The Nobel Prize in Physics this year was awarded jointly to Francois Englert and Peter W. Higgs "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles".
One of the most used methods of proof applies to those situations in which an infinite set of propositions follow one another in a sequence.
In the Australian outback there are 20 cattle ranches which are visited every month by a veterinarian. On one of these aerial trips the veterinarian took along his assistant as a copilot. When they were flying between two ranches the assistant suddenly said:
Problem 1. Suppose that $x, y, z$ are non-zero integers with no common factor except $1$ such that $$ x^2 + y^2 = z^2. $$ Prove that exactly one of these integers is a multiple of $5$
Competition Winners – Senior Division
First Prize
Leo Jiang Trinity Grammar School
Q1431. Find a four-digit number with the following property: if the last digit of the number is moved to the front and $7$ is added to the result, the answer is exactly twice the original number. Is there more than one such number?
Q1421 Find all integer solutions of the equation $2013x+49y=2$.