Baron Munchausen took several cards and wrote a positive integer on each one (some numbers may be the same). The baron reports that he has used only two distinct digits to do that. He also reports that among the leftmost digits of the sums of integers on each pair of these cards there are all the digits from 1 to 9 . Can it occur that the baron is right? Maxim Didin

Peter and Basil take turns drawing roads on a plane, Peter starts. The road is either horizontal or a vertical line along which one can drive in only one direction (that direction is determined when the road is drawn). Can Basil always act in such a way that after each of his moves one could drive according to the rules between any two constructed crossroads, regardless of Peter's actions? Alexandr Perepechko

In an acute-angled triangle ${ABC}$ , its incenter $I$ and circumcenter $O$ are marked. The lines ${AI}$ and ${CI}$ have second intersections with the circumcircle of ${ABC}$ at points $N$ and $M$ respectively. The segments ${MN}$ and ${BO}$ intersect at the point $X$ . Prove that the lines ${XI}$ and ${AC}$ are perpendicular. Fedor Ivlev

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon. Egor Bakaev

Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of two preceding ones. It is known that each of the sequences contains at least one number not present in the other sequence. What is the maximum quantity of numbers common for these two sequences? Boris Frenkin

Peter writes a positive integer on the whiteboard. Each minute Basil multiplies the last written number by 2 or by 3 and writes the product on the whiteboard too. Can Peter choose the starting integer such that, irrespective of Basil's strategy, at any given moment the number of integers on the whiteboard starting with 1 or 2 would exceed the number of the ones starting with 7, 8 or 9 ? Maxim Didin

A squared ${20} \times {20}$ board is split into two-squared dominoes. Prove that some line contains the centers of at least ten of such dominoes. Alexandr Yuran

It is known that each rectangular parallelepiped has the following property: the square of its volume is equal to the product of areas of its three faces sharing a common vertex. Does there exist a parallelepiped which has the same property but is not rectangular? Alexandr Bufetov

Does there exist an infinite sequence of real numbers ${a}_{1},{a}_{2},{a}_{3},\ldots$ such that ${a}_{1} = 1$ and for all positive integers $k$ we have the equality $$ {a}_{k} = {a}_{2k} + {a}_{3k} + {a}_{4k} + \ldots ? $$ Ilya Lobatsky

Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle. Pavel Kozhevnikov

Merlin's castle has 100 rooms and 1000 corridors. Each corridor links some two rooms. Each pair of rooms is linked by one corridor at most. Merlin has given out the plan of the castle to the wise men and declared the rules of the challenge. The wise men need to scatter across the rooms in a manner they wish. Each minute Merlin will choose a corridor and one of the wise men will have to pass along it from one of the rooms at its ends to the other one. Merlin wins when in both rooms on the ends of the chosen corridor there are no wise men. Let us call a number $m$ the magic number of the castle if $m$ wise men can pre-agree before the challenge and act in such a way that Merlin never wins, $m$ being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men). Timofey Vasilyev

Several napkins of equal size and of shape of a unit disc were placed on a table (with overlappings). Is it always possible to hammer several point-sized nails so that all the napkins will be thus attached to the table with the same number of nails? (The nails cannot be hammered into the borders of the discs). Vladimir Dolnikov, Pavel Kozhevnikov

Consider a circumscribed pentagon ${ABCDE}$ . Its incenter lies on the diagonal ${AC}$ . Prove that $$ {AB} + {BC} > {CD} + {DE} + {EA}. $$ Egor Bakaev

Pete puts 100 stones in a row: black one, white one, black one, white one, ..., black one, white one. In a single move either Pete chooses two black stones with only white stones between them, and repaints all these white stones in black, or Pete chooses two white stones with only black stones between them, and repaints all these black stones in white. Can Pete with a sequence of moves described above obtain a row of 50 black stones followed by 50 white stones? Egor Bakaev

A positive integer $M$ has been represented as a product of primes. Each of these primes is increased by 1 . The product $N$ of the new multipliers is divisible by $M$ . Prove that if we represent $N$ as a product of primes and increase each of them by 1 then the product of the new multipliers will be divisible by $N$ . Alexandr Gribalko

A mother and her son are playing. At first, the son divides a ${300}\mathrm{\;g}$ wheel of cheese into 4 slices. Then the mother divides ${280}\mathrm{\;g}$ of butter between two plates. At last, the son puts the cheese slices on those plates. The son wins if on each plate the amount of cheese is not less than the amount of butter (otherwise the mother wins). Who of them can win irrespective of the opponent's actions? Alexandr Shapovalov

The set consists of equal three-cell corners ( $L$ -triminoes), the middle cells of which are marked with paint. A rectangular board has been covered with these triminoes in a single layer so that all triminoes were entirely on the board. Then the triminoes were removed leaving the paint marks where the marked cells were. Is it always possible to know the location of the triminoes on the board using only those paint marks? Alexandr Gribalko

Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one. Sergey Yanzhinov

There are five positive integers written in a row. Each one except for the first one is the minimal positive integer that is not a divisor of the previous one. Can all these five numbers be distinct? Boris Frenkin

In an equilateral triangle ${ABC}$ the segments ${ED}$ and ${GF}$ are drawn to obtain two equilateral triangles ${ADE}$ and ${GFC}$ with sides 1 and 100 (points $E$ and $G$ are on the side ${AC}$ ). The segments ${EF}$ and ${DG}$ meet at point $O$ so that the angle ${EOG}$ is equal to ${120}^{ \circ }$ . What is the length of the side of the triangle ${ABC}$ ? Mikhail Evdokimov

There is a balance without weights and there are two piles of stones of unknown masses, 10 stones in each pile. One is allowed an unlimited number of weighing iterations, but only 9 stones at most fit on any plate of the balance. Is it always possible to determine which stone pile is heavier or establish that they are equal? Sergey Dorichenko