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2(3sec^{2} 20 sin^{2} 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:
 Three of the delegates spoke Portuguese.
 Spanish was spoken by more of the delegates than any other language.
 When Chris and Eva conversed, it had to be in Italianthe only language that both spoke.
 The only common language among Alex, Boris and Eva was French.
 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?
From Marilyn vos Savant's column in Parade magazine for April 15, 2018.
If A_{1}, A_{2}, A_{3}, . . . , A_{30} are the vertices of a regular 30gon inscribed in a unit circle, then find
A_{1}A_{2}^{2} + A_{1}A_{3}^{2} + A_{1}A_{4}^{2} + ... + A_{1}A_{30}^{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.
Find solutions for f(x) and g(x), such that:
(1): {f(x) + f'(x)}^2 = 1 + f(2x)
(2): {g(x) + g'(x)}^2 = 2*(g(x) + g(2x))
A+B=CD E+F=G H+I=J Assign 09 digits to the letters AJ to make the above equations true.
REM: Choose only the solutions in which A>B; E>F & H>I.
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X ***
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Each of the asterisks represents a prime digit.
Restore the original multiplication.
Let ABCDE be an equilateral convex pentagon such that angle ABC = 136 and angle BCD = 104. What is the measure of angle AED?
An equilateral polygon is a polygon whose sides are all of the same length.
Consider two points A (1,0), and B (5,4) and a parabola y ^{2}=4x. Find coordinates of point P on the parabola for which the sum of PA + PB is minimal.
True (albeit coded) facts:
a year ago i've visited an Europian country named 2831561, tomorrow I'm boarding a flight to another country with a lesser population i.e.
1234561
The digits that could represent the Capital letters of the corresponding capitals of those countries were not mentioned above, but enough is said for you to find the cities and the countries.
the Queen of Ruritania went shopping with her younger sister. Each bought as many lengths of silk as she is years old and as many yards of each length as she bought lengths and paid for it as many francs per yard as she bought yards of each length. The Queen spent 5803 francs more than her sister. How old is the Queen?
Source : The weekend problems book; collected by Hubert Phillips UK 1932
