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BMOS MENTORING SCHEME (Senior Level)
October 2011 (Sheet 1)
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 & J. Cranch UKMT
2011.
1. Imagine that all the natural numbers whose decimal representations include each of the
ten digits 0,1,2,3,4,5,6,7,8 and 9 exactly once are listed in ascending order of magnitude.
The first digit is not allowed to be 0; thus the first three numbers in the list are
1023456789, 1023456798 and 1023456879.
Find the millionth number in the list.
2. (a) The triangle ABC is right-angled at B. Prove that
√
|AB| + |BC| ≤ 2|AC|.
(b) Prove that if a, b > 0 and a + b = 1 then
2 2
1
1
+ b+
≥ 25/2.
a+
a
b
3. In how many ways can the letters of the word ‘orange’ be arranged such that none of the
letters are in their original place (for example, we would count ‘eorang’ but not ‘oegnar’,
because in the latter case, the ‘o’ is in its original place)?
4. D is a point on side AB of triangle ABC such that ∠ACD = ∠DCB. Prove that
|BD|
= |CB|
.
|DA|
|CA|
5. How many solutions are there (as a function of m) to the equation
x1 + x2 + · · · + xm = 2001
where the xi are all non-negative integers?
6. Each of the 64 squares of an 8×8 chessboard are coloured black or white. A move consists
of either:
(a) Selecting a 3 × 3 square within the chessboard, and then replacing black squares
with white and vice versa; or
(b) Selecting a 4 × 4 square within the chessboard, and then switching the colours
similarly.
Is there any way of colouring the squares initially such that no number of moves can
possibly result in all 64 squares being white?
7. Does there exist a positive integer, n, such that the decimal representation of 3n begins
and ends with the digits ‘2001’, i.e.
3n = 2001 · · · 2001?
Deadline for receipt of solutions: 15 November 2011
Supported by the Man Group plc Charitable Trust