Let $ p\geq 5$ be a prime number.Prove that for any partition of the set $ P\equal{}\{1,2,3,...,p\minus{}1\}$ in $ 3$ subsets there exists numbers $ x,y,z$ each belonging to a distinct subset,such that $ x\plus{}y\equiv z (mod p)$

Let $ a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that \[ ( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).\] [i]J. Suranyi[/i]

Each positive integer shall be coloured red or green such that it satisfies the following properties: - The sum of three not necessarily distinct red numbers is a red number. - The sum of three not necessarily distinct green numbers is a green number. - There are red and green numbers. Find all such colorations!

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form \[x=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $a_{1}a_{2},\cdots$ are integers, $0 \le a_{n}\le n-1$ for $n>1$, and the series terminates. Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^{6}$.

Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.

Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist infinitely many primes that are not contained in $S_{a}$.

Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$. Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points.

Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$

Here is a problem given at the mathematical test at some school: [i]The hypotenuse of the right triangle is 12 inches. The height (distance from the opposite vertex to the hypotenuse) is 12 inches. Find the area of the triangle[/i] Everybody in the class got the answer $42$ square inches, except for the two best students. Can you explain why the two best students could not get the same answer as the majority?

When Tim was Jim’s age, Kim was twice as old as Jim. When Kim was Tim’s age, Jim was 30. When Jim becomes Kim’s age, Tim will be 88. When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?

Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.

Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.

Let $ x$, $ y$, $ z$ be real numbers satisfying $ x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z)$. Prove that \[ x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27\] and determine when equality holds.

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

Hamilton Avenue has eight houses. On one side of the street are the houses numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses, finally returning to house 1 for a cup of tea. Throughout the entire journey he must observe the following rules. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout, each house except house 1 is visited exactly once (house 1 is visited twice) and the postman at no time is allowed to cross the road to the house directly opposite. How many different delivery sequences are possible?

Let $k$ be a given positive integer. Define $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.

Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$ [i]A. Hajnal[/i]

Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.

Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.

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} $