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Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Escriptions

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Quadratic Harmony

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Real(ly) Numbers

Age 16 to 18 Challenge Level:

If $x$, $y$ and $z$ are real numbers such that:

$x+y+z=5$

and $xy+yz+zx=3$,

what is the largest value that any one of these numbers can have?


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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