MathDB

3

Part of 1991 Polish MO Finals

Problems(2)

A sum which cannot be the fifth power

Source: Problem 3, Polish NO 1991

10/1/2005
Define N=∑k=160ekkkk N=\sum\limits_{k=1}^{60}e_k k^{k^k} where ek∈{−1,1}e_k \in \{-1, 1\} for each kk. Prove that NN cannot be the fifth power of an integer.
number theory unsolvednumber theory
x+y+z \leq 2 +xyz

Source: Problem 6, Polish NO 1991

10/1/2005
If x,y,zx, y, z are real numbers satisfying x2+y2+z2=2x^2 +y^2 +z^2 = 2, prove the inequality x+y+z≤2+xyz x + y + z \leq 2 + xyz When does equality occur?
inequalitiesthree variable inequalityalgebra