MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2007 Turkey Team Selection Test
2007 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
2
2
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locus of M
Two different points
A
A
A
and
B
B
B
and a circle
ω
\omega
ω
that passes through
A
A
A
and
B
B
B
are given.
P
P
P
is a variable point on
ω
\omega
ω
(different from
A
A
A
and
B
B
B
).
M
M
M
is a point such that
M
P
MP
MP
is the bisector of the angle
∠
A
P
B
\angle{APB}
∠
A
PB
(
M
M
M
lies outside of
ω
\omega
ω
) and
M
P
=
A
P
+
B
P
MP=AP+BP
MP
=
A
P
+
BP
. Find the geometrical locus of
M
M
M
.
Find all n
A number
n
n
n
is satisfying the conditions below i)
n
n
n
is a positive odd integer; ii) there are some odd integers such that their squares' sum is equal to
n
4
n^{4}
n
4
. Find all such numbers.
3
2
Hide problems
a+b+c=1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that their sum is
1
1
1
. Prove that
1
a
b
+
2
c
2
+
2
c
+
1
b
c
+
2
a
2
+
2
a
+
1
a
c
+
2
b
2
+
2
b
≥
1
a
b
+
b
c
+
a
c
.
\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.
ab
+
2
c
2
+
2
c
1
+
b
c
+
2
a
2
+
2
a
1
+
a
c
+
2
b
2
+
2
b
1
≥
ab
+
b
c
+
a
c
1
.
Write 1 or -1 on 2007x2007
We write
1
1
1
or
−
1
-1
−
1
on each unit square of a
2007
×
2007
2007 \times 2007
2007
×
2007
board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to
1
1
1
.
1
2
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graph
Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)
Two triangles
Let
A
B
C
ABC
A
BC
is an acute angled triangle and let
A
1
,
B
1
,
C
1
A_{1},\, B_{1},\, C_{1}
A
1
,
B
1
,
C
1
are points respectively on
B
C
,
C
A
,
A
B
BC,\,CA,\,AB
BC
,
C
A
,
A
B
such that
△
A
B
C
\triangle ABC
△
A
BC
is similar to
△
A
1
B
1
C
1
.
\triangle A_{1}B_{1}C_{1}.
△
A
1
B
1
C
1
.
Prove that orthocenter of
A
1
B
1
C
1
A_{1}B_{1}C_{1}
A
1
B
1
C
1
coincides with circumcenter of
A
B
C
ABC
A
BC
.