[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?

Given the larger of two circles with center $ P$ and radius $ p$ and the smaller with center $ Q$ and radius $ q$. Draw $ PQ$. Which of the following statements is false? $ \textbf{(A)}\ p\minus{}q\text{ can be equal to }\overline{PQ}\\ \textbf{(B)}\ p\plus{}q\text{ can be equal to }\overline{PQ}\\ \textbf{(C)}\ p\plus{}q\text{ can be less than }\overline{PQ}\\ \textbf{(D)}\ p\minus{}q\text{ can be less than }\overline{PQ}\\ \textbf{(E)}\ \text{none of these}$

Kathryn has a crush on Joe. Dressed as Catwoman, she attends the same school Halloween party as Joe, hoping he will be there. If Joe gets beat up, Kathryn will be able to help Joe, and will be able to tell him how much she likes him. Otherwise, Kathryn will need to get her hipster friend, Max, who is DJing the event, to play Joe’s favorite song, “Pieces of Me” by Ashlee Simpson, to get him out on the dance floor, where she’ll also be able to tell him how much she likes him. Since playing the song would be in flagrant violation of Max’s musical integrity as a DJ, Kathryn will have to bribe him to play the song. For every $\$10$ she gives Max, the probability of him playing the song goes up $10\%$ (from $0\%$ to $10\%$ for the first $\$10$, from $10\%$ to $20\%$ for the next $\$10$, all the way up to $100\%$ if she gives him $\$100$). Max only accepts money in increments of $\$10$. How much money should Kathryn give to Max to give herself at least a $65\%$ chance of securing enough time to tell Joe how much she likes him?

Let $c_i$ denote the $i$th composite integer so that $\{c_i\}=4,6,8,9,...$ Compute \[\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}\] (Hint: $\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$)

Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

Three cities that are located on the vertices of an equilateral triangle with side length $100$ units. A missile flies in a straight line in the same plane as the equilateral triangle formed by the three citiies. The radar from City $A$ reported that the closest approach of the missile was $20$ units. The radar from City $B$ reported that the closest approach of the missile was $60$ units. However, the radar for city $C$ malfunctioned and did not report a distance. Find the minimum possible distance for the closest approach of the missile to city $C$.

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

The numbers $2, 2, ..., 2$ are written on a blackboard (the number $2$ is repeated $n$ times). One step consists of choosing two numbers from the blackboard, denoting them as $a$ and $b$, and replacing them with $\sqrt{\frac{ab + 1}{2}}$. $(a)$ If $x$ is the number left on the blackboard after $n - 1$ applications of the above operation, prove that $x \ge \sqrt{\frac{n + 3}{n}}$. $(b)$ Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$. [i]Kevin Cong[/i]

Baklavas with nuts are laid out on the table in a row at the Nowruz celebration. Kosa and Kechel saw this and decided to play a game. Kosa eats one baklava from either the beginning or the end of the row in each move. Kechel either doesn't touch anything in each move or chooses the baklava he wants and just eats the nut on it. They agree that the first Kosa will start the game and make $20$ moves in each step, and the Kechel will only make $1$ move in each step. If the last baklava eaten by the Kosa is a nut, he wins the game. It is given that the number of baklavas is a multiple of $20.$ $A)$ If the number of baklavas is $400,$ prove that Kosa will win the game regardless of which strategy Kechel chooses. $B)$ Is it always true that no matter how many baklavas there are and what strategy Kechel chooses, Kosa will always win the game?

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

Let $r,s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that \[ \frac{\binom s0}{\binom tr} + \frac{\binom s1}{\binom{t}{r+1}} + \cdots + \frac{\binom ss}{\binom{t}{r+s}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}}. \]