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Find the Unique Number

This puzzle requires basic number sense and number theory. It can also be solved by mere trial and error. I tried to apply as much mathematical logic as possible.

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Solve for x
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Solve for x

Find all prime pairs (p, q) such that p^3 - q^7 = p - q
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Find all prime pairs (p, q) such that p^3 - q^7 = p - q

Find all natural numbers for which n^10 + n^5 +1 is prime
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Find all natural numbers for which n^10 + n^5 +1 is prime

For any Pythagorean triple ( a, b, c) if 3 divides c, then it divides both a and b.
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For any Pythagorean triple ( a, b, c) if 3 divides c, then it divides both a and b.

Romanian Mathematics Olympiad
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Romanian Mathematics Olympiad

Your Chess Endgame Is Horrible, Let Me Fix That
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Your Chess Endgame Is Horrible, Let Me Fix That

p² - p +1 = n³
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p² - p +1 = n³

The Differential Equations Explained Better Than School Ever Did
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The Differential Equations Explained Better Than School Ever Did

Find all n greater than 1 such that (2^n + 1)/n^2 is an integer.
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Find all n greater than 1 such that (2^n + 1)/n^2 is an integer.

Unbelievable but 95% Students Fail To Solve This is 9 seconds!
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Unbelievable but 95% Students Fail To Solve This is 9 seconds!

Find the smallest n
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Find the smallest n

The German Math Olympiad Problem That 99% of Students Get Wrong! | Math Olympiad Mathematics
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The German Math Olympiad Problem That 99% of Students Get Wrong! | Math Olympiad Mathematics

Find C if f(f(x)) has 3 distinct real roots
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Find C if f(f(x)) has 3 distinct real roots

Professor Bet $100,000 "Black Student Can't Solve This" — But Student Solved It in 90 Seconds
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Professor Bet $100,000 "Black Student Can't Solve This" — But Student Solved It in 90 Seconds

Square root of imaginary number.
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Square root of imaginary number.

Which numbers are missing? – Math puzzle
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Which numbers are missing? – Math puzzle

Harvard University Interview Tricks.
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Harvard University Interview Tricks.

Find all real x
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Find all real x

Einstein OBSERVED Ramanujan's Work And Saw Mathematics That Shouldn't Exist
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Einstein OBSERVED Ramanujan's Work And Saw Mathematics That Shouldn't Exist

Prove that p^2 -  1 is a multiple of 24
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Prove that p^2 - 1 is a multiple of 24

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