MathDB

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Part of 2021 Durer Math Competition (First Round)

Problems(2)

[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)

Source: 2021 Dürer Math Competition Regional E4 E+1 https://artofproblemsolving.com/community/c2773609_2021_

1/7/2022
Determine all triples of positive integers a,b,ca, b, c that satisfy
a) [a,b]+[a,c]+[b,c]=[a,b,c][a, b] + [a, c] + [b, c] = [a, b, c].
b) [a,b]+[a,c]+[b,c]=[a,b,c]+(a,b,c)[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c).
Remark: Here [x,y[x, y] denotes the least common multiple of positive integers xx and yy, and (x,y)(x, y) denotes their greatest common divisor.
number theoryleast common multiplegreatest common divisorLCMGCD
p(x^2) + q(x^2) = p(x)q(x) for integer polynomials

Source: 2021 Dürer Math Competition Regional E+4 https://artofproblemsolving.com/community/c2773609_2021_

1/7/2022
Find all pairs of polynomials (p,q)(p, q) with integer coefficients that satisfy the equation p(x2)+q(x2)=p(x)q(x)p(x^2) + q(x^2) = p(x)q(x) such that pp is of degree nn and has nn nonnegative real roots (with multiplicity).
algebrapolynomial