30th Eötvös Competition Problems 1926

1.  Show that for any integers m, n the equations w + x + 2y + 2z = m, 2w - 2x + y - z = n have a solution in integers.

2.  Show that the product of four consecutive integers cannot be a square.

3.  A circle or radius R rolls around the inside of a circle of radius 2R, what is the path traced out by a point on its circumference? 

Solutions

Problem 1

Show that for any integers m, n the equations w + x + 2y + 2z = m, 2w - 2x + y - z = n have a solution in integers.

Solution

w = n, x = m+n, y = m, z = -(m+n). 

Problem 2

Show that the product of four consecutive integers cannot be a square.

Solution

(n-1)n(n+1)(n+2) = (n²+n)(n²+n-2) = (n²+n-1)² - 1. Any two squares differ by more than 1, so it cannot be a square. 

Problem 3

A circle or radius R rolls around the inside of a circle of radius 2R, what is the path traced out by a point on its circumference?

Solution

The the rolling circle have center O' and the large circle have center O. Suppose the initial point of contact is A. Let A' be the point of the rolling circle that is initially at A. When the contact has moved to B, take ∠AOB = θ. Then since the small circle has half the radius, ∠A'OB = 2θ. Hence ∠O'OA' = θ, so A' lies on OA. Equally it is clear that it can reach any point on the diameter AC.