MathDB

3

Part of 2001 CentroAmerican

Problems(2)

points on a circle.

Source: Central American Olympiad 2001, problem 6

8/12/2009
In a circumference of a circle, 10000 10000 points are marked, and they are numbered from 1 1 to 10000 10000 in a clockwise manner. 5000 5000 segments are drawn in such a way so that the following conditions are met: 1. Each segment joins two marked points. 2. Each marked point belongs to one and only one segment. 3. Each segment intersects exactly one of the remaining segments. 4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment. Let S S be the sum of the products assigned to all the segments. Show that S S is a multiple of 4 4.
modular arithmetic
Number theory problem

Source: Central American Olympiad 2001, problem 3

8/12/2009
Find all the real numbers N N that satisfy these requirements: 1. Only two of the digits of N N are distinct from 0 0, and one of them is 3 3. 2. N N is a perfect square.