Mathematics

If a and b are positive integers, then a + b is also a positive integer.
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This theorem states that if a and b are positive integers, then their sum a + b is also a positive integer. This is true because both a and b are positive, so their sum must be positive as well.

Prove that if n^2 is even, then n is even.
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The given statement is true. If n^2 is even, then n is even.

A proof that 0.999... = 1
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The proof that 0.999... = 1 is elementary, using just the mathematical tools of comparison and addition of (finite) decimal numbers.

Euler's formula
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The Euler's formula states that for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. This can be proven using the Noah's Ark method, which states that if you have a convex polyhedron with V vertices, E edges, and F faces, then V - E + F = 2.

Proof that square root of 2 is irrational.
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The Pythagorean Theorem states that the length of the diagonal of a square is equal to the square root of 2. This proof shows that the square root of 2 is irrational.

Good Will Hunting Problem
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Remember the Good Will Hunting movie, where a genius solves an MIT problem? Our artificial intelligence can solve that too!