Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

Given a positive integer $n$, consider the sequence $(a_i)$, $1 \leq i \leq 2n$, defined as follows: $a_{2k-1} = -k, 1 \leq k \leq n$ $a_{2k} = n-k+1, 1 \leq k \leq n.$ We call a pair of numbers $(b,c)$ good if the following conditions are met: $i) 1 \leq b < c \leq 2n,$ $ii) \sum_{j=b}^{c}a_j = 0$ If $B(n)$ is the number of good pairs corresponding to $n$, prove that there are infinitely many $n$ for which $B(n) = n$.

Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$ [i]Dana Heuberger[/i]

Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point $C_n$. Find the sum \[\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) \]

Let $\epsilon$  be a positive real number. A positive integer will be called $\epsilon$-squarish if it is the product of two integers $a$ and $b$ such that $1 < a < b < (1 +\epsilon )a$. Prove that there are infinitely many occurrences of six consecutive $\epsilon$ -squarish integers.

Let $a_1 , a_2 , a_3 ,\ldots$ be a sequence of positive real numbers, define $s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}$ and $r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.$ Given that $\lim_{n\to \infty} s_n $ and $\lim_{n\to \infty} r_n $ exist, prove that the product of these limits is not less than $1.$

Compute $$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0}{i + j \choose i} 3^{-(i+j)}.$$

Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.

A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Alice and Bob play the following game on a $100\times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from $1$ to $100^2$ that is not written yet in any of the cells and choose an empty cell, and place it in the chosen cell. When there is no empty cell left, Alice computes the sum of the numbers in each row, and her score is the maximum of these $100$ numbers. Bob computes the sum of the numbers in each column, and his score is the maximum of these $100$ numbers. Alice wins if her score is greater than Bob's score, Bob wins if his score is greater than Alice's score, otherwise no one wins. Find if one of the players has a winning strategy, and if so which player has a winning strategy. [i]Théo Lenoir, France[/i]

Two ants, one starting at $ (-1, 1) $, the other at $ (1, 1) $, walk to the right along the parabola $ y = x^2 $ such that their midpoint moves along the line $ y = 1 $ with constant speed $1$. When the left ant first hits the line $ y = \frac {1}{2} $, what is its speed?

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

King George has decided to connect the $1680$ islands in his kingdom by bridges. Unfortunately the rebel movement will destroy two bridges after all the bridges have been built, but not two bridges from the same island. What is the minimal number of bridges the King has to build in order to make sure that it is still possible to travel by bridges between any two of the $1680$ islands after the rebel movement has destroyed two bridges?

Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .

Let $I$ be the intersection point of the bisectors of triangle $ABC$. Let us denote by $A', B', C'$ the points symmetrical to $I$ wrt the sides triangle $ABC$. Prove that if a circle circumscribes around triangle $A'B'C'$ passes through vertex $B$, then $\angle ABC = 60^o$.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}$, and $g(2020) = \text{the digit sum of } 12_\text{eight} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9.$ Find the remainder when $N$ is divided by $1000.$

Given a function $ f:\mathbb{N}\longrightarrow\mathbb{N} , $ find the necessary and sufficient condition that makes the sequence $$ \left(\left( 1+\frac{(-1)^{f(n)}}{n+1} \right)^{(-1)^{-f(n+1)}\cdot(n+2)}\right)_{n\ge 1} $$ to be monotone.

Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$.

For any $0\leq x_1,x_2,\ldots,x_{2011} \leq 1$, Find the maximum value of \begin{align*} \sum_{k=1}^{2011}(x_k-m)^2 \end{align*} where $m$ is the arithmetic mean of $x_1,x_2,\ldots,x_{2011}$.

Let $ P$ be a point in the plane of a triangle $ ABC$. Lines $ AP,BP,CP$ respectively meet lines $ BC,CA,AB$ at points $ A',B',C'$. Points $ A'',B'',C''$ are symmetric to $ A,B,C$ with respect to $ A',B',C',$ respectively. Show that: $ S_{A''B''C''}\equal{}3S_{ABC}\plus{}4S_{A'B'C'}$.