MathDB

4

Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

Stierlitz encrypts a 100-digit binary string

Source: St Petersburg 2021 10.4

12/23/2021
Stierlitz wants to send an encryption to the Center, which is a code containing 100100 characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:
i) when transmitting encryption over the radio, exactly 4949 characters should be replaced with their opposites;
ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.
Prove that Stierlitz can send 1010 encryptions, each time choosing some 4949 characters to flip, such that when the Center receives these 1010 ciphers, it may unambiguously restore the original code.
combinatorics
Writing functions on a board

Source: St Petersburg 2021 11.4

12/23/2021
The following functions are written on the board, F(x)=x2+12x2,G(x)=sin⁡(πx2),H(x)=1.F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1. If functions f,gf,g are currently on the board, we may write on the board the functions f(x)+g(x),f(x)−g(x),f(x)g(x),cf(x)f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x) (the last for any real number cc). Can a function h(x)h(x) appear on the board such that ∣h(x)−x∣<13|h(x) - x| < \frac{1}{3} for all x∈[1,10]x \in [1,10] ?
functionalgebra
Parabolas passing through pairs of points

Source: St Petersburg 2021 9.4

12/23/2021
Given are nn points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to 11. A parabola is called goodgood if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of goodgood parabolas?
algebra