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If \(\alpha\) and \(\beta\) are two distinct values of \(\theta\) lying between 0 and \(2\pi\), and they satisfy the equation \(6\cos \theta + 8\sin\theta = 9 \), find ...

Find the remainder when

\[\prod_{1 \leq i < j \leq 100} \left( i^2 + ij + j^2 \right) \]

is divided by \(101.\)

Note: \(i,j\) range over all positive ...

Given any \(9\) integers, show that it is possible to choose \(w,x,y,z\) such that \(w+x-y-z\equiv0\)\((\text{mod}\) \(20)\).

The rational polynomial \( \dfrac{ x-p} { x^2 - 3x + 2 } \) is real for all real values of \(x\), find the bounds on \(p\). We have \( l < p < m \). Find ...

How many unit circles can you fit inside a circle of diameter \(7\) such that no circle overlaps any other circle? Try to prove it.

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