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The minimum value of \[ f(x) = \int_{0}^{1} t|t-x| \ dt \] where \( x\in\mathbb{R}, \) is \( \omega\). \[\]What is the value of \( \lfloor{1000 \omega} \rfloor\)?

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I just contributed to the wiki article on the Butterfly theorem.

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\[\large {x}^{2} \frac {d^{2}y}{dx^{2}}+ x \frac{dy}{dx}+y=0 \]

where \(y=f(x),\) \(f(1)=1\) and \(y'(1)=0\)

Find ...

\[ \large \int_0^\pi \ln( 25\sin^2 x + 16\cos^2 x) \, dx = C \pi \ln\left(\frac AB\right) \]

where \(A\) and \(B\) are co-prime integers and \(C\) is ...

Consider the infinite sequence:, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ...

What is the 1000th term?

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