December Problems

It's already the last month of 2014!!! We wish you all best of luck for 2015!

To celebrate the end of 2014, try attempting the December Problems, which can be downloaded here. The problems can also be found at the bottom of this post.

Have fun, and keep watching this site for more information about the camp and training sets.

JUNIOR DIVISION

Q1: Two politicians are counting votes after an election has passed. There were 50 voters in total. If after 30 votes had been counted, Obama was leading Romney by 20 votes, what is the probability that Obama will win the election?
Q2: Let \(PA\) be a tangent to a circle with the centre \(O\). Also let \(B\) be a point on line \(PO\) so that \(AB\) is perpendicular to \(PO\). Let \(r\) be the radius of circle \(O\). Find the expression for the length of \(PO\) in terms of \(r\), \(AB\) and \(PA\).

INTERMEDIATE DIVISION

Q3: In a sequence of letters consisting of M, O, R, S, E, the number of Morse codes is the number of "MORSE" found in that order within the sequence. E.g. MOSRE has 0 Morse codes, MORSSE has 2 Morse codes because "MORSE" can be formed by the first S or the second S. What is the maximum number of Morse codes in a sequence with 2015 letters?
Q4: Let \(ABC\) be a triangle and let \(E\) be a point on line segment \(BC\) such that a circle with centre \(E\) can be drawn so that the circle is tangential to lines \(AB\) and \(AC\). Let the line \(AE\) intersect the circumcircle of triangle \(ABC\) at point \(D\) Prove that \(BD=DC\).

SENIOR DIVISION

Q5: A salesman is trying to sell maths comics to a street with 50 inhabitants. He notices that if he knocks on a house, the two adjacent residents will immediately refuse to invite him in for consideration and thus he will not bother to visit these houses. Assuming he tries to visit as many houses as possible what is the total number of sets of houses that the salesman could have visited (for instance one such set is the set of all odd numbered houses, another is all even numbered houses)?
Q6: Prove that \(a^3 + b^3 + c^3 + 6abc ≥ (ab+bc+ca)(a+b+c)\) for any real numbers \(a\), \(b\) and \(c\).
Hint: Use Schur's inequality.