Arithmetic, geometric and harmonic means

There are 7 NRICH Mathematical resources connected to Arithmetic, geometric and harmonic means
Mean Geometrically
problem
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Mean geometrically

Age
16 to 18
Challenge level
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A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?
Without Calculus
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Without calculus

Age
16 to 18
Challenge level
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Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
Pythagorean Golden Means
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Pythagorean golden means

Age
16 to 18
Challenge level
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Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
Three Ways
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Three ways

Age
16 to 18
Challenge level
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If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
AMGM
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AMGM

Age
14 to 16
Challenge level
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Can you use the diagram to prove the AM-GM inequality?

Classical Means
problem

Classical means

Age
16 to 18
Challenge level
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Use the diagram to investigate the classical Pythagorean means.
About Pythagorean Golden Means
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About Pythagorean golden means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?