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International Mathematical Arhimede Contest (IMAC)
2008 IMAC Arhimede
2008 IMAC Arhimede
Part of
International Mathematical Arhimede Contest (IMAC)
Subcontests
(6)
6
1
Hide problems
partition of {1,2,3, ..., 6024 } such that a=b+c, where a,b,c in S_1,S_2,S_3
Consider the set of natural numbers
U
=
{
1
,
2
,
3
,
.
.
.
,
6024
}
U = \{1,2,3, ..., 6024 \}
U
=
{
1
,
2
,
3
,
...
,
6024
}
Prove that for any partition of the
U
U
U
in three subsets with
2008
2008
2008
elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.
4
1
Hide problems
calculate cosa in a tetrahedron, related to 2 midpoints
Let
A
B
C
D
ABCD
A
BC
D
be a random tetrahedron. Let
E
E
E
and
F
F
F
be the midpoints of segments
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. If the angle
a
a
a
is between
A
D
AD
A
D
and
B
C
BC
BC
, determine
c
o
s
a
cos a
cos
a
in terms of
E
F
,
A
D
EF, AD
EF
,
A
D
and
B
C
BC
BC
.
3
1
Hide problems
\sqrt {\sin^{2}x/(1+\cos^{2}x)}+\sqrt {\cos^{2}x/(1 +\sin^{2}x)}>=1
Let
0
≤
x
≤
2
π
0 \leq x \leq 2\pi
0
≤
x
≤
2
π
. Prove the inequality
sin
2
x
1
+
cos
2
x
+
cos
2
x
1
+
sin
2
x
≥
1
\sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1
1
+
c
o
s
2
x
s
i
n
2
x
+
1
+
s
i
n
2
x
c
o
s
2
x
≥
1
2
1
Hide problems
(A_{1}A_{2})(BA_{2}+A_{2}C) +..+(C_{1}C_{2})(AC_{2}+C_{2}B)>3/4
In the
A
B
C
ABC
A
BC
triangle, the bisector of
A
A
A
intersects the
[
B
C
]
[BC]
[
BC
]
at the point
A
1
A_ {1}
A
1
, and the circle circumscribed to the triangle
A
B
C
ABC
A
BC
at the point
A
2
A_ {2}
A
2
. Similarly are defined
B
1
B_ {1}
B
1
and
B
2
B_ {2}
B
2
, as well as
C
1
C_ {1}
C
1
and
C
2
C_ {2}
C
2
. Prove that
A
1
A
2
B
A
2
+
A
2
C
+
B
1
B
2
C
B
2
+
B
2
A
+
C
1
C
2
A
C
2
+
C
2
B
≥
3
4
\frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}
B
A
2
+
A
2
C
A
1
A
2
+
C
B
2
+
B
2
A
B
1
B
2
+
A
C
2
+
C
2
B
C
1
C
2
≥
4
3
1
1
Hide problems
find prime p such that 1 + p\cdot 2^{p} is a perfect square
Find all prime numbers
p
p
p
for which
1
+
p
⋅
2
p
1 + p\cdot 2^{p}
1
+
p
⋅
2
p
is a perfect square.
5
1
Hide problems
For vittasko and armpist
The diagonals of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
are intersecting at the point
E
E
E
.
K
K
K
and
M
M
M
are the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. Let the points
L
L
L
on
B
C
BC
BC
and
N
N
N
on
A
D
AD
A
D
s.t.
E
L
⊥
B
C
EL\perp BC
E
L
⊥
BC
and
E
N
⊥
A
D
EN\perp AD
EN
⊥
A
D
.Prove that
K
M
⊥
L
N
KM\perp LN
K
M
⊥
L
N
.