It is known that $a+\frac{b^2}{a}=b+\frac{a^2}{b}$. Is it true that $a=b$, where $a$ and $b$ are nonzero real numbers? Proposed by R.Fedorov

Along a circle, 10 iron weights have been placed. Between every two weights there is a brass ball. The mass of each ball is equal to the difference of the masses of its neighboring weights. Prove that it is possible to divide the balls among two pans so as to make the balance in equilibrium. Proposed by V. Proizvolov

At the nodes of graph paper, gardeners live; everywhere around them grow flowers. Each flower is to be taken care of by the three gardeners nearest to it. One of the gardeners wishes to know which are the flowers (s)he has to take care of. Sketch the plot of these gardeners. Proposed by I. F. Sharygin

Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$. Proposed by V. Proizvolov

A rook stands in a corner of an $n$ by $n$ chess board. For what $n$, moving alternately along horizontals and verticals, can the rook visit all the cells of the board and return to the initial corner after $n^2$ moves? (A cell is visited only if the rook stops on it, those that the rook “flew over” during the move are not counted as visited.) Proposed by A. Spivak

Eight students solved $8$ problems. a) It turned out that each problem was solved by $5$ students. Prove that there are two students such that each problem is solved by at least one of them. b) If it turned out that each problem was solved by $4$ students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.) Proposed by S. Tokarev