Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$ \frac{a}{b},\; \frac{a-1}{b-1} $$differ by 1, then both are integers.

Let $ABCD$ be the isosceles trapezium with bases $AB$ and $CD$, such that $AC = BC$. The point $M$ is the midpoint of side $AD$. Prove that $\sphericalangle ACM = \sphericalangle CBD$.

Let $x$, $y$, $z$ be non-zero real numbers, such that $x + y + z = 0$ and the numbers $$ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \quad \text{and} \quad \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1 $$are equal. Determine their common value.

The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $$ \sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB. $$Show that $AC + BC > AB + CE$.

Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.

Let $ABCD$ be the trapezium with bases $AB$ and $CD$, such that $\sphericalangle ABC = 90^{\circ}$. The bisector of angle $BAD$ intersects the segment $BC$ in the point $P$. Show that if $\sphericalangle APD = 45^{\circ}$, then area of quadrilateral $APCD$ is equal to the area of the triangle $ABP$.

Consider the regular $101$-gon. A line $l$ does not contain any vertex of this polygon. Prove that line $l$ intersects even number of the diagonals of this polygon.

Let $ABC$ be such a triangle, that $AB = 3\cdot BC$. Points $P$ and $Q$ lies on the side $AB$ and $AP = PQ = QB$. A point $M$ is the midpoint of the side $AC$. Prove that $\sphericalangle PMQ = 90^{\circ}$.

The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$.

The integers $a, b, c$ are not $0$ such that $\frac{a}{b + c^2}=\frac{a + c^2}{b}$. Prove that $a + b + c \le 0$.

Positive integers $a, b, c$ have the property that: $\bullet$ $a$ gives remainder $2$ when divided by $b$, $\bullet$ $b$ gives remainder $2$ when divided by $c$, $\bullet$ $c$ gives remainder $4$ when divided by $a$. Prove that $c = 4$.