all positive real numbers $a$ and $b$. If such a largest positive integer $n$ exists, determine it.

Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that \[\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab} \ge \frac35\]

[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $ [b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality $$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$

As shown on the figure, square $PQRS$ is inscribed in right triangle $ABC$, whose right angle is at $C$, so that $S$ and $P$ are on sides $BC$ and $CA$, respectively, while $Q$ and $R$ are on side $AB$. Prove that $A B\geq3QR$ and determine when equality occurs. [asy] defaultpen(linewidth(0.7)+fontsize(10)); size(150); real a=8, b=6; real y=a/((a^2+b^2)/(a*b)+1), r=degrees((a,b))+180; pair A=b*dir(-r)*dir(90), B=a*dir(180)*dir(-r), C=origin, S=y*dir(-r)*dir(180), P=(y*b/a)*dir(90-r), Q=foot(P, A, B), R=foot(S, A, B); draw(A--B--C--cycle^^R--S--P--Q); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$S$", S, dir(point--S)); label("$R$", R, dir(270)); label("$Q$", Q, dir(270)); label("$P$", P, dir(point--P));[/asy]

If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to $1$. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

If $a,b,c$ are positive real numbers with $abc=1$, prove that (a) \[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}\] (b)\[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}\]

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

Given positive reals $ a, b, c $ that satisfy $ a + b + c = 1 $, show that $$ \displaystyle \sum^{}_{cyc}\frac{a^3+bc}{a^2+bc}\ge 2 $$

Prove that if m,n are nonnegative integers and 0<=x<=1 then $(1-x^n)^m + (1-(1-x)^m)^n \ge 1$

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$. Find the maximum value of $f$.

For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that $ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$