MathDB

3

Part of 2007 Serbia National Math Olympiad

Problems(2)

x^3 + 2x + 1 = 2^n

Source: Serbian Mathematical Olympiad 2007

6/9/2007
Determine all pairs of natural numbers (x;n)(x; n) that satisfy the equation x3+2x+1=2n.x^{3}+2x+1 = 2^{n}.
modular arithmeticinequalitiesquadraticsnumber theorygreatest common divisornumber theory proposed
x+y+z=1;x,y,z>0

Source: Serbian Mathematical Olympiad 2007

6/9/2007
Let kk be a given natural number. Prove that for any positive numbers x;y;zx; y; z with the sum 11 the following inequality holds: xk+2xk+1+yk+zk+yk+2yk+1+zk+xk+zk+2zk+1+xk+yk≥17.\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}. When does equality occur?
inequalitiesinequalities proposed