MathDB

8

Part of 2021 All-Russian Olympiad

Problems(3)

Strange combo game

Source: ARO 2021 9.8

4/20/2021
One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {1,2,31,2,3}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly 200200; else it is successful. Prove that the sages can create a strategy, by which they can win the game.
combinatorics
Pentagon geo from ARO

Source: ARO 2021 10.8

4/20/2021
Given is a cyclic pentagon ABCDEABCDE, inscribed in a circle kk. The line CDCD intersects ABAB and AEAE in XX and YY respectively. Segments EXEX and BYBY intersect again at PP, and they intersect kk in QQ and RR, respectively. Point A′A' is reflection of AA across CDCD. The circles (PQR)(PQR) and (A′XY)(A'XY) intersect at MM and NN. Prove that CMCM and DNDN intersect on (PQR)(PQR).
geometry
Combinatorics

Source: ARO 2021 11.8

4/20/2021
Each girl among 100100 girls has 100100 balls; there are in total 1000010000 balls in 100100 colors, from each color there are 100100 balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has 100100 balls, colored in the 100100 distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has 100100 balls, colored in the 100100 colors.
combinatorics