AM–GM Inequalities
The Arithmetic–Geometric Mean Inequality:If a ≥ 0, b ≥ 0, then(a + b)/2 ≥ √(ab) ≥ b.Equality holds if and only if a = b.
IntermediateMathematics
Algebra · AM–GM Inequalities
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12 puzzles in this collection
Intro: 1. It is known that a + b = −7, ab =
Find: a) (a + b)² b) a² + b² c) (a − b)² d) a² − ab + b² e) a³ + b³
For a real number x ≠ 0, it is known that (x − 1/x)² =
Find all values of: a) x − 1/x b) x + 1/x c) (x + 1/x)² d) x³ − 1/x³ Inequalitie
Prove that for all values of x the following inequalities hold: a) x² + 2x + 1 ≥
The Arithmetic–Geometric Mean Inequality:
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The Arithmetic–Geometric Mean Inequality: If a ≥ 0, b ≥ 0, then (a + b)/2 ≥ √(ab) ≥ b. Equality holds if and only if a = b.
All puzzles
All puzzles
Intro: 1. It is known that a + b = −7, ab =
Find: a) (a + b)² b) a² + b² c) (a − b)² d) a² − ab + b² e) a³ + b³
For a real number x ≠ 0, it is known that (x − 1/x)² =
Find all values of: a) x − 1/x b) x + 1/x c) (x + 1/x)² d) x³ − 1/x³ Inequalitie
Prove that for all values of x the following inequalities hold: a) x² + 2x + 1 ≥
Prove for all a and b the inequalities: a) a² + b² ≥ 2ab b) (c²a²)/2 + (b²)/(2c²
a) What is the minimum value of 32x² + 1/(8x²) for x ≠ 0? b) At which x is the m
Find all pairs of natural numbers x and y satisfying the equation: a) x² − y² =
For any a, b, c > 0 prove the inequalities: a) a² + b² + c² ≥ ab + bc + ac b) (a
a) Derive the formula (a + b + c)². b) It is known that a + b + c = 5 and ab + b
What can a² + b² + c² be? c) It is known that x + y + z =
Prove that two natural number squares cannot differ by 14.