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What is the sum of the digits of the sum of all the integers $x$ such that $x+123$ and $x+456$ are both square numbers?

Given the circular sector $BOA$, where $O$ is its center, can we divide it into two parts of even area, using only the compass and the ruler?

Note: This problem …

Consider an equilateral triangle wire loop with a side length $S = \sqrt{3}$ and a wire radius $R = 0.01$. Calculate the self inductance of the wire loop by making …

As shown above, the cyan polygon is surrounded by 10 unit regular heptagons. If the area of the cyan polygon can be expressed as

$A\sin\dfrac{\pi}{7} + B\sin\dfrac{2\pi}{7} + C\sin\dfrac{3\pi}{7}$

where …

Given a tetrahedron $ABCD$ with $A(0, 0, 0), B(10, 0, 0), C(0, 10, 0) , D(1, 1, 10)$, you select a point ...

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