An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Consider numbers whose fist digit is 1. Looking at each possible
value for the second digit we find 9 such numbers.
110, 121, 132, 143, 154, 165, 176, 187, 198.
Similarly there are 8 numbers starting with 2; 7 numbers
starting with 3...
Lastly there is only one number starting with 9: 990.
Hence the total is 9+8+7+6+5+4+3+2+1 = 45.
This problem is taken from the UKMT Mathematical Challenges.