MathDB

P3

Part of 2021/2022 Tournament of Towns

Problems(7)

Grasshopper Gerald hops around

Source: 43rd International Tournament of Towns, Senior A-Level P3, Fall 2021

2/18/2023
Grasshopper Gerald and his 2020 friends play leapfrog on a plane as follows. At each turn Gerald jumps over a friend so that his original point and his resulting point are symmetric with respect to this friend. Gerald wants to perform a series of jumps such that he jumps over each friend exactly once. Let us say that a point is achievable if Gerald can finish the 2020th jump in it. What is the maximum number NN{} such that for some initial placement of the grasshoppers there are just NN{} achievable points?
Mikhail Svyatlovskiy
combinatoricsTournament of Towns
Possible length of a segment

Source: 43rd International Tournament of Towns, Junior A-Level P3, Fall 2021

2/18/2023
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment?
Maxim Volchkevich
geometrylengthsTournament of Towns
Square is painted as fast as possible

Source: 43rd International Tournament of Towns, Senior O-Level P3, Fall 2021

2/18/2023
In a checkered square of size 2021×20212021\times 2021 all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take?
Ivan Yashchenko
combinatoricsboardTournament of Towns
A pirate has golden coins

Source: 43rd International Tournament of Towns, Junior O-Level P3, Fall 2021

2/18/2023
A pirate has five purses with 30 coins in each. He knows that one purse contains only gold coins, another one contains only silver coins, the third one contains only bronze coins, and the remaining two ones contain 10 gold, 10 silver and 10 bronze coins each. It is allowed to simultaneously take one or several coins out of any purses (only once), and examine them. What is the minimal number of taken coins that is necessary to determine for sure the content of at least one purse?
Mikhail Evdokimov
combinatoricscoinsTournament of Towns
2022 Tournament of towns Spring A-level JA5=SA3

Source: 2022 Tournament of towns spring A-level

5/9/2022
The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal.







Spoiler: This is one of the highlights of TT. Also SA3
geometryincircles
2022 TT Spring SO3

Source: 2022 Tournament of towns Spring Senior O level

5/11/2022
Let nn be a positive integer. Let us call a sequence a1,a2,…,ana_1,a_2,\dots,a_n interesting if for any i=1,2,…,ni=1,2,\dots,n either ai=ia_i=i or ai=i+1a_i=i+1. Let us call an interesting sequence even if the sum of its members is even, and odd otherwise. Alice has multiplied all numbers in each odd interesting sequence and has written the result in her notebook. Bob, in his notebook, has done the same for each even interesting sequence. In which notebook is the sum of the numbers greater than by how much? (The answer may depend on nn.)
algebra
Tournment of towns

Source: Tournment of towns

5/21/2022
The Fox and Pinocchio have grown a tree on the Field of Miracles with 11 golden coins. It is known that exactly 4 of them are counterfeit. All the real coins weigh the same, the counterfeit coins also weigh the same but are lighter. The Fox and Pinocchio have collected the coins and wish to divide them. The Fox is going to give 4 coins to Pinocchio, but Pinocchio wants to check whether they all are real. Can he check this using two weighings on a balance scale with no weights?
problem solving