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Let $n$ be a positive integer and
$$(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^{n-k}y^k$$
be the expansion of the binomial $(x+y)^n$.
Find the number of terms in the expansion that are not divisible by $4$,
when $n$ is a power of two. | open2math |
Find the number of divisors of 500 that are also powers of prime numbers. | open2math |
Find the value of $a$ so that the lines with the given equations are perpendicular. \begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*} | open2math |
Find all the values of $x$ that satisfy the equation
\[\sin \left(\frac{x}{2}\right) + \cos \left(\frac{x}{2}\right) = \sqrt{3}.\] | open2math |
In a right-angled triangle with sides of length $a$ and $b$, and hypotenuse of length $c$, the ratio of the area of the largest rectangle inscribed in the triangle to the area of the largest rectangle inscribed in the circle with diameter $c$ is $\frac{1}{2}$. Find the ratio of $a$ to $b$. | open2math |
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