BMOS MENTORING SCHEME (Senior Level)November 2013 (Sheet 2)Questions

The Senior Level problem sheets are ideal for students who love solving difficult mathemat-ical problems and particularly for those preparing for the British Mathematical Olympiadcompetitions. The problems get harder throughout the year and build upon ideas in earliersheets, so please try to give every problem a go. A. Rzym c©UKMT 2013.

1. Triangle ABC is right-angled at B, with a = BC and c = AB. Points B′, C ′, A′ lie onAC,AB,CB (respectively) such that BA′B′C ′ is a rectangle. For suitable choice of B′,what is the largest possible area of the rectangle (in terms of a and c)?

2. In how many ways can the letters of the word ‘magri’ be arranged such that none of theletters are in their original place (for example, we would count ‘imagr’ but not ‘miagr’,because in the latter case, the ‘m’ is in its original place)?

3. Let ABC be an equilateral triangle inscribed in a circle γ, and let X be a point on BC.Suppose that the line AX is extended to meet γ at Y . Show that 1/BY +1/CY = 1/XY .

4. A point, P , lies within triangle ABC. Lines AP,BP,CP meet BC,CA,AB at A′, B′, C ′

respectively. Let α = 4APB′, β = B′PC, γ = CA′P . Prove that

4ABC =β(α + β)(α + β + γ)

β(α + β + γ)− γ(α + β).

5. Prove that, for any positive integer p and any real number x such that p ≤ x,

1 +p

x≤

(1 +

1

x

)p

< 1 +p

x+p2

x2.

6. Do there exist integers p, q, r such that p2 + q2 − 8r = 6?

7. (a) If w, x, y, z are positive reals, prove that 4√wxyz ≥ 4

1w

+ 1x+ 1

y+ 1

z

.

(b) If x, y are non-negative reals, prove that x2012y + xy2012 ≤ x2013 + y2013.

(c) If w, x, y, z are non-negative reals and 1 ≤ wxyz, prove that16 ≤ (1 + w)(1 + x)(1 + y)(1 + z).

8. (a) Show that, given five points inside a square of side 1, some two are at distance atmost 1√

2of each other.

(b) Show that, given nine points inside a square of side 1 with no three collinear, somethree form a triangle of area at most 1

8.

9. g(w) is an integer-valued function, defined for integer w. Find g(w), given that

(a) g(g(w)) = w for all integer w

(b) g(g(w + 2) + 2) = w for all integer w

(c) g(0) = 1.

10. Does there exist a non-zero polynomial with integer coefficients, f(x), such thatf

(1 + 31/3 + 51/5 + 71/7

)= 0 ?

Deadline for receipt of solutions: 05 December 2013

For more information about the mentoring schemes, and how to join, visithttp://www.mentoring.ukmt.org.uk/