MathDB

5

Part of 2017 All-Russian Olympiad

Problems(2)

Divisors on number

Source: All Russian Olympiad 2017,Day1,grade 10,P5

5/3/2017
nn is composite. 1<a1<a2<...<ak<n1<a_1<a_2<...<a_k<n - all divisors of nn. It is known, that a1+1,...,ak+1a_1+1,...,a_k+1 are all divisors for some mm (except 1,m1,m). Find all such nn.
number theory
Triangles from polynoms

Source: All Russian Olympiad 2017,Day2,grade 11,P5

5/1/2017
P(x)P(x) is polynomial with degree n≥2n\geq 2 and nonnegative coefficients. a,b,ca,b,c - sides for some triangle. Prove, that P(a)n,P(b)n,P(c)n\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)} are sides for some triangle too.
algebrapolynomialDecreasing