Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.

For $k=1,2,\dots$, let $f_k$ be the number of times \[\sin\left(\frac{k\pi x}{2}\right)\] attains its maximum value on the interval $x\in[0,1]$. Compute \[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Consider a set of $1001$ points in the plane, no three collinear. Compute the minimum number of segments that must be drawn so that among any four points, we can find a triangle. [i]Proposed by Ahaan S. Rungta / Amir Hossein[/i]

We throw $ N$ balls into $ n$ urns, one by one, independently and uniformly. Let $ X_i\equal{}X_i(N,n)$ be the total number of balls in the $ i$th urn. Consider the random variable \[ y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|.\] Verify the following three statements: (a) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow \infty$, then \[ P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}<x \right) \rightarrow 1\minus{}e^{\minus{}x\sqrt{2/ \pi}} \;\textrm{for all}\ \; x>0 \ .\] (b) If $ n\rightarrow \infty$ and $ N/n^3 \leq K$ ($ K$ constant), then for any $ \varepsilon > 0$ there is an $ A > 0$ such that \[ P(y(N,n) < A) > 1\minus{}\varepsilon .\] (c) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow 0$ then \[ P(y(N,n) < 1) \rightarrow 1.\] [i]P. Revesz[/i]

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$ \[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]

Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

Let $x, y, z, t$ be real numbers such that $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$. Prove that $x^2 + y^2 > 1$.

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.

Ria has $4$ green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p,q$ have no common factors other than $1$, what is $p+q$?

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Fifteen boys are standing on a field, and each of them has a ball. No two distances between two of the boys are equal. Each boy throws his ball to the boy standing closest to him. (a) Show that one of the boys does not get any ball. (b) Prove that none of the boys gets more than five balls.

Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Suppose that $$f(x) = \sum_{i=0}^\infty c_ix^i$$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.

For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?

Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

What number is exactly halfway between $\frac 1 6$ and $\frac 1 4$?