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10.8

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

product a_i >= 99^{100} - VI Soros Olympiad 1999-00 Round 1 10.8

Source:

5/21/2024
There are 100100 positive numbers a1a_1, a2a_2, ......, a100a_{100} such that 1a1+1+1a2+1+...+1a100+1≤1.\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1. Prove that a1⋅a2⋅...⋅a100≥99100.a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.
inequalitiesalgebra
f(x)=sin 1998x+ sin2000x (VI Soros Olympiad 1990-00 R1 10.8)

Source:

5/28/2024
Find the smallest positive period of the function f(x)=sin⁡(1998x)+sin⁡(2000x)f(x)=\sin (1998x)+ \sin (2000x)
algebratrigonometryperiodic