October Problems

It has already been a month since the September Problems were released!!! Note that we still accept submissions until 7th of October (please keep the deadline if possible; the official deadline is the last day of the month).

The October Problems can be downloaded as a pdf file here, or see below. If you have any questions or if you want clarification on the problems, feel free to email us (nzmosa@outlook.com), message us on Facebook, or leave a comment below. At the end of the month, submit your solutions to nzmosa@outlook.com

Enjoy!  Also, on the side note, good luck to everyone who submitted the Camp Selection Problems!

JUNIOR DIVISION

Q1: We call a set of three numbers a "triplet" if we can create a triangle with sides the length of the numbers (so \({3,5,4}\) is a triplet but \({3,4,10}\) is not). If \(x\) is the smallest positive integer that will form a triplet with \(3\) and \(6\), and \(y\) is the largest such integer, then find the value of \(10x + y\).
Q2: What is the least number of \(12 \times 15 \times 20\) blocks required to make a perfect cube?

INTERMEDIATE DIVISION

Q3: What is the last digit of \(9^{9^{2014}}\)?
Q4: How many rectangles are there in the board containing the shaded circle?

SENIOR DIVISION

Q5: Suppose \(x\), \(y\) and \(z\) are positive integers such that:
\(x + \frac{1}{y+\frac{1}{z}} = 108/53\)
What is the value of \(x+y+z\)?
Q6: \(A\), \(B\), \(C\), \(D\) and \(E\) are consecutive vertices of a regular polygon. The lines \(AB\) and \(ED\) are extended to at a point \(F\), where \(∠ BFD = 170 °\). How many sides does the polygon have?