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 $\lim_{x \to +\infty} f(x) = 0$?

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.

Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).

Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?

A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

Find the tangents of the angles of a triangle knowing that they are positive integers.

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

Suppose that $ m$ and $ n$ are odd integers such that $ m^2\minus{}n^2\plus{}1$ divides $ n^2\minus{}1$. Prove that $ m^2\minus{}n^2\plus{}1$ is a perfect square.

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]

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]

At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Let $ a, b, c$ be three natural numbers such that $ a < b < c$ and $ gcd (c \minus{} a, c \minus{} b) \equal{} 1$. Suppose there exists an integer $ d$ such that $ a \plus{} d, b \plus{} d, c \plus{} d$ form the sides of a right-angled triangle. Prove that there exist integers, $ l,m$ such that $ c \plus{} d \equal{} l^{2} \plus{} m^{2} .$ [b][Weightage 17/100][/b]

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$

Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer $k$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $kN$ blocks (no matter what $N$ turns out to be)?

Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

I tried to search SRMC problems,but i didn't find them(I found only SRMC 2006).Maybe someone know where on this site i could find SRMC problems?I have all SRMC problems,if someone want i could post them, :wink: Here is one of them,this is one nice inequality from first SRMC: Let $ n$ be an integer with $ n>2$ and $ a_{1},a_{2},\dots,a_{n}\in R^{\plus{}}$.Given any positive integers $ t,k,p$ with $ 1<t<n$,set $ m\equal{}k\plus{}p$,prove the following inequalities: a) $ \frac{a_{1}^{p}}{a_{2}^{k}\plus{}a_{3}^{k}\plus{}\dots\plus{}a_{t}^{k}}\plus{}\frac{a_{2}^{p}}{a_{3}^{k}\plus{}a_{4}^{k}\plus{}\dots\plus{}a_{t\plus{}1}^{k}}\plus{}\dots\plus{}\frac{a_{n\minus{}1}^{p}}{a_{n}^{k}\plus{}a_{1}^{k}\plus{}\dots\plus{}a_{t\minus{}2}^{k}}\plus{}\frac{a_{n}^{p}}{a_{1}^{k}\plus{}a_{2}^{k}\plus{}\dots\plus{}a_{t\minus{}1}^{k}}\geq\frac{(a_{1}^{p}\plus{}a_{2}^{p}\dots\plus{}a_{n}^{p})^{2}}{(t\minus{}1) ( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ b)$ \frac{a_{2}^{k}\plus{}a_{3}^{k}\dots\plus{}a_{t}^{k}}{a_{1}^{p}}\plus{}\frac{a_{3}^{k}\plus{}a_{4}^{k}\dots\plus{}a_{t\plus{}1}^{k}}{a_{2}^{p}}\plus{}\dots\plus{}\frac{a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{t\minus{}1}^{k}}{a_{n}^{p}}\geq\frac{(t\minus{}1)(a_{1}^{k}\plus{}a_{2}^{k}\dots\plus{}a_{n}^{k})^{2}}{( a_{1}^{m}\plus{}a_{2}^{m}\plus{}\dots\plus{}a_{n}^{m})}$ :wink:

Find all $f: R\rightarrow R$ such that (i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite (ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$

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]