BMOS MENTORING SCHEME (Senior Level)
2013-2014 Sheet 5
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
2014.
1. At a local school, 53.7802% (rounded) of the children are girls. What is the minimum
number of children at the school?
2. (a) A, B, C, D are points on the circle γ, and lines AB, CD intersect at X (which may
be inside or outside γ). Show that |AX| · |BX| = |CX| · |DX|
(b) Let P be a point outside the circle γ. Let l be a line through P which is tangent to
γ at a point Q, and let l0 be another line through P which cuts γ at points R and
S. Show that |P Q|2 = |P R| · |P S|.
3. Which positive integers can be written as the sum of two or more consecutive positive
integers?
4. (a) Let a1 , a2 , . . . , an be positive real numbers, and let b1 , b2 , . . . , bn be a rearrangement
of a1 , a2 , . . . , an . Prove that
n
X
ai
i=1
bi
≥ n.
When does equality occur?
(b) For positive real numbers a, b, c prove that
a
b
c
3
+
+
≥ .
b+c c+a a+b
2
When does equality occur?
5. For −1 ≤ x ≤ 1, define pn (x) = cos(n cos−1 x). Prove that for all positive integer n, pn (x)
is a polynomial in x with rational coefficients.
6. ABC is a triangle, right-angled at C. The internal bisectors of angles BAC and ABC
meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars
from P and Q to AB. Find angle M CN .
7. Let p(x) be a polynomial with constant term 1 and all other coefficients are either 1 or 0.
Show that all real roots of p(x) are less than
√
1− 5
.
2
8. Let p1 , p2 , . . . , p6 be a permutation of the sequence 1, 2, . . . , 6 such that for all k (1 ≤ k < 6)
p1 , p2 , . . . , pk is not a permutation of 1, 2, . . . , k. How many such permutations are there?
For more information about the mentoring schemes, and how to join, visit
http://www.mentoring.ukmt.org.uk/