Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.

In any triangle $ABC$ prove that the following relationship holds: $$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

Consider the triangle $ ABC$ with side-lenghts equal to $ a,b,c$. Let $ p\equal{}\frac{a\plus{}b\plus{}c}{2}$, $ R$-the radius of circumcircle of the triangle $ ABC$, $ r$-the radius of the incircle of the triangle $ ABC$ and let $ l_a,l_b,l_c$ be the lenghts of bisectors drawn from $ A,B$ and $ C$, respectively, in the triangle $ ABC$. Prove that: $ l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr}$ [i]Proposer[/i]: [b]Baltag Valeriu[/b]

$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$. [i]Proposed by Mohammad Gharakhani[/i]

Prove that every positive $x,y$ and real $a$ satisfy inequality $x^{\sin ^2a} y^{\cos^2a} < x + y$.

Archimedes planned to count all of the prime numbers between $2$ and $1000$ using the Sieve of Eratosthenes as follows: (a) List the integers from $2$ to $1000$. (b) Circle the smallest number in the list and call this $p$. (c) Cross out all multiples of $p$ in the list except for $p$ itself. (d) Let $p$ be the smallest number remaining that is neither circled nor crossed out. Circle $p$. (e) Repeat steps $(c)$ and $(d)$ until each number is either circled or crossed out. At the end of this process, the circled numbers are prime and the crossed out numbers are composite. Unfortunately, while crossing out the multiples of $2$, Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides $2$). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from $2$ to $1000$ (including $2$), then what is the largest possible prime that Archimedes accidentally crossed out?

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ \leq \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$

Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that: $$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$

Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true: \[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\] [i]Maxim Didin[/i]

For positive numbers $x_{1},x_{2},\dots,x_{s}$, we know that $\prod_{i=1}^{s}x_{k}=1$. Prove that for each $m\geq n$ \[\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}\]

Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds: $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$

Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$

If $\alpha $ is an acute angle and $a,b\geq 0$ then show that: $\left( a+\frac{b}{\sin \alpha}\right)\left(b+\frac{a}{\cos \alpha}\right)\geq a^2+b^2+3ab$

Find the maximum value of \[abcd(a+b)(b+c)(c+d)(d+a)\] such that $a,b,c$ and $d$ are positive real numbers satisfying $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+\sqrt[3]{d}=4$

Prove that \[ 1-\frac{1}{2012}\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}\right)\ge \frac{1}{\sqrt[2012]{2013}}.\]

Given positive numbers $a,b,c,d$. Prove that the set of inequalities $$a+b<c+d$$ $$(a+b)(c+d)<ab+cd$$ $$(a+b)cd<ab(c+d)$$ contain at least one wrong.

Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/i]

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that \[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\]

In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that $$(a - b)(b - c)(a- c) \le 2.$$ When does equality hold? [i](Karl Czakler)[/i]

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a [i]local minimum[/i] in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?