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 the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$

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?

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that: \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality $$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.

The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

The vertices of two regular octagons are numbered from $1$ to $8$, in some order, which may vary between both octagons (each octagon must have all numbers from $1$ to $8$). After this, one octagon is placed on top of the other so that every vertex from one octagon touches a vertex from the other. Then, the numbers of the vertices which are in contact are multiplied (i.e., if vertex $A$ has a number $x$ and is on top of vertex $A'$ that has a number $y$, then $x$ and $y$ are multiplied), and the $8$ products are then added. Prove that, for any order in which the vertices may have been numbered, it is always possible to place one octagon on top of the other so that the final sum is at least $162$. Note: the octagons can be rotated.

Let $ a,b,c$ be real numbers such that $ a\plus{}b\plus{}c\equal{}3$. Prove that \[\frac{1}{5a^2\minus{}4a\plus{}11}\plus{}\frac{1}{5b^2\minus{}4b\plus{}11}\plus{}\frac{1}{5c^2\minus{}4c\plus{}11}\leq\frac{1}{4}\]

Denote $T$ the Toricelli point of the triangle $ABC$. Prove that $$AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)$$

Prove that for real $x\ge 1$, holds the inequality $$\frac{2^x +3^x }{3^x +4^x} \le \frac57$$

Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$. Find the maximal value of $\lambda$.

Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

Find the greatest possible value of $s>0$, such that for any positive real numbers $a,b,c$, $$(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})^2 \geq s(\frac{1}{a^2+bc}+\frac{1}{b^2+ca}+\frac{1}{c^2+ab}).$$

Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let $$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$ If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.

Given that $ \minus{} 4\le x\le \minus{} 2$ and $ 2\le y\le 4$, what is the largest possible value of $ (x \plus{} y)/x$? $ \textbf{(A)}\ \minus{}\!1\qquad \textbf{(B)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ 1$

Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.

Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n - 1$. Prove that we can cut the necklace to form a string, whose consecutive labels $x_1,x_2,...,x_n$ satisfy $\sum_{i=1}^{k} x_i \le k - 1$ for any $k = 1,...,n$

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.

Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]