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Problems
Contests
National and Regional Contests
Germany Contests
Germany Team Selection Test
2012 Germany Team Selection Test
2012 Germany Team Selection Test
Part of
Germany Team Selection Test
Subcontests
(3)
3
1
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Inequality with Sum of Squares
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
a
2
+
b
2
+
c
2
≥
3
a^2+b^2+c^2 \geq 3
a
2
+
b
2
+
c
2
≥
3
. Prove that:
(
a
+
1
)
(
b
+
2
)
(
b
+
1
)
(
b
+
5
)
+
(
b
+
1
)
(
c
+
2
)
(
c
+
1
)
(
c
+
5
)
+
(
c
+
1
)
(
a
+
2
)
(
a
+
1
)
(
a
+
5
)
≥
3
2
.
\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.
(
b
+
1
)
(
b
+
5
)
(
a
+
1
)
(
b
+
2
)
+
(
c
+
1
)
(
c
+
5
)
(
b
+
1
)
(
c
+
2
)
+
(
a
+
1
)
(
a
+
5
)
(
c
+
1
)
(
a
+
2
)
≥
2
3
.
2
1
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The Wrong Problem
Let
Γ
\Gamma
Γ
be the circumcircle of isosceles triangle
A
B
C
ABC
A
BC
with vertex
C
C
C
. An arbitrary point
M
M
M
is chosen on the segment
B
C
BC
BC
and point
N
N
N
lies on the ray
A
M
AM
A
M
with
M
M
M
between
A
,
N
A,N
A
,
N
such that
A
N
=
A
C
AN=AC
A
N
=
A
C
. The circumcircle of
C
M
N
CMN
CMN
cuts
Γ
\Gamma
Γ
in
P
P
P
other than
C
C
C
and
A
B
,
C
P
AB,CP
A
B
,
CP
intersect at
Q
Q
Q
. Prove that
∠
B
M
Q
=
∠
Q
M
N
.
\angle BMQ = \angle QMN.
∠
BMQ
=
∠
QMN
.
1
1
Hide problems
Least Changes to Make Sums Different
Find the least integer
k
k
k
such that for any
2011
×
2011
2011 \times 2011
2011
×
2011
table filled with integers Kain chooses, Abel be able to change at most
k
k
k
cells to achieve a new table in which
4022
4022
4022
sums of rows and columns are pairwise different.