In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?

Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[ \frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1 \] for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.

Let $ABCD$ be an circumscribed quadrilateral such that $m(\widehat{A})=m(\widehat{B})=120^\circ$, $m(\widehat{C})=30^\circ$, and $|BC|=2$. What is $|AD|$? $ \textbf{(A)}\ \sqrt 3 - 1 \qquad\textbf{(B)}\ \sqrt 2 - 3 \qquad\textbf{(C)}\ \sqrt 6 - \sqrt 2 \qquad\textbf{(D)}\ 2 - \sqrt 2 \qquad\textbf{(E)}\ 3 - \sqrt 3 $

For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set$\{a_1,a_2,...,a_{2006}\}$.

A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Prove that $PM = PS$ holds. [i](Karl Czakler)[/i]

Let $G$ be the centroid of triangle $ABC$ with $AB = 9$, $BC = 10,$ and $AC = 17$. Denote $D$ as the midpoint of $BC$. A line through $G$ parallel to $BC$ intersects $AB$ at $M$ and $AC$ at $N$. If $BG$ intersects $CM$ at $E$ and $CG$ intersects $BN$ at $F$, compute the area of triangle $DEF$.

Suppose that $\{D_n\}_{n\ge 1}$ is an finite sequence of disks in the plane whose total area is less than $1$. Prove that it is possible to rearrange the disks so that they are disjoint from each other and all contained inside a disk of area $4$.

Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$

Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

Equilateral triangle $ T$ is inscribed in circle $ A$, which has radius $ 10$. Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$. Circles $ C$ and $ D$, both with radius $ 2$, are internally tangent to circle $ A$ at the other two vertices of $ T$. Circles $ B$, $ C$, and $ D$ are all externally tangent to circle $ E$, which has radius $ \frac {m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. [asy]unitsize(2.2mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label("$E$",Ep,E); label("$A$",A,W); label("$B$",B,W); label("$C$",C,W); label("$D$",D,E);[/asy]

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

A sequence of natural numbers $a_1, a_2, a_3,..$ is called [i]periodic modulo[/i] $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers? b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?

Find the sum of all positive integers $n$ such that $\tau(n)^2=2n$, where $\tau(n)$ is the number of positive integers dividing $n$. [i]Proposed by Michael Kural[/i]

function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 45$