John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.

A table consisting of $9$ rows and $2001$ columns is filfed with integers $1,2,..., 2001$ in such a way that each of these integers occurs in the table exactly $9$ times and the integers in any column differ by no more than $3$. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).

A convex $n$-gon has exactly three obtuse interior angles. Find all possible values of $n$.

Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.

There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$.

We call a triple of positive integers $(a, b, c)$ harmonic if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.

A tribe called Ababab uses only letters $A$ and $B$, and they create words according to the following rules: (1) $A$ is a word; (2) if $w$ is a word, then $ww$ and $w\overline{w}$ are also words, where $\overline{w}$ is obtained from $w$ by replacing all letters $A$ with $B$ and all letters $B$ with $A$ ( $xy$ denotes the concatenation of $x$ and $y$) (3) all words are created by rules (1) and (2). Prove that any two words with the same number of letters differ exactly in half of their letters.

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

Points $D,E$ and $F$ are taken on the sides $BC,CA,AB$ of a triangle $ABC$ respectively so that the segments $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB}$ .

If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.

Consider all trapezoids in a coordinate plane with interior angles of $90^o, 90^o, 45^o$ and $135^o$ whose bases are parallel to a coordinate axis and whose vertices have integer coordinates. Define the size of such a trapezoid as the total number of points with integer coordinates inside and on the boundary of the trapezoid. (a) How many pairwise non-congruent such trapezoids of size $2001$ are there? (b) Find all positive integers not greater than $50$ that do not appear as sizes of any such trapezoid.

Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$

Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.

A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

A $3\times 3$ table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column. (a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table. (b) Show that there is such a table not all of whose entries are $0$.