BMOS MENTORING SCHEME (Senior Level)
April 2013 (Sheet 7)
Questions
The Senior Level problem sheets are ideal for students who love solving difficult mathematical problems and particularly for those preparing for the British Mathematical Olympiad
competitions. The problems get harder throughout the year and build upon ideas in earlier
c
sheets, so please try to give every problem a go. A. Rzym UKMT
2013.
1. Show that among any 18 consecutive three-digit numbers there must be one which is
divisible by the sum of its digits.
2. Consider the integers formed by taking all combinations of signs in the sequences
±12 ;
±12 ± 22 ;
±12 ± 22 ± 32 ;
±12 ± 22 ± 32 ± 42 ;
....
Prove that every integer occurs an infinite number of times.
3. We are given 6 boxes (arranged in a line from left to right) and 5 distinct balls. The balls
are placed in some or all of the first 5 boxes (and hence there are a total of 55 ways of
arranging them). We now apply the following rule repeatedly, until none of the first five
boxes has more than one ball in it:
If a box has more than one ball in it, move one ball in the leftmost such box to
the box on the right.
At the end of this process, of the 55 starting permutations how many finish with the 6’th
box empty?
4. The sides of the cyclic quadrilateral ABCD satisfy AD + BC = AB. Prove the lines
bisecting ADC and BCD meet on AB.
5. Let α and β be two positive real numbers. Consider the following two statements:
P : Both α and β are irrational and satisfy 1/α + 1/β = 1;
Q: Every positive integer occurs exactly once in the list
bαc, bβc, b2αc, b2βc, b3αc, b3βc, . . . .
Show that P is true if and only if Q is true.
6. Suppose we are given a 2k × 3 chessboard, and 3k 2 × 1 rectangular tiles.
The tiles are to be placed on the chessboard without overlap or gaps, such that the
chessboard is completely covered.
Prove that the number of distinct ways of doing so, Nk , satisfies
N2k = 4N2k−2 − N2k−4
and that
N2k
√ √ k √ √ k
= 1/2 + 1/2 3 2 + 3 + 1/2 − 1/2 3 2 − 3
Deadline for receipt of solutions: 17 May 2013
For more information about the mentoring schemes, and how to join, visit
http://www.mentoring.ukmt.org.uk/