MathDB

P1

Part of 2020/2021 Tournament of Towns

Problems(7)

Arithmetic and geometric means

Source: 42nd International Tournament of Towns, Senior A-Level P1, Fall 2020

2/18/2023
There were nn{} positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer.
Boris Frenkin
number theoryTournament of Towns
Proper intersections

Source: 42nd International Tournament of Towns, Junior A-Level P1, Fall 2020

2/18/2023
Let us say that a circle intersects a quadrilateral properly if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly?
Alexandr Perepechko
geometryTournament of Towns
Quadratic polynomials with repeated roots

Source: 42nd International Tournament of Towns, Senior O-Level P1, Fall 2020

2/18/2023
Each of the quadratic polynomials P(x),Q(x)P(x), Q(x) and P(x)+Q(x)P(x)+Q(x) with real coefficients has a repeated root. Is it guaranteed that those roots coincide?
Boris Frenkin
algebrapolynomialTournament of Towns
A hundred points on a circle

Source: 42nd International Tournament of Towns, Junior O-Level P1, Fall 2020

2/18/2023
Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points?
Sergey Dvoryaninov
geometryTournament of Towns
Inserting digits cannot make the number prime

Source: 42nd International Tournament of Towns, Junior A-Level P1, Spring 2021

2/18/2023
The number 2021=43⋅472021 = 43 \cdot 47 is composite. Prove that if we insert any number of digits “8” between 20 and 21 then the number remains composite.
Mikhail Evdikomov
number theorycomposite numbersTournament of Towns
Partition of pentagon into triangles

Source: 42nd International Tournament of Towns, Senior O-Level P1, Spring 2021

2/18/2023
[*]A convex pentagon is partitioned into three triangles by nonintersecting diagonals. Is it possible for centroids of these triangles to lie on a common straight line? [*]The same question for a non-convex pentagon.
Alexandr Gribalko
geometryTournament of Towns
Product and sum of some nine consecutive integers

Source: 42nd International Tournament of Towns, Junior O-Level P1, Spring 2021

2/18/2023
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers?
Boris Frenkin
number theoryTournament of Towns