MathDB

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Part of 2016 Saint Petersburg Mathematical Olympiad

Problems(3)

(m+n)\(a_m + a_n) for any m\ne n => n\a_n for any n

Source: St. Petersburg 2016 11.1

5/1/2019
In the sequence of integers (an)(a_n), the sum am+ana_m + a_n is divided by m+nm + n with any different mm and nn. Prove that ana_n is a multiple of nn for any nn.
number theorySequencedivisornumber theory with sequencesInteger sequence
product of (divisor +1)'s, is divided by product of divisors

Source: St. peterburg 2016 10.1

5/1/2019
Sasha multiplied all the divisors of the natural number nn. Fedya increased each divider by 11, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can nn be equal to ?
number theoryProductProductsDivisors
f (x) + cg (x) = 0 and f (x) + ch (x) = 0 have a common root, trinomials

Source: St. Petersburg 2016 9.1

5/1/2019
Given three quadratic trinomials f,g,hf, g, h without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number cc such that the equations f(x)+cg(x)=0f (x) + cg (x) = 0 and f(x)+ch(x)=0f (x) + ch (x) = 0 have a common root.
algebratrinomialquadratic trinomialcommon real root