The equations \[ax^2 + bx + c = 0 \tag{1}\] and \[- ax^2 + bx + c \tag{2}\] are given. Prove that if \(x_1\) and \(x_2\) are, respectively, any roots of the equations (1) and (2), then there is a root \(x_3\) of the equation \(\frac 12 ax^2 + bx + c\) such that either \(x_1 \leq x_3 \leq x_2\) or \(x_1 \geq x_3 \geq x_2\).
Prove that if \(x_0^4 + a_1x_0^3 + a_2x_0^2 + a_3x_0 + a_4\) and \(4x_0^3 + 3a_1x_0^2 + 2a_2x_0 + a_3 = 0\) then \(x^4 + a_1x^3 + a_2x^2 + a_3x + a_4\) is divisible by \((x - x_0)^2\).
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
Two different numbers \(x\) and \(y\) (not necessarily integers) are such that \(x^2-2000x=y^2-2000y\). Find the sum of \(x\) and \(y\).
Find \(x^3 +y^3\) if \(x+y=5\) and \(x+y+x^2 y +xy^2 =24\).
Is it true that, if \(b>a+c>0\), then the quadratic equation \(ax^2 +bx+c=0\) has two roots?
Compute the following: \[\frac{(2001\times 2021 +100)(1991\times 2031 +400)}{2011^4}.\]
Prove that, if \(b=a-1\), then \[(a+b)(a^2 +b^2)(a^4 +b^4)\dotsb(a^{32} +b^{32})=a^{64} -b^{64}.\]
Prove the following formulae are true: \[\begin{aligned} a^{n + 1} - b^{n + 1} &= (a - b) (a^n + a^{n-1}b + \dots + b^n);\\ a^{2n + 1} + b^{2n + 1} &= (a + b) (a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - \dots + b^{2n}). \end{aligned}\]
Derive from the theorem in question 61013 that \(\sqrt{17}\) is an irrational number.