Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

A coin is biased in such a way that on each toss the probability of heads is $\frac{2}{3}$ and the probability of tails is $\frac{1}{3}$. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B? $\textbf{(A)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ less than the probability of winning Game B.} $ $\textbf{(B)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ less than the probability of winning Game B.}$ $\textbf{(C)} \text{ The probabilities are the same.}$ $\textbf{(D)} \text{ The probability of winning Game A is }\frac{2}{81}\text{ greater than the probability of winning Game B.}$ $\textbf{(E)} \text{ The probability of winning Game A is }\frac{4}{81}\text{ greater than the probability of winning Game B.}$

Polynomial $ P(t)$ is such that for all real $ x$, \[ P(\sin x) \plus{} P(\cos x) \equal{} 1. \] What can be the degree of this polynomial?

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:

A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_0, P_1, P_2, \ldots, \ldots, P_{2n}$ such that $P_0$ and $P_{2n}$ are both the origin and $P_{i}P_{i+1}=1$ for each $i.$ A closed lattice loop of length $2026$ is chosen uniformly at random from all such loops. Let $k$ be the maximum integer such that the line $\ell$ with equation $x+y=k$ passes through at least one point of the loop. Compute the expected number of indices $i$ such that $0 \le i \le 2025$ and $P_i$ lies on $\ell.$ (A lattice point is a point with integer coordinates.)

A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$? $ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$

On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$. (a) Prove that $P$, $T$, $S$ are collinear. (b) Prove that $P$, $K$, $L$ are collinear. [i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}$$ prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$

Ten coins are placed in a circle, showing “heads” (the tails are down). Two moves are allowed: (a) turn over four consecutively placed coins; (b) turn over four coins placed as $XX OXX$ ($X$ is one of the coins to be turned over, $O$ is not touched). Is it possible to have all ten coins showing “tails” after a finite sequence of such moves? (A Tolpygo)

Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence: $\bullet$ $n_{i+1}\geq n_i$ $\bullet$ There is at least one number $i$, such that $n_i=2022$ $\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.