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Puzzle 156: Three simultaneous equations

Find all positive real solutions of the simultaneous equations: x + y^2 + z^3 = 3, y + z^2 + x^3 = 3, z + x^2 + y^3 = 3.

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Puzzle 157: Trigonometric product

Compute the infinite product [sin(x) cos(x/2)]^(1/2) * [sin(x/2) cos(x/4)]^(1/4) * [sin(x/4) cos(x/8)]^(1/8) * ... , where 0

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Puzzle 158: Fermat squares

By Fermat's Little Theorem, the number x = (2^(p-1) - 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?

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Puzzle 159: Eight odd squares

Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares. For example, 6 = 2^2 + 1^2 + 1^2 is the sum of three squares. Given this theorem,...

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Puzzle 160: Absolute maximum

The absolute value of a real number is defined as its numerical value without regard for sign. So, for example, abs(2) = abs(-2) = 2. The maximum of two real numbers is defined as the numerically...

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