Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$

Of the following statements, the only one that is incorrect is: $ \textbf{(A)}$ An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity. $ \textbf{(B)}$ The arithmetic mean of two unequal positive quantities is greater than their geometric mean. $ \textbf{(C)}$ If the sum of two positive quantities is given, ther product is largest when they are equal. $ \textbf{(D)}$ If $ a$ and $ b$ are positive and unequal, $ \frac {1}{2}(a^2 \plus{} b^2)$ is greater than $ [\frac {1}{2}(a \plus{} b)]^2$. $ \textbf{(E)}$ If the product of two positive quantities is given, their sum is greatest when they are equal.

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.

Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that \[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\] Prove that $a + b + c + d > 1$. [i]Proposed by Victor Domínguez[/i]

Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$. Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.

Find the minimum value of real number $m$, such that inequality \[m(a^3+b^3+c^3) \ge 6(a^2+b^2+c^2)+1\] holds for all positive real numbers $a,b,c$ where $a+b+c=1$.

Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$

Let $x, y, z$ be non-negative real numbers whose sum is $ 1$. Prove that: $$\sqrt[3]{x - y + z^3} + \sqrt[3]{y - z + x^3} + \sqrt[3]{z - x + y^3} \le 1$$

Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose \[a^2+b^2+c^2+d^2=ab+bc+cd+da,\] and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

If $x_k >0$ ($k=1, 2, \cdots , n$), then $$\sum \limits_{k=1}^n \left ( \frac{x_k}{1+x_1^2+x_2^2+\cdots +x_k^2} \right )^2 \leq \frac{\sum \limits_{k=1}^n x_k^2}{1+\sum \limits_{k=1}^n x_k^2} $$

For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that \[ \left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2. \]

Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that \[\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.\]

Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities $$ 4016016\le x<y<z\le 4020025. $$

(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively. (2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$: $$f(f(n)) \le \frac{f(n) + n}{2}$$

Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

Let $a_1, a_2, ..., a_9$ be non-negative real numbers such that $a_1 = a_9 = 0$ and at least one of the remaining terms is different from $0$. a) Prove that for some $i$ $(i = 2, ..., 8$) ,holds that $a_{i-1} + a_{i+1} < 2a_i.$ b) Will the previous statement be true, if we change the number $2$ for $1.9$ in the inequality?

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies \[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \] Prove that, for all $x\in [0,\infty)$, \[ f(x)\ge 3e^{2x}-2e^{3x} \]

Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$.

Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$; ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$. Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.