Contest time again! How excited it is to see balloons floating around. But to tell you a secret, the judges' favorite time is guessing the most popular problem. When the contest is over, they will count the balloons of each color and find the result. This year…

<p>Julius Caesar lived in a time of danger and intrigue. The hardest situation Caesar ever faced was keeping himself alive. In order for him to survive, he decided to create one of the first ciphers. This cipher was so incredibly sound, that no one could…

<p>Among grandfather��s papers a bill was found:</p> <p align="center">72 turkeys $_679_</p> <p>The first and the last digits of the number that obviously represented the total price of those turkeys are replaced here by blanks (d…

Give a natural number n (1 <= n <= 500000), please tell the summation of all its proper divisors. Definition: A proper divisor of a natural number is the divisor that is strictly less than the number. e.g. number 20 has 5 proper divisors: 1, 2, 4, 5, 10,…

Have you ever used file searching tools provided by an operating system? For example, in DOS, if you type "dir *.exe", the OS will list all executable files with extension "exe" in the current directory. These days, you are so mad with the crappy operating sys…

<p>As part of an arithmetic competency program, your students will be given randomly generated lists of from 2 to 15 unique positive integers and asked to determine how many items in each list are twice some other item in the same list. You will need a p…