Problem 1.
Consider the following operation on positive real numbers written on a
blackboard: choose a number r written on the blackboard, erase that number,
and then write a pair of positive real numbers a and b satisfying the
condition 2*r^2=ab on the board.
Assume that you start out with just one positive real number r on the
blackboard, and apply this operation (k^2 -1) times to end up with k^2
positive real numbers, not necessarily distinct. Show that there exists
a number on the board which does not exceed k*r.
Problem 2.
Let a_1, a_2, a_3, a_4, a_5 be real numbers satisfying the following
equations:
a_1 a_2 a_3 a_4 a_5 1
─── + ─── + ─── + ─── + ─── = ─── for k=1,2,3,4,5.
k^2 +1 k^2 +2 k^2 +3 k^2 +4 k^2 +5 k^2
a_1 a_2 a_3 a_4 a_5
Find the value of ── + ── + ── + ── + ──.
37 38 39 40 41
(Express the value in a single fraction.)
Problem 3.
Let three circles Γ_1,Γ_2,Γ_3, which are non-overlapping and mutually
external, be given in the plane. For each point P in the plane, outside
the three circles, construct six points A_1, B_1, A_2, B_2, A_3, B_3 as
follows: For each i=1,2,3, A_i, B_i are distinct points on the circle Γ_i
such that the lines PA_i and PB_i are both tangents to Γ_i. Call the point
P exceptional if, from the construction, three lines A_1B_1, A_2B_2, A_3B_3
are concurrent. Show that every exceptional point of the plane, if exists,
lies on the same circle.
Problem 4.
prove that for any positive integer k, there exists an arithmetic sequence
a_1 a_2 a_k
──, ──, ……, ── of rational numbers, where a_i, b_i are relatively
b_1 b_2 b_k
prime positive integers for each i=1,2,...,k, such that the positive
integers a_1, b_1, a_2, b_2,...,a_k, b_k are all distinct.
Problem 5.
Larry and Rob are two robots travelling in one car from Argovia to Zillis.
Both robots have control over the steering and steer according to the
following algorithm: Larry makes a 90°left turn after every l kilometer
driving from start' Rob makes a 90°right turn after every r kilometer
driving from start, where l and r are relatively prime positive integers.
In the event of both turns occurring simultaneously, the carwill keep going
without changing direction. Assume that the ground is flat and the car can
move in any direction.
Let the car start from Argovia facing towards Zillis. For which choices of
the pair (l,r) is the car guaranteed to reach Zillis, regardless of how far
it is from Argovia?