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Problems
Contests
National and Regional Contests
Romania Contests
IMAR Test
2005 IMAR Test
2005 IMAR Test
Part of
IMAR Test
Subcontests
(3)
2
2
Hide problems
Find the image of the function M:X->R
Let
n
≥
3
n \geq 3
n
≥
3
be an integer and let
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
such that
n
b
≥
a
2
nb\geq a^2
nb
≥
a
2
. We consider the set
X
=
{
(
x
1
,
x
2
,
…
,
x
n
)
∈
R
n
∣
∑
k
=
1
n
x
k
=
a
,
∑
k
=
1
n
x
k
2
=
b
}
.
X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} .
X
=
{
(
x
1
,
x
2
,
…
,
x
n
)
∈
R
n
∣
k
=
1
∑
n
x
k
=
a
,
k
=
1
∑
n
x
k
2
=
b
}
.
Find the image of the function
M
:
X
→
R
M: X\to \mathbb{R}
M
:
X
→
R
given by
M
(
x
1
,
x
2
,
…
,
x
n
)
=
max
1
≤
k
≤
n
x
k
.
M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k .
M
(
x
1
,
x
2
,
…
,
x
n
)
=
1
≤
k
≤
n
max
x
k
.
Dan Schwarz
$Q$ and $R$ are respectively the incenters in the triangles
Let
P
P
P
be an arbitrary point on the side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
and let
D
D
D
be the tangency point between the incircle of the triangle
A
B
C
ABC
A
BC
and the side
B
C
BC
BC
. If
Q
Q
Q
and
R
R
R
are respectively the incenters in the triangles
A
B
P
ABP
A
BP
and
A
C
P
ACP
A
CP
, prove that
∠
Q
D
R
\angle QDR
∠
Q
D
R
is a right angle. Prove that the triangle
Q
D
R
QDR
Q
D
R
is isosceles if and only if
P
P
P
is the foot of the altitude from
A
A
A
in the triangle
A
B
C
ABC
A
BC
.
1
2
Hide problems
abc \geq 1 then \sum \frac 1{1+b+c} \leq 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
c
≥
1
abc\geq 1
ab
c
≥
1
. Prove that
1
1
+
b
+
c
+
1
1
+
c
+
a
+
1
1
+
a
+
b
≤
1.
\frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1.
1
+
b
+
c
1
+
1
+
c
+
a
1
+
1
+
a
+
b
1
≤
1.
[hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5.
Some incircles...
The incircle of triangle
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at the points
D
,
E
,
F
D,E,F
D
,
E
,
F
, respectively. Let
K
K
K
be a point on the side
B
C
BC
BC
and let
M
M
M
be the point on the line segment
A
K
AK
A
K
such that
A
M
=
A
E
=
A
F
AM=AE=AF
A
M
=
A
E
=
A
F
. Denote by
L
,
N
L,N
L
,
N
the incenters of triangles
A
B
K
,
A
C
K
ABK,ACK
A
B
K
,
A
C
K
, respectively. Prove that
K
K
K
is the foot of the altitude from
A
A
A
if and only if
D
L
M
N
DLMN
D
L
MN
is a square. [hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2 Bogdan Enescu
3
2
Hide problems
angles
A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals 180 degrees.
A flea moves in a positive direction on the Ox axis
A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with
2
\sqrt 2
2
or
2005
\sqrt{2005}
2005
. Prove that there exists
n
0
n_0
n
0
such that the flea can reach any interval
[
n
,
n
+
1
]
[n,n+1]
[
n
,
n
+
1
]
with
n
≥
n
0
n\geq n_0
n
≥
n
0
.