| A1. Find all real solutions to x2/(x+1-√(x+1))2 < (x2+3x+18)/(x+1)2. |
|
| A2. The Fibonacci sequence is defined by F0 = 1, F1 = 1, Fn+2 = Fn+1 + Fn. Show that some Fibonacci number is divisible by 1000. |
|
| A3. In the triangle ABC, AD is an altitude, BE is an angle bisector and CF is a median. Show that they are concurrent iff a2(a-c) = (b2-c2)(a+c). |
|
| A4. Show that a 10000 x 10000 board can be tiled by 1 x 3 tiles and a 2 x 2 tile placed centrally, but not if the 2 x 2 tile is placed in a corner. |
|
| A5. The sequence u0, u1, u2, ... is defined as follows. u0 = 0, u1 = 1, and un+1 is the smallest integer > un such that there is no arithmetic progression ui, uj, un+1 with i < j < n+1. Find u100. |
|
| B1. Solve: y2 = (x+8)(x2+2) and y2 - (8+4x)y + (16+16x-5x2) = 0. |
|
| B2. f(n) is a function defined on the positive integers with positive integer values such that f(ab) = f(a)f(b) when a, b are relatively prime and f(p+q) = f(p)+f(q) for all primes p, q. Show that f(2) = 2, f(3) = 3 and f(1999) = 1999. |
|
| B3. Show that a2/(a+b) + b2/(b+c) + c2/(c+d) + d2/(d+a) ≥ 1/2 for positive reals a, b, c, d such that a + b + c + d = 1, and that we have equality iff a = b = c = d. |
|
| B4. Let d(n) be the number of positive divisors of n. Find all n such that n = d(n)4. |
|
| B5. ABCDEF is a convex hexagon such that AB = BC, CD = DE, EF = FA and ∠B + ∠D + ∠F = 360o. Show that the perpendiculars from A to FB, C to BD, and E to DF are concurrent. |
|
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Irish home
© John Scholes
jscholes@kalva.demon.co.uk
12 Oct 2003
Last updated/corrected 12 Oct 03