In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.

Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that \[ \frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|. \] When does equality hold?

Positive integers $m, n$ are given such that $\sqrt{2}<\frac{m}{n}<\sqrt{2}+\frac{1}{2}$ and $m$ is even. Prove that there exist positive integers $k<m$ and $l<n$ such that $$|\frac{k}{l}-\sqrt{2}|<\frac{m}{n}-\sqrt{2}$$

Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$ When does equality hold? [i](Karl Czakler)[/i]

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$

Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$

Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$. [list] a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1. b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that \[|a_j-a_k|<\frac{1}{n-1}.\] [/list]

Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$. Find the maximum area of $S(a)$.