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What is the smallest perfect square that ends with the four digits 9009?

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Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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Mod 7

Find the remainder when 3^{2001} is divided by 7.

Rational Round

Age 16 to 18 Challenge Level:

Show that for every integer $k$ the point $(x, y)$, where

$$x = {2k\over k^2 + 1}, \ y = {k^2 - 1\over k^2 + 1},$$

lies on the unit circle, $x^2 + y^2 =1$. That is, there are infinitely many rational points on this circle.

Show that there are no rational points on the circle $x^2 + y^2 =3$.