MathDB
Problems
Contests
International Contests
APMO
1991 APMO
1991 APMO
Part of
APMO
Subcontests
(5)
2
1
Hide problems
Colouring points
Suppose there are
997
997
997
points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least
1991
1991
1991
red points in the plane. Can you find a special case with exactly
1991
1991
1991
red points?
4
1
Hide problems
Hands out candies
During a break,
n
n
n
children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule: He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of
n
n
n
for which eventually, perhaps after many rounds, all children will have at least one candy each.
5
1
Hide problems
Construct a circle tangent to other two circles
Given are two tangent circles and a point
P
P
P
on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point
P
P
P
.
1
1
Hide problems
Similar triangles involve centroid
Let
G
G
G
be the centroid of a triangle
A
B
C
ABC
A
BC
, and
M
M
M
be the midpoint of
B
C
BC
BC
. Let
X
X
X
be on
A
B
AB
A
B
and
Y
Y
Y
on
A
C
AC
A
C
such that the points
X
X
X
,
Y
Y
Y
, and
G
G
G
are collinear and
X
Y
XY
X
Y
and
B
C
BC
BC
are parallel. Suppose that
X
C
XC
XC
and
G
B
GB
GB
intersect at
Q
Q
Q
and
Y
B
YB
Y
B
and
G
C
GC
GC
intersect at
P
P
P
. Show that triangle
M
P
Q
MPQ
MPQ
is similar to triangle
A
B
C
ABC
A
BC
.
3
1
Hide problems
Sum a_n = sum b_n
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
⋯
\cdots
⋯
,
a
n
a_n
a
n
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,
⋯
\cdots
⋯
,
b
n
b_n
b
n
be positive real numbers such that
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n
a
1
+
a
2
+
⋯
+
a
n
=
b
1
+
b
2
+
⋯
+
b
n
. Show that
a
1
2
a
1
+
b
1
+
a
2
2
a
2
+
b
2
+
⋯
+
a
n
2
a
n
+
b
n
≥
a
1
+
a
2
+
⋯
+
a
n
2
\frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2}
a
1
+
b
1
a
1
2
+
a
2
+
b
2
a
2
2
+
⋯
+
a
n
+
b
n
a
n
2
≥
2
a
1
+
a
2
+
⋯
+
a
n