MathDB

1

Part of 2020 OMpD

Problems(2)

Diophantine with polynomial and exponential

Source: 2020 1st OMpD L3 P1 - Brazil - Olimpíada Matemáticos por Diversão

7/8/2023
Determine all pairs of positive integers (x,y)(x, y) such that: x4−6x2+1=7⋅2yx^4 - 6x^2 + 1 = 7\cdot 2^y
number theoryDiophantine equationExponential equationpolynomialalgebra
find all possible values of a division

Source: 2020 1st OMpD L2 P1 - Brazil - Olimpíada Matemáticos por Diversão

7/8/2023
Let a,b,ca, b, c be real numbers such that a+b+c=0a + b + c = 0. Given that a3+b3+c3≠0a^3 + b^3 + c^3 \neq 0, a2+b2+c2≠0a^2 + b^2 + c^2 \neq 0, determine all possible values for: a5+b5+c5(a3+b3+c3)(a2+b2+c2)\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}
algebrafactoring polynomials