MathDB

4

Part of 2019 Tuymaada Olympiad

Problems(2)

calculator operations limited, if S > x^n + 1 then S > x^n + x - 1

Source: Tuymaada 2019 p4

7/22/2019
A calculator can square a number or add 11 to it. It cannot add 11 two times in a row. By several operations it transformed a number xx into a number S>xn+1S > x^n + 1 (x,n,Sx, n,S are positive integers). Prove that S>xn+xāˆ’1S > x^n + x - 1.
Operationalgebracombinatoricsinequalities
diplomas at the All-Russian Olympiad and percentages problem

Source: Tuymaada Olympiad 2019 junior p4

7/22/2019
A quota of diplomas at the All-Russian Olympiad should be strictly less than 45%45\%. More than 2020 students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than 45%45\%. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to 45%45\%. Prove that the number of participants should be at least doubled.
combinatorics