Problem Solving

These are examples of problems we have done from Math Olympiad (Volume 1) by Dr. George Lenchner:

If we count by 3s starting with 1, the following sequence is obtained: 1,4,7,10.....

What is the 100th number in the sequence?

A book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit 1 appear in the page numbers?

In three bowling games, Alice scores 139, 143, and 144. What score will Alice need in a fourth game in order to have an average score of 145 for all four games?

I have 30 coins consisting of nickels and quarters. The total value of the coins is $4.10. How many of each kind do I have?

Find the sum of the counting numbers from 1 to 25 inclusive. In other words, if S=1 + 2 + 3 + ... + 24 + 25, find the value of S.