MathDB

4

Part of 2012 Saint Petersburg Mathematical Olympiad

Problems(2)

Sequences

Source: St Petersburg Olympiad 2012, Grade 11, P4

9/29/2017
x1,...,xnx_1,...,x_n are reals and x12+...+xn2=1x_1^2+...+x_n^2=1 Prove, that exists such y1,...,yny_1,...,y_n and z1,...,znz_1,...,z_n such that ∣y1∣+...+∣yn∣≤1|y_1|+...+|y_n| \leq 1; max(∣z1∣,...,∣zn∣)≤1max(|z_1|,...,|z_n|) \leq 1 and 2xi=yi+zi2x_i=y_i+z_i for every ii
algebra
Numbers on circle

Source: St Petersburg Olympiad 2012, Grade 10, P4

9/29/2017
Some notzero reals numbers are placed around circle. For every two neighbour numbers a,ba,b it true, that a+ba+b and 1a+1b\frac{1}{a}+\frac{1}{b} are integer. Prove that there are not more than 44 different numbers.
number theory