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International Mathematical Arhimede Contest (IMAC)
2010 IMAC Arhimede
2010 IMAC Arhimede
Part of
International Mathematical Arhimede Contest (IMAC)
Subcontests
(6)
1
1
Hide problems
n distinct triangles from 3n points
3
n
3n
3
n
points are given (
n
≥
1
n\ge 1
n
≥
1
) in the plane, each
3
3
3
of them are not collinear. Prove that there are
n
n
n
distinct triangles with the vertices those points.
2
1
Hide problems
f(x + y) = f(x) + f(y) + f(xy) , real
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that we have
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
f(x + y) = f(x) + f(y) + f(xy)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
5
1
Hide problems
n points create always obtuse triangles, prove n+1 do it also
Different points
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2,..., A_n
A
1
,
A
2
,
...
,
A
n
in the plane (
n
>
3
n> 3
n
>
3
) are such that the triangle
A
i
A
j
A
k
A_iA_jA_k
A
i
A
j
A
k
is obtuse for all the different
i
,
j
,
k
∈
{
1
,
2
,
.
.
.
,
n
}
i,j,k \in\{1,2,...,n\}
i
,
j
,
k
∈
{
1
,
2
,
...
,
n
}
. Prove that there is a point
A
n
+
1
A_{n + 1}
A
n
+
1
in the plane, such that the triangle
A
i
A
j
A
n
+
1
A_iA_jA_{n + 1}
A
i
A
j
A
n
+
1
is obtuse for all different
i
,
j
∈
{
1
,
2
,
.
.
.
,
n
}
i,j \in\{1,2,...,n\}
i
,
j
∈
{
1
,
2
,
...
,
n
}
4
1
Hide problems
segment divides a square in 2 tangential polygons, computational
Let
M
M
M
and
N
N
N
be two points on different sides of the square
A
B
C
D
ABCD
A
BC
D
. Suppose that segment
M
N
MN
MN
divides the square into two tangential polygons. If
R
R
R
and
r
r
r
are radii of the circles inscribed in these polygons (
R
>
r
R> r
R
>
r
), calculate the length of the segment
M
N
MN
MN
in terms of
R
R
R
and
r
r
r
.(Moldova)
3
1
Hide problems
IMAC 2010 seniors, first day, third problem.
Let
A
B
C
ABC
A
BC
be a triangle and let
D
∈
(
B
C
)
D\in (BC)
D
∈
(
BC
)
be the foot of the
A
A
A
- altitude. The circle
w
w
w
with the diameter
[
A
D
]
[AD]
[
A
D
]
meet again the lines
A
B
AB
A
B
,
A
C
AC
A
C
in the points
K
∈
(
A
B
)
K\in (AB)
K
∈
(
A
B
)
,
L
∈
(
A
C
)
L\in (AC)
L
∈
(
A
C
)
respectively. Denote the meetpoint
M
M
M
of the tangents to the circle
w
w
w
in the points
K
K
K
,
L
L
L
. Prove that the ray
[
A
M
[AM
[
A
M
is the
A
A
A
-median in
△
A
B
C
\triangle ABC
△
A
BC
(Serbia).
6
1
Hide problems
IMAC inequality
Consider real numbers
a
,
b
,
c
≥
0
a, b ,c \ge0
a
,
b
,
c
≥
0
with
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
. Prove that:
b
c
3
a
2
+
4
4
+
c
a
3
b
2
+
4
4
+
a
b
3
c
2
+
4
4
≤
2
∗
3
4
3
\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}
4
3
a
2
+
4
b
c
+
4
3
b
2
+
4
c
a
+
4
3
c
2
+
4
ab
≤
3
2
∗
4
3