Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.

A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.

A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?

Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.

prove that the number of permutations such that the order of each element is a multiple of $d$ is $\frac{n!}{(\frac{n}{d})!d^{\frac{n}{d}}} \prod_{i=0}^{\frac{n}{d}-1} (id+1)$. [i]proposed by Mohammad Mansouri[/i]

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a function having the following property: For any two points $A$ and $B$ in $\mathbb{R}^2$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$. Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$ (a) Suppose that $C,D$ are two fixed points in $\mathbb{R}^2$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$. (b) Consider two more point $E$ and $F$ in $\mathbb{R}^2$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\alpha$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.

Let be a number $ n\ge 2, $ a binary funcion $ b:\mathbb{Z}\rightarrow\mathbb{Z}_2, $ and $ \frac{n^3+5n}{6} $ consecutive integers. Show that among these consecutive integers there are $ n $ of them, namely, $ b_1,b_2,\ldots ,b_n, $ that have the properties: $ \text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right) $ $ \text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1} $

Triangle $ABC$ is inscribed in circle $\gamma_1$ with radius $r_1$. Let $\gamma_2$ (with radius $r_2$) be the circle internally tangent to $\gamma_1$ at $A$ and tangent to $BC$ at $D$. Let $I$ be the incenter of $ABC$, and $P$ and $Q$ be the intersection of $\gamma_2$ with $AB$ and $AC$ respectively. Given that $P$, $I$, and $Q$ are collinear, $AI=25$, and the circumradius of triangle $BIC$ is $24$, compute the ratio of the radii $\tfrac{r_2}{r_1}$.

We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

Find all positive integers $n$ such that the number $$n^6 + 5n^3 + 4n + 116$$ is the product of two or more consecutive numbers.

Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties: (1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s. (2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.

Given a triangle $ABC$, points $D,E,F$ lie on sides $BC,CA,AB$ respectively. Moreover, the radii of incircles of $\triangle AEF, \triangle BFD, \triangle CDE$ are equal to $r$. Denote by $r_0$ and $R$ the radii of incircles of $\triangle DEF$ and $\triangle ABC$ respectively. Prove that $r+r_0=R$.

Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]

Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.