Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

$n$ is called [i]diamond 2005[/i] if $n=\overline{...ab999...99999cd...}$, e.g. $2005 \times 9$. Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite [i]diamond 2005[/i]. Compare with [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url]

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality $$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$

Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.

A circle with center O is inscribed in a quadrilateral ABCD and touches its non-parallel sides BC and AD at E and F respectively. The lines AO and DO meet the segment EF at K and N respectively, and the lines BK and CN meet at M. Prove that the points O,K,M and N lie on a circle.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

Find all polynomials $P(x)$ with real coefficients which satisfies \\ $P(2015)=2025$ and $P(x)-10=\sqrt{P(x^{2}+3)-13}$ for every $x\ge 0$ .

A $\textit{lattice point}$ of the coordinate plane is a point $(x,y)$ in which both $x$ and $y$ are integers. Let $k\geq2$ be a positive integer. Find the smallest positive integer $c_k$ (which may depend on $k$) such that every lattice point can be colored with one of $c_k$ colors, subject to the following two conditions: [list=1] [*] If $(x,y)$ and $(a,b)$ are two distinct neighboring points; that is, $|x-a|\leq1$ and $|y-b|\leq1$, then $(x,y)$ and $(a,b)$ must be different colors. [/*] [*] If $(x,y)$ and $(a,b)$ are two lattice points such that $x\equiv a\pmod{k}$ and $y\equiv b\pmod{k}$, then $(x,y)$ and $(a,b)$ must be the same color. [/*] [/list]

Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that: For each point $P\in \Gamma$, there exists a partition of $S$ in sets $A$ and $B$ ($A\cup B=S$, $A\cap B=\phi$) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$.

The quadrilateral $ABCD$ has $AB$ parallel to $CD$. $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$. M is the intersection of $AQ$ and $DP$, and $N$ is the intersection of $PC$ and $QB$. Find $MN$ in terms of $AB$ and $CD$.

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

In a regular pyramid with a regular $n$-gon as a base, the dihedral angle between a lateral face and the base is equal to $\alpha$, and the angle between a lateral edge and the base is equal to $\beta$. Prove that $sin^2 \alpha - sin^2 \beta \leq tg^2 \frac{\pi}{2n}$.

Several small villages are situated on the banks of a straight river. On one side, there are 20 villages in a row, and on the other there are 15 villages in a row. We would like to build bridges, each of which connects a village on the one side with a village on the other side. The bridges must not cross, and it should be possible to get from any village to any other village using only those bridges (and not any roads that might exist between villages on the same side of the river). How many different ways are there to build the bridges.

Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.

Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality $$ f(f(x)+f(y))=(x+y)f(x+y) $$ for all real $x$ and $y$. [i](B. Serankou)[/i]

On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 49 \qquad \textbf{(E)}\ 112$

A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$

Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$. [i]Proposed by Tony Wang[/i]

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

In the future, each country in the world produces its Olympic athletes via cloning and strict training programs. Therefore, in the finals of the 200 m free, there are two indistinguishable athletes from each of the four countries. How many ways are there to arrange them into eight lanes?

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[ (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2. \] Prove that $ABCD$ is an isosceles trapezoid.