From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

Let $ a_0,a_1,\dots,a_{n \plus{} 1}$ be natural numbers such that $ a_0 \equal{} a_{n \plus{} 1} \equal{} 1$, $ a_i>1$ for all $ 1\leq i \leq n$, and for each $ 1\leq j\leq n$, $ a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}$. Prove that there exist one $ 2$ in the sequence.

Let $ A \equal{} (a_{ik})$ be an $ n\times n$ matrix with nonnegative elements such that $ \sum_{k \equal{} 1}^n a_{ik} \equal{} 1$ for $ i \equal{} 1,...,n$. Show that, for every eigenvalue $ \lambda$ of $ A$, either $ |\lambda| < 1$ or there exists a positive integer $ k$ such that $ \lambda^k \equal{} 1$

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.

An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will [i]not[/i] have distinct positive real roots? $\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$

A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]

The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$. [i]O. Mushkarov, N. Nikolov[/i] [hide]No-one in the competition scored more than 2 points[/hide]

The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $X$ be a point on the incircle, different from the points $D,E,F$. The lines $XD$ and $EF,XE$ and $FD,XF$ and $DE$ meet at points $J,K,L$, respectively. Let further $M,N,P$ be points on the sides $BC,CA,AB$, respectively, such that the lines $AM,BN,CP$ are concurrent. Prove that the lines $JM,KN$ and $LP$ are concurrent. [i]Dinu Serbanescu[/i]

Find all pairs $(a, b)$ of positive integers such that \[ 3^a = 2b^2 + 1. \]

Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that \[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\] Prove that the points $E,F,G,H,K,L$ all lie on a sphere.

On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$. [i]Nicolae Bourbacut[/i]