MathDB

5

Part of 2000 All-Russian Olympiad

Problems(3)

a_(n+1) = a_n - 2 if new, else a_n + 3

Source: All-Russian MO 2000

12/30/2012
The sequence a1=1a_1 = 1, a2,a3,⋯a_2, a_3, \cdots is defined as follows: if an−2a_n - 2 is a natural number not already occurring on the board, then an+1=an−2a_{n+1} = a_n-2; otherwise, an+1=an+3a_{n+1} = a_n + 3. Prove that every nonzero perfect square occurs in the sequence as the previous term increased by 33.
inductionalgebra
Every three elements have a sum in M

Source: All-Russian MO 2000

12/30/2012
Let MM be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to MM. Find the greatest possible number of elements of MM.
absolute valuecombinatorics unsolvedcombinatorics
sin^n(2x) + (sin^n x - cos^n x)^2 <= 1

Source: All-Russian MO 2000

12/30/2012
Prove the inequality sin⁡n(2x)+(sin⁡nx−cos⁡nx)2≤1. \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1.
inequalitiestrigonometryinequalities unsolved