Algebra Warmups
\[f(x)=x^{3}-x^{2}-2x+1\]
Let the zeros of the function above be \(\alpha,\beta,\) and \(\gamma. \)
Find \( f'(\alpha)\times f'(\beta)\times f'(\gamma). \)
Note: \(f'(x)\) denotes the derivative of \( f(x).\)
by
Ronak Agarwal
Let \(x, y, z\geq 0\) be reals such that \(x+y+z=1\). Find the maximum possible value of
\[x (x+y)^{2}(y+z)^{3}(x+z)^{4}.\]
by
Joel Tan
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? \[x^{64}-2x^{52}+3x^{43}+2x^{38}-2x^{29}+5x^{17}+5x^{10}-7x^7+7\]
by
Sanjeet Raria
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
\[ \left[ 9 \sum_{i=1}^{81} a_i ^2 \right] + \left[ \sum_{1 \leq j < k \leq 81 } ( a_k - a_j + 1)^2 \right] ? \]
by
Calvin Lin
\[ \begin{array} { l l} a + b + c & \in \mathbb{N} \\ a^4+b^4+c^4 & =32\\ a^5+b^5+c^5& =186\\ a^6+b^6+c^6& =803\\ \end{array} \]
What is the value of \(a+b+c\)?
by
Trevor Arashiro