MathDB

5

Part of 2013 Saudi Arabia BMO TST

Problems(2)

X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0

Source: 2013 Saudi Arabia BMO TST I p5

7/24/2020
Let kk be a real number such that the product of real roots of the equation X4+2X3+(2+2k)X2+(1+2k)X+2k=0X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 is −2013-2013. Find the sum of the squares of these real roots.
polynomialrootsalgebra
sum of the squares of its digits is a perfect square

Source: 2013 Saudi Arabia BMO TST II p5

7/24/2020
We call a positive integer good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, 122122 and 3434 are good and 304304 and 1212 are not not good. Prove that there exists a nn-digit good number for every positive integer nn.
number theorySum of SquaresDigitsPerfect Square