MathDB

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Part of 2015 China National Olympiad

Problems(2)

Inequality on Complex Numbers

Source: China Mathematical Olympiad 2015 Q1

12/20/2014
Let z1,z2,...,znz_1,z_2,...,z_n be complex numbers satisfying ∣zi−1∣≤r|z_i - 1| \leq r for some rr in (0,1)(0,1). Show that
∣∑i=1nzi∣⋅∣∑i=1n1zi∣≥n2(1−r2). \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).
inequalitiestrigonometrycomplex numbersinequalities proposed
Divisibility condition on Binomial coefficient

Source: China Mathematical Olympiad 2015 Q4

12/21/2014
Determine all integers kk such that there exists infinitely many positive integers nn not satisfying n+k∣(2nn)n+k |\binom{2n}{n}
logarithmsnumber theory proposednumber theory