# Answer to STEP I 1999 Q.2

I/1999/2 – A point moves in the x,y plane so that the sum of the squares of its distances from the three fixed points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is always $a^{2}$.

Find the equation of the locus of the point and interpret it geometrically.

Explain why $a^{2}$ cannot be less than the sum of the squares of the distances of the three points from their centroid. Continue reading

# Answer to STEP I 2007 Q.1

I/2007/1 – A positive integer with 2n digits (the first of which must not be 0) is called a balanced number if the sum of the first n digits equals the sum of the last n digits. For example, 1634 is a 4-digit balanced number, but 123401 is not a balanced number. (i) Show that seventy 4-digit balanced numbers can be made using the digits 0, 1, 2, 3 and 4. (ii) Show that $\frac{1}{6}k(k+1)(4k+5)$ 4-digit balanced numbers can be made using the digits 0 to k. You may use the identity $\sum_1^n{r^2} \equiv \frac{1}{6}n(n+1)(2n+1)$ Continue reading