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2(3sec² 20 sin² 10)^1/3

The expression above can be expressed in the form a + bsec 20. Find a and b.

DO+SI=MI
RE+MI=FA
LA+SI=SOL

Solve to get the unique answer.

A tetrahedron with five edges of unit length is inscribed in a unit sphere. How long is the sixth edge?

primer+rimer+imer+mer+er+r=malaga

There is only one valid solution for the above alphametic.

Find it.

Five delegates to an international conference, Alex, Boris, Chris, Daniel and Eva, among them spoke five languages. In fact one of them spoke only one of the languages; one spoke only two of them; another, only three; another spoke four of the languages and finally one spoke all five of the languages.

The following are true about which languages were spoken:

1. Three of the delegates spoke Portuguese.
2. Spanish was spoken by more of the delegates than any other language.
3. When Chris and Eva conversed, it had to be in Italian--the only language that both spoke.
4. The only common language among Alex, Boris and Eva was French.
5. Boris and Chris both spoke English, but when Daniel joined them, they had to switch to Spanish, the only language common to all three.

What languages did each of the delegates speak?

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From Marilyn vos Savant's column in Parade magazine for April 15, 2018.

If A₁, A₂, A₃, . . . , A₃₀ are the vertices of a regular 30-gon inscribed in a unit circle, then find

|A₁A₂|² + |A₁A₃|² + |A₁A₄|² + ... + |A₁A₃₀|²

A certain device consists of 24 equally reliable components. It is known that one (and only one) is faulty. The location of the failed component is performed by replacement - one after another.

1. Find the probabilty of restoring the functioning of the system within 3 trials.
2. Same question for n trials.