MathDB

P2

Part of 2022/2023 Tournament of Towns

Problems(7)

Nice lcm number theory

Source: 44th International Tournament of Towns, Senior A-Level P2, Fall 2022

2/16/2023
Consider two coprime integers pp{} and qq{} which are greater than 11{} and differ from each other by more than 11{}. Prove that there exists a positive integer nn{} such that lcm(p+n,q+n)<lcm(p,q).\text{lcm}(p+n, q+n)<\text{lcm}(p,q).
number theoryTournament of Towns
Numbers on cards

Source: 44th International Tournament of Towns, Junior A-Level P2, Fall 2022

2/16/2023
The numbers 1,19,199,1999,…1, 19, 199, 1999,\ldots are written on several cards, one card for each number.
[*]Is it possible to choose at least three cards so that the sum of the numbers on the chosen cards equals a number in which all digits, except for a single digit, are twos? [*]Suppose you have chosen several cards so that the sum of the numbers on the chosen cards equals a number, all of whose digits are twos, except for a single digit. What can this single different digit be?
number theoryTournament of Towns
Circles in rhombus

Source: 44th International Tournament of Towns, Senior O-Level P2, Fall 2022

2/16/2023
A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.
geometryTournament of Towns
Product of palindromes

Source: 44th International Tournament of Towns, Junior O-Level P2, Fall 2022

2/16/2023
Does there exist a natural number that can be represented as the product of two numeric palindromes in more than 100100{} ways?
Tournament of Townsnumber theorypalindromes
Bacteria multiplication on board

Source: 44th International Tournament of Towns, Junior A-Level P2, Spring 2023

1/9/2024
There is a bacterium in one of the cells of a 10×1010 \times 10{} checkered board. At the first move, the bacterium shifts to a cell adjacent by side to the original one, and divides into two bacteria (both stay in the same cell). Then again, one of the bacteria on the board shifts to a cell adjacent by side and divides into two bacteria, and so on. Is it possible that after some number of such moves the number of bacteria in each cell of the board is the same?
Alexandr Gribalko
combinatoricsboard
Sums of floor functions

Source: 44th International Tournament of Towns, Senior O-Level P2 and Junior O-Level P4, Spring 2023

1/9/2024
А positive integer nn{} is given. For every xx{} consider the sum Q(x)=∑k=110n⌊xk⌋.Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.Find the difference Q(10n)−Q(10n−1)Q(10^n)-Q(10^n-1).
Alexey Tolpygo
floor functionnumber theory
How many sides can have length 1?

Source: 44th International Tournament of Towns, Junior O-Level P2, Spring 2023

1/9/2024
Medians BKBK{} and CNCN{} of triangle ABCABC intersect at M.M{}. Consider quadrilateral ANMKANMK and find the maximum possible number of its sides having length 1.
Egor Bakaev
geometrysidelengths