Here are some sample problems from the Problem Solvers Club
Junior
1. Which multiple of 11 is nearest to 1000?
2. My children are all at school. The product of their ages if 60,060. How many children have I got?
3. Does there exist a two-digit number ‘ab’ such that the difference between ‘ab’ and its reverse ‘ba’ is a prime number?
4. What is the surface area of a cube which just fits inside a sphere of radius 1 cm?
5. Which primes can be expressed as the difference of two squares? Which primes can be expressed as the difference of two squares in more than one way?
Intermediate
1. When a barrel is 30% empty it contains 30 litres more than when it is 30% full? How many litres does the barrel hold when full? ?
2. In the multiplication sum, p, q, r and s stand for different digits. p q
Find the digit which each letter represents, explaining how x r q
you know that you have found all possible solutions. s s s
3. A is the number 11111…11111 formed with all 2003 digits equal to 1. What is the sum of the digits of the product 2003 x A?
4. Add together all the different four-digit numbers that can be made using each of the digits of 2008 exactly once. (None of these numbers start with 0.)
Senior
1. How many odd numbers are there in the nth row of Pascal’s triangle?
2. What is the smallest number of weeks equal to a square number of seconds?
3. Find (with proof) the units digit of 252007 + 762007.
4. 100 children stand holding hands to form a circle. There are 19 ‘handholds’ where both hands belong to a boy. If there are 60 boys and 40 girls, how many ‘handholds’ are there?
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