For arbitrary positive integers $a, b$, denote $a \ominus b = \frac{a-b}{\text{gcd}(a,b)}$.
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $\text{gcd}(n, n \ominus m) = 1$ for every positive integer $m \lt n$;
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
Four musketeers together bought a plot of rectangular shape and paid for it equally. They divided the plot by two cuts into four pieces of rectangular shape, from which every musketeer got one. It turned out that one musketeer obtained as much land as the other three in total. Prove that the price per acre of one musketeer's piece turned out as large as the sum of the prices per acre of the other three musketeers' pieces. (Juniors.)
In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.
Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$. (Seniors.)
A ball bearing consists of two cylinders with the same axis and $n$ equal balls between them. The centers of all the balls are on the same plane perpendicular to the axis of the cylinders and each ball touches both cylinders and two adjacent balls. Let $r$ be the radius of the balls and let $R$ be the radius of the outer cylinder. Prove that $r/R \lt \frac{\pi}{n+\frac{\pi}{n}}$. (Grade 11.)
Let the angles of a triangle be $\alpha$, $\beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality
$$
cot^2 \alpha + cot^2 \beta + cot^2 \gamma \ge 3 \left( \frac{9R^2}{p^2} - 1 \right).
$$
When is the equality achieved?
In a coordinate city there are $n \ge 3$ tramlines parallel to the $x$-axis such that each line begins from $x$-coordinate $0$ and ends at $x$-coordinate $n$. Exactly one tram of length $1$ is moving on each line: on the first line with speed $1$, on the second line with speed $2$ etc, until on the last line with the speed $n$. When a tram reaches the end of its line it instantly starts moving back without turning around. In the morning all trams start moving at the same time from the starting position where the $x$-coordinate of the back end of the tram is $0$. Prove that the trams' projections onto the $x$-axis never cover the whole interval from $0$ to $n$. (Grade 12.)
Call a positive integer $n$ prime-prone if there exist at least three prime numbers from which we can get $n$ by removing the last digit. Prove that every two prime-prone positive integers differ from each other by at least $3$. (Juniors.)
Every unit square of a $n \times n$ board is colored either red or blue so that among all $2 \times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different).
a) Find the least possible value of $n$.
b) For the least possible value of $n$ find the least possible number of red unit squares.
Let $a$ and $b$ be the lengths of the legs of a given right triangle. Prove that angle $\varphi$, where $0 \lt \varphi \lt 90^\circ$, is an acute angle of this triangle if and only if $(a \cos \varphi + b \sin \varphi)(a \sin \varphi + b \cos \varphi) = 2ab$. (Seniors.)
Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.
Let $n \ge 2$. Positive integers $a_1, a_2, \dots, a_n$ whose sum is even and which satisfy $a_i \le i$ for every $i = 1, 2, \dots, n$, are given. Prove that it is possible to choose signs in the expression $a_1 \pm a_2 \pm \dots \pm a_n$ in such a way that its value becomes $0$. (Seniors.)
Juku drew a regular hexagon and chose three triangles with different areas whose vertices were among the vertices of the hexagon. Prove that the sum of the areas of the triangles is equal to the area of the hexagon.
Let $a$, $b$ and $c$ be positive integers such that $ab$ is divisible by $2c$, $bc$ is divisible by $3a$ and $ca$ is divisible by $5b$. Find the least possible value of $abc$.
Points $A'$, $B'$ and $C'$ are chosen correspondingly on the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$ so that $\frac{|A'B|}{|AB|} = \frac{|B'C|}{|BC|} = \frac{|C'A|}{|CA|} = k$. Find all positive real numbers $k$ for which the area of the triangle $A'B'C'$ is exactly half of the area of the triangle $ABC$.
In a regular $n$-gon, either $0$ or $1$ is written at each vertex. Using non-intersecting diagonals, Juku divides this polygon into triangles. Then he writes into each triangle the sum of the numbers at its vertices. Prove that Juku can choose the diagonals in such a way that the maximal and minimal number written into the triangles differ by at most $1$. (Seniors.)
In the buffet of the kitchen, there are three candy boxes, each containing the same number of candies. Every time when Juku goes into the kitchen, he takes either three candies from one box or one candy from every box. Prove that irrespectively of how Juku takes the candies, he always retains the possibility to completely clean out all candy boxes. (Juniors.)
Call *pure* any positive integer $n$ that does not occur in any integer sequence $c_0, c_1, c_2, \dots$, where $0 \lt c_0 \lt n$ and
$$
c_i = \begin{cases} \frac{1}{2}c_{i-1} & \text{if } c_{i-1} \text{ is even,} \\ 3c_{i-1} - 1 & \text{if } c_{i-1} \text{ is odd,} \end{cases}
$$
for every $i \ge 1$. (For instance, $10$ is not pure since it occurs in the sequence $5, 14, 7, 20, 10, \ldots$ ...)
a) Is every positive multiple of $3$ pure?
b) Prove that if an integer $n \gt 1$ is pure but not divisible by $3$, then $n + 1$ is divisible by $6$.