MathDB

P2

Part of 2020/2021 Tournament of Towns

Problems(6)

Strange algebraic geometry

Source: Tournament of Towns 2020 Senior A-level

10/25/2020
Baron Munchausen presented a new theorem: if a polynomial xn−axn−1+bxn−2+…x^{n} - ax^{n-1} + bx^{n-2}+ \dots has nn positive integer roots then there exist aa lines in the plane such that they have exactly bb intersection points. Is the baron’s theorem true?
geometryalgebrapolynomialVietaTournament of TownsToTBaron Munchausen
Apexes of triangles lie on semicircle

Source: 42nd International Tournament of Towns, Senior O-Level P2, Fall 2020

2/18/2023
There were ten points X1,…,X10X_1, \ldots , X_{10} on a line in this particular order. Pete constructed an isosceles triangle on each segment X1X2,X2X3,…,X9X10X_1X_2, X_2X_3,\ldots, X_9X_{10} as a base with the angle α\alpha{} at its apex. It so happened that all the apexes of those triangles lie on a common semicircle with diameter X1X10X_1X_{10}. Find α\alpha{}.
Egor Bakaev
geometryTournament of Towns
Tennis tournament bracket

Source: 42nd International Tournament of Towns, Junior O-Level P2, Fall 2020

2/18/2023
A group of 8 players played several tennis tournaments between themselves using the single-elimination system, that is, the players are randomly split into pairs, the winners split into two pairs that play in semifinals, the winners of semifinals play in the final round. It so happened that after several tournaments each player had played with each other exactly once. Prove that
[*]each player participated in semifinals more than once; [*]each player participated in at least one final.
Boris Frenkin
combinatoricsTournament of Towns
Representing pairs of real numbers in a certain way

Source: 42nd International Tournament of Towns, Senior A-Level P2, Spring 2021

2/18/2023
Does there exist a positive integer nn{} such that for any real xx{} and yy{} there exist real numbers a1,…,ana_1, \ldots , a_n satisfying x=a1+⋯+an and y=1a1+⋯+1an?x=a_1+\cdots+a_n\text{ and }y=\frac{1}{a_1}+\cdots+\frac{1}{a_n}? Artemiy Sokolov
algebraTournament of Towns
Masses of weights

Source: 42nd International Tournament of Towns, Junior O-Level P3 & Senior O-Level P2, Spring 2021

2/18/2023
Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known.
The Jury
(For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan.
Alexey Tolpygo
combinatoricsTournament of Towns
Must the triangle be isosceles?

Source: 42nd International Tournament of Towns, Junior O-Level P2, Spring 2021

2/18/2023
Let AXAX and BZBZ be altitudes of the triangle ABCABC. Let AYAY and BTBT be its angle bisectors. It is given that angles XAYXAY and ZBTZBT are equal. Does this necessarily imply that ABCABC is isosceles?
The Jury
geometryTournament of Towns