July problems

Here's the second installment in our series of monthly competitions! You can download the PDF here. The problems are also repeated below. If you have any questions or if you want clarification on the problems, feel free to email us, message us on Facebook, or leave a comment below.

Good luck, have fun, and send your solutions in to nzmosa@outlook.com! Remember to include your name, school and school year in your email!

-NZMOSA Team

(P.S: If you've just joined us, you can check out the rules about the competition.)

Junior division

Q1:  Two years from now, my mother's age will be three times my age. Ten years from now, she will be four years more than twice my age. How old is each of us now?

Q2:  In the diagram below, \(ABCD\) is a square. There are four identical quarter-circles in the corners, leaving a star shape in the middle. \(AC\) and \(BD\) are the diagonals of this square.
 
 
Find the ratio of the shaded area to the area of square \(ABCD\).

Intermediate division

Q3:  A jar contains \(8\) red lollies and \(12\) purple lollies. A second jar contains \(20\) red lollies and \(N\) purple lollies. A single lolly is picked at random (without looking) from each jar. The probability that both lollies are of the same colour is \(0.44\). What is \(N\)?
Q4:  Let \(n\) be a positive integer (a natural number). Prove that \(n^5-n\) is divisible by \(30\) no matter what \(n\) is.

Senior division

Q5:  Given that \(\dfrac{1}{x}+x=-2\), what is the value of \(\dfrac{1}{x^5}+x^5\)?
Q6:  In the diagram below, \(AB\) is the diameter of a circle, and \(ABCD\) is a quadrilateral whose vertices all lie on that circle. \(S\) is the intersection point between \(BD\) and \(AC\), and \(T\) is a point on line \(AB\) so that \(ST\) is perpendicular to \(AB\).
 
 
Prove that angle \(CTS\) and angle \(DTS\) are equal regardless of where \(C\) and \(D\) are drawn (as long as they are on the same side of \(AB\)).