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Note: \({ a }^{ \star }\) means \(0\) or more number of \(a\)'s\(,\) \({ a }^{ + }=a{ a }^{ \star }\) and ...

A circle is tangent to both the negative \(x\)-axis and positive \(y\)-axis, as shown below.

A point on the circle is chosen at random and its coordinates are ...

If \(a + \dfrac 1a = 2\cos 6^\circ \), find \(a^{ 1000 } +\dfrac 1{a^{1000}}+1\).

Find the number of triples of integers \(a<b<c\) with \(a\geq -10\), such that \(a(b-c)^3+b(c-a)^3+c(a-b)^3=0.\)

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