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Algebra Warmups

From cracking cryptograms to calculating the top speed of a rocket, algebra gives you tools to apply mathematical reasoning to a wide range of problems. Dive in and see what you already know! See more

Level 5

         

\[f(x)=x^{3}-x^{2}-2x+1\]

Let the zeros of the function above be \(\alpha,\beta,\gamma \).

Find \( f'(\alpha)\times f'(\beta)\times f'(\gamma) \).

Details and Assumptions:

  • \(f'(x)\) denote the derivative of \( f(x)\).

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

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

If a complex number \(\alpha\) satisfies the equation \( \alpha^3-\alpha^2-2\alpha+1=0,\) where \(\alpha=x+\frac{1}{x}\) for some complex number \(x\), then what is the value of the expression below? \[x^{64}-2x^{52}+3x^{43}+2x^{38}-2x^{29}+5x^{17}+5x^{10}-7x^7+7\]

Given 81 variables that satisfy

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

what is the maximum value of

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

\[ \begin{array} { l l} a + b + c & \in \mathbb{N} \\ a^4+b^4+c^4 & =32\\ a^5+b^5+c^5& =186\\ a^6+b^6+c^6& =803\\ \end{array} \]

What is the value of \(a+b+c\)?

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