MathDB

Problem 3

Part of 2014 VJIMC

Problems(2)

another tanh inequality

Source: VJIMC 2014 1.3

5/20/2021
Let n≥2n\ge2 be an integer and let x>0x>0 be a real number. Prove that (1−tanh⁡x)n+tanh⁡(nx)<1.\left(1-\sqrt{\tanh x}\right)^n+\sqrt{\tanh(nx)}<1.
inequalities
a binomial sum, alternating

Source: VJIMC 2014 2.3

5/21/2021
Let kk be a positive even integer. Show that ∑n=0k/2(−1)n(k+2n)(2(k−n)+1k+1)=(k+1)(k+2)2.\sum_{n=0}^{k/2}(-1)^n\binom{k+2}n\binom{2(k-n)+1}{k+1}=\frac{(k+1)(k+2)}2.
Summationbinomial coefficientsalgebra