2009 APMO

Part of APMO

Subcontests

(5)

1

Find pairs (l,r) such that (0,i) is reachable for all i>0

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 degrees left turn after every

\ell

kilometer driving from start, Rob makes a 90 degrees right turn after every

r

kilometer driving from start, where

\ell

r

are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will 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 (

\ell

r

) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

1

Existence of a rational arithmetic sequence

Prove that for any positive integer

k

, there exists an arithmetic sequence

\frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}

of rational numbers, where

a_i, b_i

are relatively prime positive integers for each i \equal{} 1,2,...,k such that the positive integers

a_1, b_1, a_2, b_2, ..., a_k, b_k

are all distinct.

1

Show that exceptional points lie on the same circle

Let three circles

\Gamma_1, \Gamma_2, \Gamma_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 \equal{} 1, 2, 3,

A_i, B_i

are distinct points on the circle

\Gamma_i

such that the lines

PA_i

PB_i

are both tangents to

\Gamma_i

. Call the point

P

exceptional if, from the construction, three lines

A_1B_1, A_2 B_2, A_3 B_3

are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.

1

Sum of 5 fractions

a_1

a_2

a_3

a_4

a_5

be real numbers satisfying the following equations: \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2} for k \equal{} 1, 2, 3, 4, 5 Find the value of \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41} (Express the value in a single fraction.)

1

Splitting numbers on the board with 2r^2 = ab

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

b

satisfying the condition 2 r^2 \equal{} 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 \minus{} 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 kr.