# Problems from Russian Mathematical Circles

I  have culled these problems for fun with mathematics or for practising for Pre-RMO. (The source is Russian Mathematical Circles):

Problem 1:

A number of bacteria are placed in a glass. One second later each bacterium divides in two, the next second each of the resulting bacteria divides in two again, etc. After one minute, the glass is full. When was the glass half-full?

Problem 2:

Ann, John, and Alex took a bus tour of Disneyland. Each of them must pay 5 plastic chips for the ride, but they have only plastic coins of values 10, 15 and 20 chips (each has an unlimited number of each type of coin). How can they pay for the ride?

Problem 3:

Jack tore out several successive pages from a book. The number of the first page he tore out was 183, and it is known that the number of the last page is written with the same digits in some order. How many pages did Jack tear out of the book?

Problem 4:

There are 24 pounds of nails in a sack. Can you measure out 9 pounds of  nails using only a balance with two pans?

Problem 5:

A caterpillar crawls up a pole 75 inches high, starting from the ground. Each day it travels up 5 inches, and each night it slides down 4 inches. When will it reach the top of the pole?