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National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2012 Estonia Team Selection Test
2012 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(5)
5
1
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max (x^2+y^2+z^2)(x^3+y^3+z^3)/(x^4+y^4+z^4), if x+y+z=2012, x,y,z>0
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers whose sum is
2012
2012
2012
. Find the maximum value of
(
x
2
+
y
2
+
z
2
)
(
x
3
+
y
3
+
z
3
)
(
x
4
+
y
4
+
z
4
)
\frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}
(
x
4
+
y
4
+
z
4
)
(
x
2
+
y
2
+
z
2
)
(
x
3
+
y
3
+
z
3
)
4
1
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circumcircle of APQ passes through circumcenter of ABC iff AP = CQ
Let
A
B
C
ABC
A
BC
be a triangle where
∣
A
B
∣
=
∣
A
C
∣
|AB| = |AC|
∣
A
B
∣
=
∣
A
C
∣
. Points
P
P
P
and
Q
Q
Q
are different from the vertices of the triangle and lie on the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Prove that the circumcircle of the triangle
A
P
Q
APQ
A
PQ
passes through the circumcenter of
A
B
C
ABC
A
BC
if and only if
∣
A
P
∣
=
∣
C
Q
∣
|AP| = |CQ|
∣
A
P
∣
=
∣
CQ
∣
.
3
1
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acute angle wanted, cyclic ABCD related
In a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
we have
∣
A
D
∣
>
∣
B
C
∣
|AD| > |BC|
∣
A
D
∣
>
∣
BC
∣
and the vertices
C
C
C
and
D
D
D
lie on the shorter arc
A
B
AB
A
B
of the circumcircle. Rays
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
K
K
K
, diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
P
P
P
. Line
K
P
KP
K
P
intersects the side
A
B
AB
A
B
at point
L
L
L
. Prove that
∠
A
L
K
\angle ALK
∠
A
L
K
is acute.
2
1
Hide problems
a_i = a_{i+n}, a_i not divisible by n, a_{i+a_i} divisible by a_i for any i
For a given positive integer
n
n
n
one has to choose positive integers
a
0
,
a
1
,
.
.
.
a_0, a_1,...
a
0
,
a
1
,
...
so that the following conditions hold: (1)
a
i
=
a
i
+
n
a_i = a_{i+n}
a
i
=
a
i
+
n
for any
i
i
i
, (2)
a
i
a_i
a
i
is not divisible by
n
n
n
for any
i
i
i
, (3)
a
i
+
a
i
a_{i+a_i}
a
i
+
a
i
is divisible by
a
i
a_i
a
i
for any
i
i
i
. For which positive integers
n
>
1
n > 1
n
>
1
is this possible only if the numbers
a
0
,
a
1
,
.
.
.
a_0, a_1, ...
a
0
,
a
1
,
...
are all equal?
1
1
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sum of perfect squares = sum of their perfect cubes
Prove that for any positive integer
k
k
k
there exist
k
k
k
pairwise distinct integers for which the sum of their squares equals the sum of their cubes.