MathDB

5

Part of 2023/2024 Tournament of Towns

Problems(4)

four points concyclic

Source: 45th International Tournament of Towns, Senior A-Level P5, Fall 2023

12/11/2023
Chord DED E of the circumcircle of the triangle ABCA B C intersects sides ABA B and BCB C in points PP and QQ respectively. Point PP lies between DD and QQ. Angle bisectors DFD F and EGE G are drawn in triangles ADPA D P and QECQ E C. It turned out that the points DD, F,G,EF, G, E are concyclic. Prove that the points A,P,Q,CA, P, Q, C are concyclic. Azamat Mardanov
geometry
balance weighting.

Source: 45th International Tournament of Towns, Junior A-Level P5, Fall 2023

12/16/2023
5. Tom has 13 weight pieces that look equal, however 12 of them weigh the same and the 13th piece is fake and weighs more than the others. He also has two balances: one shows correctly which pan is heavier or that their weights are equal, the other one gives the correct result when the weights on the pans differ, and gives a random result when the weights are equal. (Tom does not know which balance is which). Tom can choose the balance before each weighting. Prove that he can surely determine the fake weight piece in three weighings. Andrey Arzhantsev
combinatorics
Nine farmers and a crown

Source: 45th International Tournament of Towns, Senior O-Level P5, Fall 2023

12/16/2023
5. Nine farmers have a checkered 9×99 \times 9 field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 9 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 8 berries from the field so that for sure no farmer will notice this? Tatiana Kazitsina
combinatorics
brick game

Source: 45th International Tournament of Towns, Junior O-Level P5, Fall 2023

12/16/2023
5. Alice and Bob have found 100 bricks of the same size, 50 white and 50 black. They came up with the following game. A tower will mean one or several bricks standing on top of one another. At the start of the game all bricks lie separately, so there are 100 towers. In a single turn a player must put one of the towers on top of another tower (no flipping towers allowed) so that the resulting tower has no same-colored bricks next to each other. The players make moves in turns, Alice starts first. The one unable to make the next move loses the game. Who can guarantee the win regardless of the opponent's strategy?
combinatorics