MathDB

3

Part of 2021 South East Mathematical Olympiad

Problems(2)

China South East Mathematical Olympiad 2021 Grade10 Q3

Source: China Nanchang, Jul 28,2021

7/28/2021
Let pp be an odd prime and {ui}i≥0\{u_i\}_{i\ge 0}be an integer sequence. Let vn=∑i=0nCnipiuiv_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i where CniC_n^i denotes the binomial coefficients. If vn=0v_n=0 holds for infinitely many nn , prove that it holds for every positive integer nn.
algebranumber theoryChina
China South East Mathematical Olympiad 2021 Grade11 Q3

Source: China Nanchang, Jul 28, 2021

7/28/2021
Let a,b,c≥0a,b,c\geq 0 and a2+b2+c2≤1.a^2+b^2+c^2\leq 1. Prove thataa2+bc+1+bb2+ca+1+cc2+ab+1+3abc<3\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3
inequalitiesalgebraChina