MathDB

2.3

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

(x+y)^2/( x^3 + xy^2 - x^2y - y^3) not an integer

Source: 2011 Saudi Arabia Pre-TST November 2.3

1/1/2022
Let x,yx, y be distinct positive integers. Prove that the number (x+y)2x3+xy2−x2y−y3\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3} is not an integer
number theoryInteger
f = g^2 if f = aX^2 + bX+ c is a perfect square for every n

Source: 2011 Saudi Arabia Pre-TST February 2.3

1/1/2022
Let f=aX2+bX+c∈Z[X]f = aX^2 + bX+ c \in Z[X] be a polynomial such that for every positive integer nn,f(n) f(n ) is a perfect square. Prove that f=g2f = g^2 for some polynomial g∈Z[X]g \in Z[X].
algebrapolynomialPerfect Square