kyary
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I can't comment so this answer is more of a comment. You use stars and bars. You have three bars | | | where each space represents one of x, y, w, or z. And you have 16 stars. So you count the number ...

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 ...

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 ...

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: $a+b+c+d+e = 0$ $ab+ac+ad+ae+...+de = -11$ So $22 = (\text{... View answer Accepted answer 2 votes There are a variety of techniques to show that$\sum_{i=1}^k c_i^2$is minimized when all the$c_i$are as close together as possible. Not sure about what degree of rigor you want, but one easy method ... View answer Accepted answer 1 votes Let M be the MULTISET union of A1,...,A6 Suppose$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}}$different possible ... View answer 1 votes Expanding both sides and moving positive terms to their respective sides, we get:$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\;$and$\;...

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 ...

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}}...