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### Number and algebra

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### For younger learners

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# Without Calculus

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### Mean Geometrically

### Pythagorean Golden Means

### Three Ways

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

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Age 16 to 18

Challenge Level

- Problem
- Student Solutions

Given that $u> 0$ and $v> 0$ what is the smallest possible value of $1/u + 1/v$ given that $u + v = 5$?

Can you find this value by more than one method (not involving trial and error) without using calculus?

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?

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.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.