kyary
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How many are the non-negative integer solutions of $𝑥 + 𝑦 + 𝑤 + 𝑧 = 16$ where $x < y$?
7 votes

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 ...

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Probability that the sum of the numbers shown is a multiple of $5$
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 ...

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Find all $\alpha$ such that $\frac{\cos \alpha - a \sin \alpha}{a \cos \alpha + \sin \alpha}$ is rational (given $a$ rational)
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 ...

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Prove that for any real values of $c$, the equation $x(x^2-1)(x^2-10)=c$ can't have $5$ integer solutions.
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: $a+b+c+d+e = 0$ $ab+ac+ad+ae+...+de = -11$ So $22 = (\text{...

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Optimization over combinations of $n$ balls over $k$ bins.
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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 ...

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Partition of a set into some subsets which are not disjoint necessary
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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 ...

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Proving $\left(a^2+b^2\right)^2\geqslant(a+b+c)(a+b-c)(b+c-a)(c+a-b)$ for positive reals $a$, $b$, $c$
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 $\;...

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How to calculate green area plus red area without using integral calculus?
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 ...

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Solving $P_y\frac{\sin{\theta_1}}{\sqrt{1-\sin^2{\theta_1}}}+F_y\frac{m\sin{\theta_1}}{\sqrt{1-m^2\sin^2{\theta_1}}}=F_x - P_x$ for $\sin\theta_1$
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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}}...

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An IMO training problem: Living in an overpopulated apartment
1 votes

At least for 1 and 2: Why is S needed to be considered? At a larger-scale level, the solution attempts to find a property of the system that is: a nonnegative integer will decrease to a smaller ...

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