iBug
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Find $f(x)$ so that $\sum\limits_{cyc}a^{2}-f(x)\left(\prod_{sym}a-\prod_{sym}(1-a)\right)\geqq3(\frac{x}{2})^{2}$ .
2 votes

You're already on your way to success. Because the expression $$ a^2 + b^2 + c^2 \geqslant X^2 $$ is a homogeneous expression, we can easily do the following substitution: $$ a'= \cfrac aX ,\, b'= ...

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Prove that $2^n +1$ is divisible by $3$ for all positive integers $n$.
2 votes

$2^n+1$ is divisible by $3$ only when $n = 2k+1, (k∈\mathbb{Z}^*)$. Use modulus: For every odd number $n$, we have $$2^n≡2 (\mod n)$$ and for every even number $n$, we have $$2^n≡1 (\mod n)$$

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$\cfrac{2x+3y}{a-2b} = \cfrac{4y+7z}{3b-c} = \cfrac{6z+5x}{2c-3a}$. Find, $11x+17y+20z$
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1 votes

Consider them as vectors. $$ \begin{cases} \vec i = 2x + 3y \\ \vec j = 4y + 7z \\ \vec k = 6z + 5x \\ \vec v = 11x + 17y + 20z \end{cases} $$ Notice that they're orthogonal, so there's only one way ...

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Counting the probability of # of boxes having different colors of balls
1 votes

This is what I've worked out and the result is the same as @saulspatz's answer so I'm thinking it's correct. The expression is: $$ P(X = x) = \begin{cases} \frac{2^xC^x_{200}C^\frac{100 - x}{2}_{200 ...

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Does the word 'distinct' in the definition of Set implies an equivalence relation between the objects of the collection?
Accepted answer
1 votes

A set is defined as a distinct collection of objects. The relation between an object and a set has only two cases, belongs or does not belong. Any additional relation is outside a set's scope, such as ...

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Graph Theory: How do we know Hamiltonian Path exists in graph where every vertex has degree ≥3?
0 votes

Here's a simple graph for you:

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