Prove that each of the numbers $1, 2, 3, ..., 2^n$ can be written in one of two colors (red and blue) such that no non-constant $2n$-term arithmetic sequence chosen from these numbers is monochromatic .

The graphs of $ y\equal{}\frac{x^2\minus{}4}{x\minus{}2}$ and $ y\equal{}2x$ intersect in: $ \textbf{(A)}\ \text{1 point whose abscissa is 2} \qquad \textbf{(B)}\ \text{1 point whose abscissa is 0}\\ \textbf{(C)}\ \text{no points} \qquad \textbf{(D)}\ \text{two distinct points} \qquad \textbf{(E)}\ \text{two identical points}$

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

Let $u$ and $v$ be real numbers such that \[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \] Determine, with proof, which of the two numbers, $u$ or $v$, is larger.

Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player increases the chosen entry by $k$ and decreases the other two entries by $k$. A player loses if, on their turn, some entry on the board becomes negative. Find the number of initial triples $(a, b, c)$ for which Tadashi has a winning strategy.

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$

Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways: 1) take a coin from an arbitrary pile; 2) select a pile and divide it into two non-empty piles. The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.

Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. [i]Proposed by Bojan Bašić, Serbia[/i]

Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.

Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$ \[f(f(x)^2 + f(y)) = xf(x) + y.\]

For $x,y\in\mathbb{R}$, solve the system of equations \[ \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases} \]

Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as $\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad \textbf{(B)}\ -f(y)\qquad \textbf{(C)}\ -f(-y)\qquad \textbf{(D)}\ f(-y)\qquad \textbf{(E)}\ f(y)$

Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.

Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?

Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's [b]not[/b] known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least $1000$ points b) Is it possible that Luffy reaches at least $1000000$ points, [b]without[/b] have less than $-42$ points in any moment?

Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers. For $m \geq 1$, we define: \[ X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}. \] Show that \[\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.\]

Let $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$. If the maximum element of $S$ is $p_0/q_0$ in reduced form, find $p_0+q_0$.

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]