Summary: MATH BATTLE
Problem 1 A thin rectangular piece of tin is 26 inches long and 21 inches
wide. A thin square piece of metal with side length of 4 inches is cut from
each corner, and the tin is folded to form an open rectangular box. How many
cubic inches are in the volume of the box?
Problem 2 What is the smallest positive integer greater than 1 such that
division by each element of the set 4,5,6,9,10 gives a remainder of 1?
Problem 3 The numbers 1447, 1005, and 1231 have something in common.
Each is a four-digit number beginning with 1 that has exactly two identical
digits. How many such numbers are there?
Problem 4 John and Pete have three pieces of paper. Each of the boys
picks one piece, tears it up, and puts the smaller pieces back. They repeat
this process several times. John always tears a piece of paper into 3 smaller
pieces while Pete always tears a piece of paper into 5 smaller pieces. After a
few minutes can there be exactly 100 pieces of paper?
Problem 5 The increasing sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all
positive integers that are neither the square nor the cube of a positive integer.
Find the 500th term of this sequence.
Problem 6 A biologist wants to calculate the number of fish in a lake. On
May 1 she catches a random sample of 60 fish, tags them, and releases them.