MathDB

P2

Part of Russian TST 2016

Problems(5)

Participants at an Olympiad stand in a circle

Source: Russian TST 2016, Day 10 P2 (Group A), P3 (Group B)

4/19/2023
An Olympiad has 99 tasks. Several participants of the Olympiad are standing in a circle. They all solved different sets of tasks. Any two participants standing side by side do not have a common solved problem, but have a common unsolved one. Prove that the number of participants in the circle does not exceed 299−(9950).2^{99}-\binom{99}{50}.
combinatorics
Weird strict inequality

Source: Russian TST 2016, Day 10 P2 (Group NG)

4/19/2023
Let x,y,zx,y,z{} be positive real numbers. Prove that (xy+yz+zx)(1x2+y2+1y2+z2+1z2+x2)>52.(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.
algebrainequalities
Inequality with powers of two

Source: Russian TST 2016, Day 11 P2 (Group A), P3 (Group B)

4/19/2023
Prove that 1+211−21+22(1−21)(1−22)+⋯+22016(1−21)⋯(1−22016)>0.1+\frac{2^1}{1-2^1}+\frac{2^2}{(1-2^1)(1-2^2)}+\cdots+\frac{2^{2016}}{(1-2^1)\cdots(1-2^{2016})}>0.
algebrainequalities
Children with different heights

Source: Russian TST 2016, Day 13 P2

4/20/2023
In a class, there are nn{} children of different heights. Denote by AA{} the number of ways to arrange them all in a row, numbered 1,2,…,n1,2,\ldots,n from left to right, so that each person with an odd number is shorter than each of his neighbors. Let BB{} be the number of ways to organize n−1n-1 badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that A=BA = B.
combinatorics
Functional equation happens with a certain condition

Source: Russian TST 2016, Day 12 P2

4/19/2023
Prove that a function f:R+→Rf:\mathbb{R}_+\to\mathbb{R} satisfies f(x+y)−f(x)−f(y)=f(1x+1y)f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)if and only if it satisfies f(xy)=f(x)+f(y)f(xy)=f(x)+f(y).
algebrafunctional equation