MathDB

P4

Part of 2021/2022 Tournament of Towns

Problems(6)

Minimal number with certain property

Source: 43rd International Tournament of Towns, Senior A-Level P4, Fall 2021

2/18/2023
What is the minimum kk{} for which among any three nonzero real numbers there are two numbers aa{} and bb{} such that either ∣a−b∣⩽k|a-b|\leqslant k or ∣1/a−1/b∣⩽k|1/a-1/b|\leqslant k?
Maxim Didin
algebraTournament of Towns
3D geo with locus

Source: 43rd International Tournament of Towns, Senior O-Level P4, Fall 2021

2/18/2023
Given is a segment ABAB. Three points X,Y,ZX, Y, Z are picked in the space so that ABXABX is an equilateral triangle and ABYZABYZ is a square. Prove that the orthocenters of all triangles XYZXYZ obtained in this way belong to a fixed circle.
Alexandr Matveev
geometry3D geometryLocusTournament of Towns
Diagonals yield isosceles triangles

Source: 43rd International Tournament of Towns, Junior O-Level P4, Fall 2021

2/18/2023
A convex nn{}-gon with n>4n > 4 is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the nn{}-gon.
Maxim Didin
geometryTournament of Townspolygon
2022TTsJA4

Source:

5/18/2022
Consider a white 100×100 square. Several cells (not necessarily neighbouring) were painted black. In each row or column that contains some black cells their number is odd. Hence we may consider the middle black cell for this row or column (with equal numbers of black cells in both opposite directions). It so happened that all the middle black cells of such rows lie in different columns and all the middle black cells of the columns lie in different rows.
a) Prove that there exists a cell that is both the middle black cell of its row and the middle black cell of its column.
b) Is it true that any middle black cell of a row is also a middle black cell of its column?
geometry
Tournament of Towns [Junior O-Level Paper, Spring 2022]

Source: Tournament of Towns [Junior O-Level Paper, Spring 2022]

5/14/2022
Let us call a 1×3 rectangle a tromino. Alice and Bob go to different rooms, and each divides a 20 × 21 board into trominos. Then they compare the results, compute how many trominos are the same in both splittings, and Alice pays Bob that number of dollars. What is the maximal amount Bob may guarantee to himself no matter how Alice plays?
combinatorics
43rd ToT O level Q4

Source:

5/14/2022
Consider a square ABCD. A point P was selected on its diagonal AC. Let H be the orthocenter of the triangle APD, let M be the midpoint of AD and N be the midpoint of CD. Prove that PN is orthogonal to MH.
geometry