Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant. [i]V. Totik[/i]

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.

The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?

If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)

Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$. Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$. What is the greatest distance between Alex and Farmer Tim while they are walking in these paths?

Find the value of $x+y$ for which the expression \[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\] is minimized.

Find all prime numbers $p$ such that $1 +p+p^2 +p^3 +p^4$ is a perfect square.

Prove that for any non-zero real numbers $a, b, c,$ \[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

Given a set of n elements, find the largest number of subsets such that no subset is contained in any other

The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$

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.

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]

If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

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])

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

Prove that if $a,b,c$ are positive integers such that $c^2 = a^2+b^2$, then both $c^2+ab$ and $c^2-ab$ are also expressible as the sums of squares of two positive integers.

For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$. [b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.