MathDB

3

Part of 2019 Tournament Of Towns

Problems(5)

fixed point for angle bisector of gray and black sides, 2 equal wooden disks

Source: Tournament of Towns, Junior A-Level , Spring 2019 p3

5/9/2020
Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane.
(Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev) (Senior version[url=https://artofproblemsolving.com/community/c6h2102856p15209040] here)
geometryangle bisectorcirclesdisksfixedFixed pointColoring
m+n divides mn, then m + n <= n^2

Source: Tournament of Towns, Junior O-Level , Spring 2019 p3

5/9/2020
The product of two positive integers mm and nn is divisible by their sum. Prove that m+n≤n2m + n \le n^2.
(Boris Frenkin)
number theoryTournament of TownsToTDivisibility
any triangle can be cut into bicentric 2019 bicentric quadrilaterals

Source: Tournament of Towns, Senior O-Level , Spring 2019 p3

5/11/2020
Prove that any triangle can be cut into 20192019 quadrilaterals such that each quadrilateral is both inscribed and circumscribed.
(Nairi Sedrakyan)
bicentric quadrilateralTilingcombinatoricscombinatorial geometrycut
row of 100 cells each containing a token, for 1 dollar interchange neighbour

Source: Tournament of Towns, Junior O-Level , Fall 2019 p3

4/19/2020
There is a row of 100100 cells each containing a token. For 11 dollar it is allowed to interchange two neighbouring tokens. Also it is allowed to interchange with no charge any two tokens such that there are exactly 33 tokens between them. What is the minimum price for arranging all the tokens in the reverse order?
(Egor Bakaev)
combinatorics
100 coins and you have to weigh them

Source:

11/29/2019
There are 100 visually identical coins of three types: golden, silver and copper. There is at least one coin of each type. Each golden coin weighs 3 grams, each silver coins weighs 2 grams and each copper coin weighs 1 gram. How to find the type of each coin performing no more than 101 measurements on a balance scale with no weights.
combinatoricscoinsalgorithmStrategy