Math Problems for OSN Contest Practice

1 To make ‘short’ pastry one uses flour to fat (butter, margarine, etc.) in the ratio 2 : 1. To make ‘flaky’ pastry requires a ratio 4 : 3. I have 3 kg of flour and 2 kg of fat and wish to use all the ingredients making some of each type of pastry. How much flaky pastry do I make?

2 What is the 2011^th character in the sequence ABCDEDCBABCDEDCBABCDEDCBAB…?

3 A polygon of 100 sides has all possible diagonals drawn from a chosen vertex to the others. How many triangles are formed? What is the sum of the interior angles of the 100 sided polygon?

4 Write down the units digits of all the square numbers from 1² to at least 20².

How can you tell that 573 is not a square number?

5 The exterior angles of a triangle are in the ratio 4 : 5 : 6.

What is the least (interior) angle of the triangle?

6 You tell your friend that there is the same chance of obtaining an even number of heads when tossing four coins as of a head showing when tossing just one coin.

She doesn’t believe you; how do you convince her?

7 In a quadrilateral ABCD, side AB = side BC = diagonal CA = 2 and side AD = 1. D is located so that a circle using diagonal BD as a diameter will pass through A. Prove that BD = √5.

8 p and q are two prime numbers chosen so that p + q and p - q are also prime.

Prove that p² - q is a prime number.

9 Work out the day of the week for 14 Sep 1752, the first day of the new calendar in England when an Act of Parliament brought the calendar into alignment with the Gregorian calendar used in most of Western Europe by that time.

[Leap years occur every 4 years with the exception that every 100th year is not a leap year but every 400th year is a leap year. Look up Gregorian calendar on Wikipedia.]

10 Construct a sequence of five single digit positive integers a, b, c, d, e so that :

b divides a + b but a does not do so;

c divides a + b + c but a and b do not do so;

d divides a + b + c + d but a, b and c do not do so;

e divides a + b + c + d + e but a, b, c and d do not do so;

[Taken from JMO Mentoring Scheme – October 2011 paper]