MathDB

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Part of 2018 Czech-Polish-Slovak Junior Match

Problems(2)

ad + bc = x, ac + bd = y, ab + cd = z, one of a, b,c,d equals to sum of other 3

Source: Czech-Polish-Slovak Junior Match 2018, individual p1 CPSJ

2/1/2020
Are there four real numbers a,b,c,da, b, c, d for every three positive real numbers x,y,zx, y, z with the property ad+bc=xad + bc = x, ac+bd=yac + bd = y, ab+cd=zab + cd = z and one of the numbers a,b,c,da, b, c, d is equal to the sum of the other three?
Sumsystem of equationsalgebra
(a + b + c) | (a - b)^2 + (b - c)^2 + (c - a)^2

Source: Czech-Polish-Slovak Junior Match 2018, Team p1 CPSJ

3/7/2020
For natural numbers a,bca, b c it holds that (a+b+c)2∣ab(a+b)+bc(b+c)+ca(c+a)+3abc(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc. Prove that (a+b+c)∣(a−b)2+(b−c)2+(c−a)2(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2
number theorydividesdivisible