APMO 2009!!

[tatari/nightmare]

1.Consider the following operation on positive real numbers written on a black-
board:
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 $2r^2 = ab$ 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 $kr$.

2.Let $a_1,a_2,a_3,a_4,a_5$ be reals numbers satisfying the equation
$$\sum_{j = 1}^{5}\dfrac{a_j}{k^2+j}=\dfrac{1}{k^2}$$ for $k=1,2,3,4,5$
Evaluate $\dfrac{a_1}{37}+\dfrac{a_2}{38}+\dfrac{a_3}{39}+\dfrac{a_4}{40}+\dfrac{a_5}{41}$

3.Let three circles $O_1, O_2, O_3$, 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 $O_i$ such that the lines $PA_i$ and $PB_i$ are both tangents to $O_i$. 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.

4.Prove that for any $k\in\mathbb{N}$,there exist an arithmetic sequence
$$\dfrac{a_1}{b_1},\dfrac{a_2}{b_2},......,\dfrac{a_k}{b_k}$$
of rational numbers ,where $a_i,b_i$ are relatively prime natural numbers for $i=1,2,...,k$
such that $a_1,a_2,....,a_k,b_1,b_2,....,b_k$ are all distrinct

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 $90^{\circ}$ left turn after every $L$ kilometer driving from start; Rob makes a $90^{\circ}$ 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 car will keep going without changing direction. Assume that the ground is
at 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?

Ps. ผมลองทำดูบ้างแล้ว ก้อทำได้แค่ 2 ข้อคือข้อ 1 กับข้อ 2 กับข้อ 3 อีกครึ่งข้อ แต่ยอมรับว่าปีนี้ง่ายกว่า
ปีที่แล้ว(มั้ง?)

[littledragon]

ขอบคุณมากครับ

[คusักคณิm]

http://www.kms.or.kr/competitions/ap...MO2009_PRB.pdf