MathDB

2014 Tournament of Towns.

Part of Tournament Of Towns

Subcontests

(7)
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TOT 2014 Spring A J7, grashoppers and 10 points on circle

Points A1,A2,...,A10A_1, A_2, ..., A_{10} are marked on a circle clockwise. It is known that these points can be divided into pairs of points symmetric with respect to the centre of the circle. Initially at each marked point there was a grasshopper. Every minute one of the grasshoppers jumps over its neighbour along the circle so that the resulting distance between them doesn't change. It is not allowed to jump over any other grasshopper and to land at a point already occupied. It occurred that at some moment nine grasshoppers were found at points A1,A2,...,A9A_1, A_2, ... , A_9 and the tenth grasshopper was on arc A9A10A1A_9A_{10}A_1. Is it necessarily true that this grasshopper was exactly at point A10A_{10}?
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TOT 2014 Spring O J5, Ali Baba and 40 thieves, one boat can carry 2 or 3

Ali Baba and the 4040 thieves want to cross Bosporus strait. They made a line so that any two people standing next to each other are friends. Ali Baba is the first, he is also a friend with the thief next to his neighbour. There is a single boat that can carry 22 or 33 people and these people must be friends. Can Ali Baba and the 4040 thieves always cross the strait if a single person cannot sail?

TOT 2014 Spring A J5, bw points, positive inegers on segments

There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?
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TOT 2014 Spring O J2, 15 pasties on a round plate in a circular way

Mother baked 1515 pasties. She placed them on a round plate in a circular way: 77 with cabbage, 77 with meat and one with cherries in that exact order and put the plate into a microwave. All pasties look the same but Olga knows the order. However she doesn't know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Can Olga eat her favourite pasty for sure if she is not allowed to try more than three other pasties?

TOT 2014 Spring A J2, two player game in 5x5 board

Peter marks several cells on a 5×55\times 5 board. Basil wins if he can cover all marked cells with three-cell corners. The corners must be inside the board and not overlap. What is the least number of cells Peter should mark to prevent Basil from winning? (Cells of the corners must coincide with the cells of the board).
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Median-- geo (easy).

Point L is marked on side BC of triangle ABC so that AL is twice as long as the median CM. Given that angle ALC is equal to 45 degrees prove that AL is perpendicular to CM.

TOT 2014 Spring A J4, one king, two wizards, 100 pos. integers

The King called two wizards. He ordered First Wizard to write down 100100 positive integers (not necessarily distinct) on cards without revealing them to Second Wizard. Second Wizard must correctly determine all these integers, otherwise both wizards will lose their heads. First Wizard is allowed to provide Second Wizard with a list of distinct integers, each of which is either one of the integers on the cards or a sum of some of these integers. He is not allowed to tell which integers are on the cards and which integers are their sums. If Second Wizard correctly determines all 100100 integers the King tears as many hairs from each wizard's beard as the number of integers in the list given to Second Wizard. What is the minimal number of hairs each wizard should sacri ce to stay alive?