MathDB

5

Part of 2016 Tournament Of Towns

Problems(4)

Concatenating Multiples of a Prime

Source: Tournament of Towns Spring 2016 Junior A-Level

2/23/2017
Let pp be a prime integer greater than 10k10^k. Pete took some multiple of pp and inserted a k−k-digit integer AA between two of its neighbouring digits. The resulting integer C was again a multiple of pp. Pete inserted a k−k-digit integer BB between two of neighbouring digits of CC belonging to the inserted integer AA, and the result was again a multiple of pp. Prove that the integer BB can be obtained from the integer AA by a permutation of its digits. (8 points) Ilya Bogdanov
number theory
Manipulating Polynomials

Source: Tournament of Towns Sprin 2016

2/22/2017
On a blackboard, several polynomials of degree 3737 are written, each of them has the leading coefficient equal to 11. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials f,gf, g and replace it by another pair of polynomials f1,g1f_1, g_1 of degree 3737 with the leading coefficients equal to 11 such that either f1+g1=f+gf_1+g_1 = f+g or f1g1=fgf_1g_1 = fg. Prove that it is impossible that after some move each polynomial on the blackboard has 3737 distinct positive roots. (8 points)
Alexandr Kuznetsov
combinatoricsalgebrapolynomialinvariant
Hexagonal pyramid with equal edges

Source: Tournament of Towns oral round p5

3/21/2016
In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.
pyramidanalytic geometrygeometry
Cut-paste square to cover circle

Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #5

4/22/2017
Is it possible to cut a square of side 11 into two parts and rearrange them so that one can cover a circle having diameter greater than 11?
(Note: any circle with diameter greater than 11 suffices)
(A. Shapovalov)
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
combinatorial geometrygeometrycombinatorics