MathDB

1

Part of 2018 All-Russian Olympiad

Problems(3)

Sequence is all prime

Source: All-Russia 2018 Grade 9 P1

4/28/2018
Suppose a1,a2,…a_1,a_2, \dots is an infinite strictly increasing sequence of positive integers and p1,p2,…p_1, p_2, \dots is a sequence of distinct primes such that pn∣anp_n \mid a_n for all n≥1n \ge 1. It turned out that an−ak=pn−pka_n-a_k=p_n-p_k for all n,k≥1n,k \ge 1. Prove that the sequence (an)n(a_n)_n consists only of prime numbers.
number theoryalgebraprimeprime numbers
All Russian Olympiad Day 1 P1

Source: Grade 11 P1

4/28/2018
The polynomial P(x)P (x) is such that the polynomials P(P(x))P (P (x)) and P(P(P(x)))P (P (P (x))) are strictly monotone on the whole real axis. Prove that P(x)P (x) is also strictly monotone on the whole real axis.
algebra
All-Russian Olympiad Day 1 Problem 10.1.

Source: All-Russian Olympiad 2018

4/24/2018
Determine the number of real roots of the equation ∣x∣+∣x+1∣+⋯+∣x+2018∣=x2+2018x−2019|x|+|x+1|+\cdots+|x+2018|=x^2+2018x-2019
algebra