Algebra

Algebra Warmups

Algebra Warmups: Level 5 Challenges

         

f(x)=x3x22x+1f(x)=x^{3}-x^{2}-2x+1

Let the zeros of the function above be α,β,\alpha,\beta, and γ.\gamma.

Find f(α)×f(β)×f(γ). f'(\alpha)\times f'(\beta)\times f'(\gamma).

Note: f(x)f'(x) denotes the derivative of f(x). f(x).

Let x,y,z0x, y, z\geq 0 be reals such that x+y+z=1x+y+z=1. Find the maximum possible value of

x(x+y)2(y+z)3(x+z)4.x (x+y)^{2}(y+z)^{3}(x+z)^{4}.

If a complex number α\alpha satisfies the equation α3α22α+1=0, \alpha^3-\alpha^2-2\alpha+1=0, where α=x+1x\alpha=x+\frac{1}{x} for some complex number xx, then what is the value of the expression below? x642x52+3x43+2x382x29+5x17+5x107x7+7x^{64}-2x^{52}+3x^{43}+2x^{38}-2x^{29}+5x^{17}+5x^{10}-7x^7+7

Given 81 variables that satisfy

0a1a2a811, 0 \leq a_1 \leq a_2 \leq \ldots \leq a_{81} \leq 1,

what is the maximum value of

[9i=181ai2]+[1j<k81(akaj+1)2]? \left[ 9 \sum_{i=1}^{81} a_i ^2 \right] + \left[ \sum_{1 \leq j < k \leq 81 } ( a_k - a_j + 1)^2 \right] ?

a+b+cNa4+b4+c4=32a5+b5+c5=186a6+b6+c6=803 \begin{array} { l l} a + b + c & \in \mathbb{N} \\ a^4+b^4+c^4 & =32\\ a^5+b^5+c^5& =186\\ a^6+b^6+c^6& =803\\ \end{array}

What is the value of a+b+ca+b+c?

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