Prove that $y\sqrt{3-2x}+x\sqrt{3-2y}\le x^2+y^2$ for any number $x,y\in\left[1,\frac32\right]$. When does equality occur?

Let $S(n)$ denote the sum of digits of a natural number $n$. Find all $n$ for which $n+S(n)=2004$.

A line $d_i~(i=1,2,3)$ intersects two opposite sides of a square $ABCD$ at points $M_i$ and $N_i$. Prove that if $M_1N_1=M_2N_2=M_3N_3$, then two of the lines $d_i$ are either parallel or perpendicular.

Find all permutations of the numbers $1,2,\ldots,9$ in which no two adjacent numbers have a sum divisible by $7$ or $13$.

Let $a,b,c,d$ be real numbers. Prove that the set $M=\left\{ax^3+bx^2+cx+d|x\in\mathbb R\right\}$ contains no irrational numbers if and only if $a=b=c=0$ and $d$ is rational.

Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. Prove that if $AP=4PN$, then $ABCD$ is a parallelogram.

Let $n$ be a positive integer. We denote by $S$ the sum of elements of the set $M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}$. (a) Show that $S$ is divisible by $6$. (b) Find all $n\in\mathbb N$ for which $S+(1-n)(1+n)=2001$.

Prove that every positive integer $k$ can be written as $k=\frac{mn+1}{m+n}$, where $m,n$ are positive integers.

Prove that $\frac1{2002}<\frac12\cdot\frac34\cdot\frac56\cdots\frac{2001}{2002}<\frac1{44}$.

If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

In a triangle $ABC$, the line symmetric to the median through $A$ with respect to the bisector of the angle at $A$ intersects $BC$ at $M$. Points $P$ on $AB$ and $Q$ on $AC$ are chosen such that $MP\parallel AC$ and $MQ\parallel AB$. Prove that the circumcircle of the triangle $MPQ$ is tangent to the line $BC$.

Find all integers that can be written as $\frac{(a+b)(b+c)(c+a)}{abc}$, where $a,b,c$ are pairwise coprime positive integers.

Consider all quadratic trinomials $x^2+px+q$ with $p,q\in\{1,\ldots,2001\}$. Which of them has more elements: those having integer roots, or those having no real roots?

The incircle of a triangle $ABC$ is centered at $I$ and touches $AC,AB$ and $BC$ at points $K,L,M$, respectively. The median $BB_1$ of $\triangle ABC$ intersects $MN$ at $D$. Prove that the points $I,D,K$ are collinear.

Find the intersection of all sets of consecutive positive integers having at least four elements and the sum of elements equal to $2001$.

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.

Prove that the sum of two consecutive prime numbers is never a product of two prime numbers.

In a triangle $ABC$ the altitude $AD$ is drawn. Points $M$ on side $AC$ and $N$ on side $AB$ are taken so that $\angle MDA=\angle NDA$. Prove that the lines $AD,BM$ and $CN$ are concurrent.

Show that there are nine distinct nonzero integers such that their sum is a perfect square and the sum of any eight of them is a perfect cube.

Prove that for any integer $n>1$ there are distinct integers $a,b,c$ between $n^2$ and $(n+1)^2$ such that $c$ divides $a^2+b^2$.

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

Suppose that $a,b,c$ are real numbers such that $\left|ax^2+bx+c\right|\le1$ for $-1\le x\le1$. Prove that $\left|cx^2+bx+a\right|\le2$ for $-1\le x\le1$.

Find all real solutions of the equation $$x^2+y^2+z^2+t^2=xy+yz+zt+t-\frac25.$$

Prove that there are no $2003$ odd positive integers whose product equals their sum. Is the previous proposition true for $2001$ odd positive integers?

During a fight, each of the $2001$ roosters has torn out exactly one feather of another rooster, and each rooster has lost a feather. It turned out that among any three roosters there is one who hasn’t torn out a feather from any of the other two roosters. Find the smallest $k$ with the following property: It is always possible to kill $k$ roosters and place the rest into two henhouses in such a way that no two roosters, one of which has torn out a feather from the other one, stay in the same henhouse.

In a triangle $ABC$, the angle bisector at $A$ intersects $BC$ at $D$. The tangents at $D$ to the circumcircles of the triangles $ABD$ and $ACD$ meet $AC$ and $AB$ at $N$ and $M$, respectively. Prove that the quadrilateral $AMDN$ is inscribed in a circle tangent to $BC$.

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.

Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.

A box $3\times5\times7$ is divided into unit cube cells. In each of the cells, there is a cockchafer. At a signal, every cockchafer moves through a face of its cell to a neighboring cell. (a) What is the minimum number of empty cells after the signal? (b) The same question, assuming that the cockchafers move to diagonally adjacent cells (sharing exactly one vertex).

Consider the set $M=\{1,2,...,n\},n\in\mathbb N$. Find the smallest positive integer $k$ with the following property: In every $k$-element subset $S$ of $M$ there exist two elements, one of which divides the other one.

Let $m\ge2$ be an integer. The sequence $(a_n)_{n\in\mathbb N}$ is defined by $a_0=0$ and $a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor}$ for all $n$. Determine $\lim_{n\to\infty}\frac{a_n}n$.

For an arbitrary point $D$ on side $BC$ of an acute-angled triangle $ABC$, let $O_1$ and $O_2$ be the circumcenters of the triangles $ABD$ and $ACD$, and $O$ be the circumcenter of the triangle $AO_1O_2$. Find the locus of $O$ when $D$ moves across $BC$.

Let $P(x)=x^n+a_1x^{n-1}+\ldots+a_n$ ($n\ge2$) be a polynomial with integer coefficients having $n$ real roots $b_1,\ldots,b_n$. Prove that for $x_0\ge\max\{b_1,\ldots,b_n\}$, $$P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.$$

Prove that the sum of the numbers $1,2,\ldots,n$ divides their product if and only if $n+1$ is a composite number.

For a positive integer $n$, denote $A_n=\{(x,y)\in\mathbb Z^2|x^2+xy+y^2=n\}$. (a) Prove that the set $A_n$ is always finite. (b) Prove that the number of elements of $A_n$ is divisible by $6$ for all $n$. (c) For which $n$ is the number of elements of $A_n$ divisible by $12$?

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.

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

The sequence of functions $f_n:[0,1]\to\mathbb R$ $(n\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$. Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$, $x=1$, and $y=0$. Compute $$\lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}}n\right)^n.$$

A regular $n$-gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.

Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.

In a triangle $ABC$, $BC=a$, $AC=b$, $\angle B=\beta$ and $\angle C=\gamma$. Prove that the bisector of the angle at $A$ is equal to the altitude from $B$ if and only if $b=a\cos\frac{\beta-\gamma}2$.

For each integer $n\ge2$ prove the inequality $$\log_23+\log_34+\ldots+\log_n(n+1)<n+\ln n-0.9.$$

Prove that if a positive integer $n$ divides the five-digit numbers $\overline{a_1a_2a_3a_4a_5}$, $\overline{b_1b_2b_3b_4b_5}$, $\overline{c_1c_2c_3c_4c_5}$, $\overline{d_1d_2d_3d_4d_5}$, $\overline{e_1e_2e_3e_4e_5}$, then it also divides the determinant $$D=\begin{vmatrix}a_1&a_2&a_3&a_4&a_5\\b_1&b_2&b_3&b_4&b_5\\c_1&c_2&c_3&c_4&c_5\\d_1&d_2&d_3&d_4&d_5\\e_1&e_2&e_3&e_4&e_5\end{vmatrix}.$$

Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$

Let $P$ be the midpoint of the arc $AC$ of a circle, and $B$ be a point on the arc $AP$. Let $M$ and $N$ be the projections of $P$ onto the segments $AC$ and $BC$ respectively. Prove that if $D$ is the intersection of the bisector of $\angle ABC$ and the segment $AC$, then every diagonal of the quadrilateral $BDMN$ bisects the area of the triangle $ABC$.