kyary
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Let M be the MULTISET union of A1,...,A6 Suppose $M$ has $i_1$ 1's, $i_2$ 2's,..., $i_{10}$ 10's. Then there are $\binom{6}{i_1} * \binom{6}{i_2} * ... * \binom{6}{i_{10}}$ different possible ...

Expanding both sides and moving positive terms to their respective sides, we get: $a^4 +b^4+\dfrac{c^4}{2}\geqslant a^2 c^2 + b^2 c^2$ By AMGM we have: $a^4 + \dfrac{c^4}{4}\geqslant a^2 c^2\;$ and $\;... View answer 2 votes Using a little trickery with complex numbers, we can arrive at the sum of all coefficients of terms of degree a multiple of 5 of the above polynomial. Consider the 5th roots of unity (i.e. the roots ... View answer 2 votes Let$\beta = arctan(a)$Then the expression becomes $$\frac{\cos \alpha - \tan \beta \sin \alpha}{\tan \beta \cos \alpha + \sin \alpha}$$ Divide both the numerator and denominator by$\cos \alpha$to ... View answer 1 votes Here is a solution to the question involving extended sine law and circle angle properties. Edit: you don't need to find theta for this question, can use trig identities instead to get an exact ... View answer Accepted answer 1 votes There's an easier way to do it, assuming you have a way of solving quartics. By the way quartics can be solved systematically. Plugging our values into snell's law, we get$$\frac{x}{\sqrt{d^2 + x^2}}... View answer Accepted answer 2 votes Here's a more "elementary" way. Let a,b,c,d,e be integer roots. We attempt to find a contradiction. By Vieta's formulas, we have:$a+b+c+d+e = 0ab+ac+ad+ae+...+de = -11$So$22 = (\text{...

There are a variety of techniques to show that $\sum_{i=1}^k c_i^2$ is minimized when all the $c_i$ are as close together as possible. Not sure about what degree of rigor you want, but one easy method ...