Solve the equation: $\sin ^3x+\sin ^32x+\sin ^33x=(\sin x + \sin 2x + \sin 3x)^3$.

The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?

Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers. [i]Proposed by Huawei Zhu[/i]

Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$

The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). [center][img]https://cdn.artofproblemsolving.com/attachments/b/1/e9245179b7e0f5acb98b226bdc6db87fd72ad5.png[/img] [/center] Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .

All positive integers are coloured either red or green, such that the following conditions are satisfied: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

We call a sequence of integers a [i]Fibonacci-type sequence[/i] if it is infinite in both ways and $a_{n}=a_{n-1}+a_{n-2}$ for any $n\in\mathbb{Z}$. How many [i]Fibonacci-type sequences[/i] can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than $N$ ? (two sequences are considered to be the same if they differ only by shifting of indices) [i]Proposed by I. Pevzner[/i]

Let $u_n$ denote the ramp function $$ u_n (x) =\begin{cases} -n \;\; \text{for} \;\; x \leq -n, \\ \; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\ \;n \;\; \; \text{for} \;\; n \leq x, \end{cases}$$ and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)

Given a positive integer $n \geq 3$, let $C_{n}$ be the collection of all $n$-tuples $a=(a_{1},a_{2},...,a_{n})$ of nonnegative reals $a_i$, $i=1,...,n$, such that $a_{1}+a_{2}+...+a_{n}=1$. For $k \in \left \{ 1,...,n-1 \right \}$ and $a \in C_{n}$, consider the sum set $\sigma_{k}(a) = \left \{a_{1}+...+a_{k},a_{2}+...+a_{k+1},...,a_{n-k+1}+...+a_{n} \right \}$. Show the following. (a) There exist $m_k=\max\{\min\sigma_k(a):a\in\mathcal{C}_n\}$ and $M_k=\min\{\max\sigma_k(a):a\in\mathcal{C}_n\}$. (b) It holds that $\displaystyle{1\leq\sum_{k=1}^{n-1}(\frac{1}{M_k}-\frac{1}{m_k})\leq n-2}$. Moreover, on the left side, equality is attained only for finitely many values of $n$, whereas on the right side, equality holds for infinitely values of $n$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$, there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$). For any set $A$ composed of airports $(1\le | A| \le n-1)$, there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$. Prove that: For any airport $x$, we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.

Find the all $(m,n)$ integer pairs satisfying $m^4+2n^3+1=mn^3+n$.

Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions: $(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$ $(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\] then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.

If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)? (A) at B (B) at C (C) at D (D) at both B and D (E) From C to D

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

There is a bottle with a flat and circular bottom, closed and partially filled of wine, so that its level does not exceed the cylindrical part. Discuss in which cases the capacity of the bottle can be calculated without opening it, having only one double graduated decimeter; and if possible, describe how it would be calculated. (Problem of the Italian [i]Gara Mathematica[/i]).

$O$ is the centre of the circumcircle of triangle $ABC$, and $M$ is its orthocentre. Point $A$ is reflected in the perpendicular bisector of the side $BC$,$ B$ is reflected in the perpendicular bisector of the side $CA$, and finally $C$ is reflected in the perpendicular bisector of the side $AB$. The images are denoted by $A_1, B_1, C_1$ respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that $O$ bisects the line segment $MK$.

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

How many ways are there to permute the letters $\{S,C,R, A,M,B,L,E\}$ without the permutation containing the substring $L AME$?

A triangle $ABC$ is located in a cartesian plane $\pi$ and has a perimeter of $3 + 2\sqrt3$. It is known that the triangle $ABC$ has the property that any triangle in the plane $\pi$, congruent with it, contains inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that the triangle $ABC$ is equilateral.

Show that, for any positive integer $n$, \[ \sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1}, \] where $[x]$ means the greatest integer not exceeding $x$, and $\textstyle\binom nr$ is the binomial coefficient "$n$ choose $r$", with the convention $\textstyle\binom n0 = 1$.

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]