MathDB

2

Part of 1996 China National Olympiad

Problems(2)

Sum of x is 1

Source: CMO-1996-5

1/3/2006
Let nn be a natural number. Suppose that x0=0x_0=0 and that xi>0x_i>0 for all i∈{1,2,…,n}i\in\{1,2,\ldots ,n\}. If ∑i=1nxi=1\sum_{i=1}^nx_i=1 , prove that 1≤∑i=1nxi1+x0+x1+…+xi−1xi+…+xn<π21\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2}
inequalitiestrigonometrycalculusderivativeintegration
a+b|ab

Source:

5/26/2009
Find the smallest positive integer K K such that every K K-element subset of {1,2,...,50} \{1,2,...,50 \} contains two distinct elements a,b a,b such that a\plus{}b divides ab ab.
number theoryrelatively primecombinatorics unsolvedcombinatorics