MathDB

3

Part of 2011 Polish MO Finals

Problems(2)

Number of Real Solutions of The System of The Equations

Source: LXII Polish Olympiad 2011, Problem 3

5/24/2011
Let n≥3n\geq 3 be an odd integer. Determine how many real solutions there are to the set of nn equations {x1(x1+1)=x2(x2−1)x2(x2+1)=x3(x3−1)⋮xn(xn+1)=x1(x1−1)\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.
algebra unsolvedalgebra
Sum of polynomials is not equal to x^2+7

Source: LXII Polish Olympiad 2011, Problem 6

5/24/2011
Prove that it is impossible for polynomials f1(x),f2(x),f3(x),f4(x)∈Q[x]f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x] to satisfy f12(x)+f22(x)+f32(x)+f42(x)=x2+7.f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.
algebrapolynomialalgebra unsolved