Melbourne University Mathematics and Statistics Society

Sample Questions

In addition to these sample questions, you may also wish to look over the questions asked in previous Schools Maths Olympics or University Maths Olympics.


Questions

Question 1. (5 marks)

Each of the angles of a triangle when expressed in degrees is a perfect square. Find the three angles.

Question 2. (5 marks)

Call a four-digit number balanced if the sum of the two left-most digits equals the sum of the two right-most digits. For example, 2754 is balanced. How many balanced numbers are there between 1000 and 2000?

Question 3. (5 marks)

There are exactly four positive integers n such that (n+7) is a factor of (n+1)2. Find the largest such n.

Question 4. (10 marks)

Ten (not necessarily distinct) integers have the property that if all but one of them are added the possible results are: 82, 83, 84, 85, 87, 89, 90, 91, 92. What is the smallest of the integers?

Question 5. (10 marks)

A circle is inscribed in a triangle whose sides are 40, 40 and 48 cm respectively. A smaller circle is tangent to the two equal sides of the triangle and to the first circle. Find the radius, in cm, of the smaller circle.

Question 6. (10 marks)

Find the sum of all values of x which satisfy the following system of equations:

  • y = x2 - 5x + 5
  • z = x2 - 12x + 35
  • yz = 1

where x, y, z are real numbers.

Answers

  1. 100, 64, 16
  2. 63
  3. 29
  4. 5
  5. 3
  6. 20