Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

Let $a_{i}$, $i = 1,2, \dots ,n$, $n \geq 3$, be positive integers, having the greatest common divisor 1, such that \[a_{j}\textrm{ divide }\sum_{i = 1}^{n}a_{i}\] for all $j = 1,2, \dots ,n$. Prove that \[\prod_{i = 1}^{n}a_{i}\textrm{ divides }\Big{(}\sum_{i = 1}^{n}a_{i}\Big{)}^{n-2}.\]

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$? $\bullet$ In how many of these triangles is the perimeter divisible by $3$?

Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.

In square $ABCD$ with side length $2$, let $P$ and $Q$ both be on side $AB$ such that $AP=BQ=\frac{1}{2}$. Let $E$ be a point on the edge of the square that maximizes the angle $PEQ$. Find the area of triangle $PEQ$.

Given $ x,y,z\in (0,1)$ satisfying that $ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$. Find the maximum value of $ xyz$.

Oleksiy placed positive integers in the cells of the $8\times 8$ chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to $1$. Prove that at least one of the boys made a mistake in the calculation. For example, for a square $3\times 3$ and the arrangement of numbers shown below, Fedir would write the following numbers: $2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42$, and their sum ends with a digit $6$; Oleksiy would write the following numbers: $3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13$, and their product ends with a digit $0$. \begin{tabular}{| c| c | c |} \hline 1 & 2 & 3 \\ \hline 2 & 4 & 6 \\ \hline 3 & 5 & 7 \\ \hline \end{tabular} [i]Proposed by Oleksiy Masalitin and Fedir Yudin[/i]

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.

Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations: $a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$.

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

If $\alpha $ is the real root of the equation \[E(x) = x^3 - 5x -50 = 0\] such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$, where $n$ is a positive integer, prove that: [b](a)[/b] $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$ [b](b)[/b] $\alpha < x_{n+1} < x_n.$

Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that: [b]a)[/b] $ f $ is nondecreasing. [b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $ [i]Vasile Solovăstru[/i]

For $x\ge 0$, define \[f(x)=\frac1{x+2\cos x}\] Find the set $\{ y \in \mathbb{R}: y=f(x), x\ge 0\}$

You have four fair $6$-sided dice, each numbered $1$ to $6$ (inclusive). If all four dice are rolled, the probability that the product of the rolled numbers is prime can be written as $\tfrac{a}{b}$, where $a$ and $b$ are relatively prime. What is $a+b$?

Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that \[ xf(x+y)=(x+\alpha y)f(x)+xf(y) \] for all $x,y\in\mathbb{R}$.

Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

For pairwise distinct nonnegative reals $a,b,c$, prove that $$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.

Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

A tank has the shape of a regular hexagonal prism, whose bases are $1$ m on a side and its height is $10$ m. The lateral edges are placed in an oblique position and is partially filled with $9$ m$^3$ of water. The plane of the free surface of the water cuts to all lateral edges. One of them is left with a part of $2$ m under water. What part is under water on the opposite side edge of the prism?

A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$. [i]A corollary of a theorem from ergodic theory[/i]

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]