There are two types of items in Alik's collection: badges and bracelets and there are more badges than bracelets. Alik noticed that if he increases the number of bracelets some (not necessarily integer) number of times without changing the number of icons, then in its collection will be $100$ items. And if, on the contrary, he increases the initial number of badges by the same number of times, leaving the same number of bracelets, then he will have $101$ items. How many badges and how many bracelets could there be in Alik's collection?

In a Cartesian coordinate system (with the same scale on the x and y axes)there is a graph of the exponential function $y=3^x$. Then the y-axis and all marks on the x-axis erased. Only the graph of the function and the x-axis remained without a scale and a mark of $0$. How can you restore the y-axis using a compass and ruler?

Bisector $AL$ is drawn in an acute triangle $ABC$. On the line $LA$ beyond the point $A$, the point K is chosen with $AK = AL$. Circumcirles of triangles $BLK$ and $CLK$ intersect segments $AC$ and $AB$ at points $P$ and $Q$ respectively. Prove that lines $PQ$ and $BC$ are parallel.

The starship is in a half-space at a distance $a$ from its boundary. The crew knows about it, but has no idea in which direction to move in order to reach the boundary plane. The starship can fly in space along any trajectory, measuring the length of the path traveled, and has a sensor that sends a signal when the border has been reached. Can a starship be guaranteed to reach the border with a path no longer than $14a$?

The Sultan gathered $300$ court sages and offered them a test. There are caps of $25$ different colors, known in advance to the sages. The Sultan said that one of these caps will be put on each of the sages, and if for each color write the number of caps worn, then all numbers will be different. Every sage can see the caps of the other sages, but not own cap. Then all the sages will simultaneously announce the supposed color of their cap. Can sages advance agree to act in such a way that at least $150$ of them are guaranteed to name a color right?

$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$. Prove $bc \geq b+c$

The volleyball championship with $16$ teams was held in one round (each team played with each exactly one times, there are no draws in volleyball). It turned out that some two teams won the same number of matches. Prove there are the three teams that beat each other in a round robin (i.e. A beat B, B beat C, and C beat A).

In a convex $12$-gon, all angles are equal. It is known that the lengths of some $10$ of its sides are equal to $1$, and the length of one more equals $2$. What can be the area of this $12$-gon?

A diagonal is drawn in an isosceles trapezoid. By the contour of each of the resulting two triangles creeps its own beetle. The velocities of the beetles are constant and identical. Beetles don't change directions around their contours, and along the diagonal of the trapezoid they crawl in different directions. Prove that for any starting positions of the beetles they will ever meet.

Tanya wrote numbers in forms $n^7-1$ for $n=2,3,...$ and noticed that for $n=8$ she got number divisible by $337$. For what minimal $n$ did she get number divisible by $2022$?