MathDB

2

Part of 2000 All-Russian Olympiad

Problems(3)

Sasha guessing X <= 100 in 7 questions

Source: All-Russian MO 2000

12/30/2012
Tanya chose a natural number X≤100X\le100, and Sasha is trying to guess this number. He can select two natural numbers MM and NN less than 100100 and ask about gcd⁡(X+M,N)\gcd(X+M,N). Show that Sasha can determine Tanya's number with at most seven questions.
ceiling functionlogarithmsmodular arithmeticnumber theory unsolvednumber theory
Inequality in 2n variables with 13th powers

Source: All-Russian MO 2000

12/30/2012
Let −1<x1<x2,⋯<xn<1-1 < x_1 < x_2 , \cdots < x_n < 1 and x113+x213+⋯+xn13=x1+x2+⋯+xnx_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n. Prove that if y1<y2<⋯<yny_1 < y_2 < \cdots < y_n, then x113y1+⋯+xn13yn<x1y1+x2y2+⋯+xnyn. x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n.
inequalitiesinequalities unsolvedn-variable inequalityabel formula
Partition into 100 sets with condition a+99b=c

Source: All-Russian MO 2000

12/30/2012
Prove that one can partition the set of natural numbers into 100100 nonempty subsets such that among any three natural numbers aa, bb, cc satisfying a+99b=ca+99b=c, there are two that belong to the same subset.
number theory unsolvednumber theory