Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. [i]Proposed by Finbarr Holland, Ireland[/i]

There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$.

Using the digits: $ 1,2,3,4,5,$ players $ A$ and $ B$ compose a $ 2005$-digit number $ N$ by selecting one digit at a time: $ A$ selects the first digit, $ B$ the second, $ A$ the third and so on. Player $ A$ wins if and only if $ N$ is divisible by $ 9$. Who will win if both players play as well as possible?

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$. a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic. b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.

Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive). 2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd. 3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

What is the sum of all positive integers less than $30$ divisible by $2, 3$, or $5$?

a,b,c>0 and $abc\ge 1$.Prove that: $\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}$

The square $DEFG$ is contained in equilateral triangle $ABC$, with $E$ on $\overline{AC}$, $G$ on $\overline{AD}$, and $F$ as the midpoint of $\overline{BC}$. Find $AD$ if $DE = 6$.

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

Find all positive integers $n$ such that the number \[A=\sqrt{\frac{9n-1}{n+7}}\] is rational.

For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?

Let $P,Q,R$ be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds: $P(Q(x))\equal{}Q(R(x))\equal{}R(P(x))$. Show that the polynomials $P,Q,R$ are equal.

Using several white edge cubes of side $ 1$, Guille builds a large cube. Then he chooses $4$ faces of the big cube and paints them red. Finally, he takes apart the large cube and observe that the cubes with at least a face painted red is $431$. Find the number of cubes that he used to assemble the large cube. Analyze all the possibilities.

Two disjoint circles are inscribed in an angle with vertex $A$, whose measure is equal to $a$. The distance between their centers is $d$. A straight line tangent to both circles and not passing through $A$ intersects the sides of the angle at points $B$ and $C$. Find the radius of the circle circumscribed about triangle $ABC$.

A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus. Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$. [i]Proposed by Federico Ardila, Colombia [/i]

Prove that for $n>1$ and real numbers $a_0,a_1,\dots, a_n,k$ with $a_1=a_{n-1}=0$, \[|a_0|-|a_n|\leq \sum_{i=0}^{n-2}|a_i-ka_{i+1}-a_{i+2}|.\]

An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$. $\textbf{(A) }\dfrac{\sqrt[4]3}2\hspace{14em}\textbf{(B) }\dfrac{\sqrt[4]3}{\sqrt2}\hspace{14em}\textbf{(C) }1$ $\textbf{(D) }\dfrac34\hspace{14.9em}\textbf{(E) }\dfrac43\hspace{14.9em}\textbf{(F) }\sqrt3$ $\textbf{(G) }\dfrac{\sqrt6}2$

There is a lighthouse on a small island. Its lamp enlights a segment of a sea to the distance $a$. The light is turning uniformly, and the end of the segment moves with the speed $v$. Prove that a ship, whose speed doesn't exceed $v/8$ cannot arrive to the island without being enlightened.

Let $ n$ be a positive integer and let $ a_i$ be real numbers, $ i \equal{} 1,2,\ldots,n$ such that $ |a_i|\leq 1$ and $ \sum_{i\equal{}1}^n a_i \equal{} 0$. Show that $ \sum_{i\equal{}1}^n |x \minus{} a_i|\leq n$, for every $ x\in \mathbb{R}$ with $ |x|\le 1$.

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$. [i]Proposed by Aaron Lin[/i]

Let $ABCD$ be a convex trapezoid such that $\angle BAD = \angle ADC = 90^o$, $AB = 20$, $AD = 21$, and $CD = 28$. Point $P \ne A$ is chosen on segment $AC$ such that $\angle BPD = 90^o$. Compute $AP$.

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.