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Problems
Contests
International Contests
APMO
1999 APMO
1999 APMO
Part of
APMO
Subcontests
(5)
5
1
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Circle and points
Let
S
S
S
be a set of
2
n
+
1
2n+1
2
n
+
1
points in the plane such that no three are collinear and no four concyclic. A circle will be called
Good
\text{Good}
Good
if it has 3 points of
S
S
S
on its circumference,
n
−
1
n-1
n
−
1
points in its interior and
n
−
1
n-1
n
−
1
points in its exterior. Prove that the number of good circles has the same parity as
n
n
n
.
3
1
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Prove circumcircle tangent to segments
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two circles intersecting at
P
P
P
and
Q
Q
Q
. The common tangent, closer to
P
P
P
, of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
touches
Γ
1
\Gamma_1
Γ
1
at
A
A
A
and
Γ
2
\Gamma_2
Γ
2
at
B
B
B
. The tangent of
Γ
1
\Gamma_1
Γ
1
at
P
P
P
meets
Γ
2
\Gamma_2
Γ
2
at
C
C
C
, which is different from
P
P
P
, and the extension of
A
P
AP
A
P
meets
B
C
BC
BC
at
R
R
R
. Prove that the circumcircle of triangle
P
Q
R
PQR
PQR
is tangent to
B
P
BP
BP
and
B
R
BR
BR
.
2
1
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Prove sum a_i/i >=a_n
Let
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
be a sequence of real numbers satisfying
a
i
+
j
≤
a
i
+
a
j
a_{i+j} \leq a_i+a_j
a
i
+
j
≤
a
i
+
a
j
for all
i
,
j
=
1
,
2
,
…
i,j=1,2,\dots
i
,
j
=
1
,
2
,
…
. Prove that
a
1
+
a
2
2
+
a
3
3
+
⋯
+
a
n
n
≥
a
n
a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n
a
1
+
2
a
2
+
3
a
3
+
⋯
+
n
a
n
≥
a
n
for each positive integer
n
n
n
.
4
1
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Determine pairs (a,b)
Determine all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers with the property that the numbers
a
2
+
4
b
a^2+4b
a
2
+
4
b
and
b
2
+
4
a
b^2+4a
b
2
+
4
a
are both perfect squares.
1
1
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Find smallest n
Find the smallest positive integer
n
n
n
with the following property: there does not exist an arithmetic progression of
1999
1999
1999
real numbers containing exactly
n
n
n
integers.