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by math154, Feb 3, 2012, 5:16 AM
Hmm a bit on the easy side I guess, but still nice.
1. (Russia 2011, 11.4) Ten cars, which do not necessarily start at the same place, are all going one way on a highway which does not loop around. The highway goes through several towns. Every car goes with some constant speed in those towns and with some other constant speed out of those towns. For different cards these speeds can be different. 2011 flags are put in different places next to the highway. Every car went by every flag, and no car passed another right next to any of the flags. Prove that there are at least two flags at which all cars went by in the same order.
2. (Russia 2002, 11.8) Show that the numerator of
(in reduced form) is infinitely often not a prime power.
1. (Russia 2011, 11.4) Ten cars, which do not necessarily start at the same place, are all going one way on a highway which does not loop around. The highway goes through several towns. Every car goes with some constant speed in those towns and with some other constant speed out of those towns. For different cards these speeds can be different. 2011 flags are put in different places next to the highway. Every car went by every flag, and no car passed another right next to any of the flags. Prove that there are at least two flags at which all cars went by in the same order.
denote the time it takes for car 
to get to the first flag (choose a starting time arbitrarily). Suppose car 
when in and out of town, respectively; let 
(not both zero) be the distances from flag 
to flag 
along in and out of town routes. Then car 
to get to flag 
for some 
,
for all 
.
pairwise distinct cars (WLOG cars 
)
for all 
and 
.
iff![\[(x_k,y_k)\in S_{i,j}=\{(x,y)\mid L_{i,j}(x,y)=(a_i-a_j)x+(b_i-b_j)y+(t_i-t_j)>0\},\]](http://latex.artofproblemsolving.com/f/9/f/f9f56603a6e3049e2438f5de19aa67b1403e9a91.png)
and 
iff 
.
in the Cartesian plane for 
and 
)
(where 
is the number of lines) distinct regions (well-known, e.g. 
and 
are in the same region iff the orderings at flags 
are equal, so we're done by pigeonhole.2. (Russia 2002, 11.8) Show that the numerator of
(in reduced form) is infinitely often not a prime power.
and primes from 
to 
.
for 
,
,
as well. Subtracting dudes, get a contradiction.This post has been edited 6 times. Last edited by math154, Apr 21, 2012, 4:07 AM
July 2014
June 2014