MathDB
Problems
Contests
International Contests
APMO
2006 APMO
2006 APMO
Part of
APMO
Subcontests
(5)
5
1
Hide problems
Each clown is required to use at least five different colour
In a circus, there are
n
n
n
clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number
n
n
n
of clowns so as to make the ringmaster's order possible.
4
1
Hide problems
Let P be the midpoint of the line segment AB
Let
A
,
B
A,B
A
,
B
be two distinct points on a given circle
O
O
O
and let
P
P
P
be the midpoint of the line segment AB. Let
O
1
O_1
O
1
be the circle tangent to the line
A
B
AB
A
B
at
P
P
P
and tangent to the circle
O
O
O
. Let
l
l
l
be the tangent line, different from the line
A
B
AB
A
B
, to
O
1
O_1
O
1
passing through
A
A
A
. Let
C
C
C
be the intersection point, different from
A
A
A
, of
l
l
l
and
O
O
O
. Let
Q
Q
Q
be the midpoint of the line segment
B
C
BC
BC
and
O
2
O_2
O
2
be the circle tangent to the line
B
C
BC
BC
at
Q
Q
Q
and tangent to the line segment
A
C
AC
A
C
. Prove that the circle
O
2
O_2
O
2
is tangent to the circle
O
O
O
.
2
1
Hide problems
Finite sum of distinct integral powers of the golden ratio
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
1
1
Hide problems
Real numbers such that a_1+a_2+...+a_n is an integer
Let
n
n
n
be a positive integer. Find the largest nonnegative real number
f
(
n
)
f(n)
f
(
n
)
(depending on
n
n
n
) with the following property: whenever
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are real numbers such that
a
1
+
a
2
+
⋯
+
a
n
a_1+a_2+\cdots +a_n
a
1
+
a
2
+
⋯
+
a
n
is an integer, there exists some
i
i
i
such that
∣
a
i
−
1
2
∣
≥
f
(
n
)
\left|a_i-\frac{1}{2}\right|\ge f(n)
a
i
−
2
1
≥
f
(
n
)
.
3
1
Hide problems
Not all checkers are in the same row
Let
p
≥
5
p\ge5
p
≥
5
be a prime and let
r
r
r
be the number of ways of placing
p
p
p
checkers on a
p
×
p
p\times p
p
×
p
checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that
r
r
r
is divisible by
p
5
p^5
p
5
. Here, we assume that all the checkers are identical.