BMOS MENTORING SCHEME (Senior Level)
March 2012 (Sheet 6)
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
2012.
1. Let ABCDEF be a regular hexagon with sides of length 2. Let P , Q, R, S, T and U be
the midpoints of the sides AB, BC, CD, DE, EF and F A respectively. Find the area
of the hexagon P QRST U .
2. Let ABCDE be a convex pentagon. Suppose that each of the triangles ABC, BCD,
CDE, DEA and EAB has area 1. What is the area of the pentagon ABCDE?
3. A square ABCD is inscribed in a circle γ. P is a point on arc CD of γ. Prove that
|P A| × (|P A| + |P C|) = |P B| × (|P B| + |P D|).
4. Let n be a positive integer. Let d(n) denote the number of divisors of n, and let σ(n) be
the sum of these divisors (in each case including n itself). Prove that
σ(n) √
≥ n.
d(n)
5. Let a, b and c be distinct integers. Show that there cannot exist a polynomial p(x) with
integer coefficients such that p(a) = b, p(b) = c and p(c) = a.
6. (a) For real a1 , a2 , a3 , b1 , b2 , b3 prove that (a21 +a22 +a23 )(b21 +b22 +b23 ) ≥ (a1 b1 +a2 b2 +a3 b3 )2 .
When does equality occur?
(b) For positive real x, y, z prove that
x+y
x+y+z
1/2
+
y+z
x+y+z
1/2
+
z+x
x+y+z
1/2
≤ 61/2 .
7. Does there exist a set A of positive integers in which each distance occurs precisely once,
i.e. such that for every positive integer n, there are unique x and y in A with x − y = n?
Deadline for receipt of solutions: 05 April 2012
Supported by the Man Group plc Charitable Trust