3

Part of 2008 Bulgaria National Olympiad

Problems(2)

Source: BMO 2008

n\in\mathbb{N}

0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi

b_1,b_2,\ldots ,b_n

are real numbers for which the following inequality is satisfied : \left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k} for all

k\in\mathbb{N}

. Prove that b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0.

inequalitiestrigonometrymodular arithmeticinductionceiling functionpigeonhole principlealgebra unsolved

base subsets -- Bulgaria MO

Source:

M

be the set of the integer numbers from the range

[-n, n]

P

M

is called a base subset if every number from

M

can be expressed as a sum of some different numbers from

P

. Find the smallest natural number

k

such that every

k

numbers that belongs to

M

form a base subset.

inductioncombinatorics unsolvedcombinatorics