August problems!!

Again, another month has passed (already! I know, right?), and here the August problems 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.)

(P.P.S: We'll still accept July problem solutions! Submitting will put you in a pretty good position for a prize. We will announce winners as soon as we have enough people)

Junior division

Q1:  Tom's number is 50% larger than Alice's, and Sally's number is 25% larger than Alice's. What percent larger is Tom's number compared to Sally's number?

Q2:  What is the area of a rhombus which has diagonals of length 15 cm and 7 cm?
A rhombus is a quadrilateral with four equal sides.

Intermediate division

Q3:  Lorde, Kimbra and Stan Walker are singing together in a concert. Each song is a solo, a duet or a trio. Lorde sings 4 songs in total, Kimbra sings 6 in total and Stan sings 7 in total. Lorde sings 1 solo piece, Kimbra sings 2, and Stan sings 3 solo pieces. As the finale, they sing one trio song. How many songs do they sing in total?

Q4:  What is the sum of all the digits of \(2^{361} \times 5^{365}\)?

Senior division

Q5:  In the diagram below, all three circles are identical with radius \(r\) and are tangent to each other and to an equilateral triangle.
 
 
What is the area of the entire equilateral triangle in terms of \(r\)?
Q6:  If \(\dfrac{1}{\sin{x}} - \dfrac{1}{\tan{x}} = 5\), prove that the value of \(\dfrac{1}{\sin{x}} + \dfrac{1}{\tan{x}}\) is \(\dfrac{1}{5}\).

We will not accept a calculator solution for Q6.