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National and Regional Contests
France Contests
France Team Selection Test
2003 France Team Selection Test
2003 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
2
1
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France TST 2003 D2 Q2
10
10
10
cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let
n
n
n
be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly
n
n
n
depends on the flight schedule. Find the largest
n
n
n
and the corresponding flight schedule.
3
1
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France TST 2003 D1 Q3
M
M
M
is an arbitrary point inside
△
A
B
C
\triangle ABC
△
A
BC
.
A
M
AM
A
M
intersects the circumcircle of the triangle again at
A
1
A_1
A
1
. Find the points
M
M
M
that minimise
M
B
⋅
M
C
M
A
1
\frac{MB\cdot MC}{MA_1}
M
A
1
MB
⋅
MC
.
1
1
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France TST 2003 D1 Q2
A lattice point in the coordinate plane with origin
O
O
O
is called invisible if the segment
O
A
OA
O
A
contains a lattice point other than
O
,
A
O,A
O
,
A
. Let
L
L
L
be a positive integer. Show that there exists a square with side length
L
L
L
and sides parallel to the coordinate axes, such that all points in the square are invisible.