MathDB

1

Part of 1989 Vietnam National Olympiad

Problems(2)

Inequality

Source: Vietnam NMO 1989 Problem 1

2/1/2009
Let n n and N N be natural number. Prove that for any α \alpha , 0≤α≤N 0\le\alpha\le N, and any real x x, it holds that { |\sum_ {k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}
inequalitiestrigonometryinequalities unsolved
Integer equation

Source: Vietnam NMO 1989 Problem 4

2/1/2009
Are there integers x x, y y, not both divisible by 5 5, such that x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}?
number theory unsolvednumber theory