MathDB

5

Part of 2010 Dutch IMO TST

Problems(2)

3(x^2 + y^2 + z^2) = 1 , x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3, system 3x3

Source: 2010 Dutch IMO TST1 P5

1/10/2020
Find all triples (x,y,z)(x,y, z) of real (but not necessarily positive) numbers satisfying 3(x2+y2+z2)=13(x^2 + y^2 + z^2) = 1 , x2y2+y2z2+z2x2=xyz(x+y+z)3x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3.
system of equationsalgebra
for each prime p there is k : A(k),A(k + 1) are divisible by p, trinomial

Source: 2010 Dutch IMO TST2 p5

1/10/2020
The polynomial A(x)=x2+ax+bA(x) = x^2 + ax + b with integer coefficients has the following property: for each prime pp there is an integer kk such that A(k)A(k) and A(k+1)A(k + 1) are both divisible by pp. Proof that there is an integer mm such that A(m)=A(m+1)=0A(m) = A(m + 1) = 0.
number theorytrinomial