MathDB

P5

Part of 2022/2023 Tournament of Towns

Problems(6)

Easy nonagon geo

Source: 44th International Tournament of Towns, Junior A-Level P5, Fall 2022

2/16/2023
On the sides of a regular nonagon ABCDEFGHIABCDEFGHI, triangles XAB,YBC,ZCDXAB, YBC, ZCD and TDETDE are constructed outside the nonagon. The angles at X,Y,Z,TX, Y, Z, T in these triangles are each 20∘20^\circ. The angles XAB,YBC,ZCDXAB, YBC, ZCD and TDETDE are such that (except for the first angle) each angle is 20∘20^\circ greater than the one listed before it. Prove that the points X,Y,Z,TX, Y , Z, T lie on the same circle.
geometrynonagonTournament of Towns
What a lame rook

Source: 44th International Tournament of Towns, Senior O-Level P5, Fall 2022

2/16/2023
A 2N×2N2N\times2N board is covered by non-overlapping dominos of 1×21\times2 size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is:
[*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves?
combinatoricsdominoesboardTournament of Towns
Counterfeit coins are no good!

Source: 44th International Tournament of Towns, Junior O-Level P5, Fall 2022

2/16/2023
Alice has 8 coins. She knows for sure only that 7 of these coins are genuine and weigh the same, while the remaining one is counterfeit and is either heavier or lighter than any of the other 7. Bob has a balance scale. The scale shows which plate is heavier but does not show by how much. For each measurement, Alice pays Bob beforehand a fee of one coin. If a genuine coin has been paid, Bob tells Alice the correct weighing outcome, but if a counterfeit coin has been paid, he gives a random answer. Alice wants to identify 5 genuine coins and not to give any of these coins to Bob. Can Alice achieve this result for sure?
combinatoricsgameTournament of Towns
Integers coloured in four colours

Source: 44th International Tournament of Towns, Junior A-Level P5, Spring 2023

1/9/2024
The positive integers from 1 to 100 are painted into three colors: 50 integers are red, 25 integers are yellow and 25 integers are green. The red and yellow integers can be divided into 25 triples such that each triple includes two red integers and one yellow integer which is greater than one of the red integers and smaller than another one. The same assertion is valid for the red and green integers. Is it necessarily possible to divide all the 100 integers into 25 quadruples so that each quadruple includes two red integers, one yellow integer and one green integer such that the yellow and the green integer lie between the red ones?
Alexandr Gribalko
combinatoricscolorings
Five points with distnaces exceeding 2

Source: 44th International Tournament of Towns, Senior O-Level P5, Spring 2023

1/9/2024
The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space?
Alexey Tolpygo
geometrydistances
Weighting problem with coins on a board

Source: 44th International Tournament of Towns, Junior O-Level P5, Spring 2023

1/9/2024
There is a single coin on each square of a 5×55 \times 5 board. All the coins look the same. Two of them are fakes and have equal weight. Genuine coins are heavier than fake ones and also weigh the same. The fake coins are on the squares sharing just one vertice. Is it possible to determine for sure a) 13 genuine coins; b) 15 genuine coins; and c) 17 genuine coins in a single weighing on a balance with no unit weights?
Rustem Zhenodarov, Alexandr Gribalko, Sergey Tokarev
combinatoricsweights