MathDB

2

Part of 1982 Vietnam National Olympiad

Problems(2)

Solve for x: x(x + 1)(x + 2)(x + 3) + 1 - m = 0.

Source: Vietnam MO 1982 P2

3/17/2011
For a given parameter mm, solve the equation x(x+1)(x+2)(x+3)+1−m=0.x(x + 1)(x + 2)(x + 3) + 1 - m = 0.
parameterizationalgebra proposedalgebra
Inequality in three variables and product.

Source: Vietnam MO 1982 P5

3/17/2011
Let pp be a positive integer and q,zq, z be real numbers with 0\le q\le 1 and q^{p+1}\le z\le 1. Prove that ∏k=1p∣z−qkz+qk∣≤∏k=1p∣1−qk1+qk∣.\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.
inequalitiesinequalities unsolved