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Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2008 Iran Team Selection Test
2008 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(12)
12
1
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Bisection of an angle
In the acute-angled triangle
A
B
C
ABC
A
BC
,
D
D
D
is the intersection of the altitude passing through
A
A
A
with
B
C
BC
BC
and
I
a
I_a
I
a
is the excenter of the triangle with respect to
A
A
A
.
K
K
K
is a point on the extension of
A
B
AB
A
B
from
B
B
B
, for which \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C.
I
a
K
I_aK
I
a
K
intersects the extension of
A
D
AD
A
D
at
L
L
L
. Prove that
D
I
a
DI_a
D
I
a
bisects the angle
∠
A
I
a
B
\angle AI_aB
∠
A
I
a
B
iff AL\equal{}2R. (
R
R
R
is the circumradius of
A
B
C
ABC
A
BC
)
11
1
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Find all functions
k
k
k
is a given natural number. Find all functions
f
:
N
→
N
f: \mathbb{N}\rightarrow\mathbb{N}
f
:
N
→
N
such that for each
m
,
n
∈
N
m,n\in\mathbb{N}
m
,
n
∈
N
the following holds: f(m)\plus{}f(n)\mid (m\plus{}n)^k
10
1
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Find Angle A
In the triangle
A
B
C
ABC
A
BC
,
∠
B
\angle B
∠
B
is greater than
∠
C
\angle C
∠
C
.
T
T
T
is the midpoint of the arc
B
A
C
BAC
B
A
C
from the circumcircle of
A
B
C
ABC
A
BC
and
I
I
I
is the incenter of
A
B
C
ABC
A
BC
.
E
E
E
is a point such that \angle AEI\equal{}90^\circ and
A
E
∥
B
C
AE\parallel BC
A
E
∥
BC
.
T
E
TE
TE
intersects the circumcircle of
A
B
C
ABC
A
BC
for the second time in
P
P
P
. If \angle B\equal{}\angle IPB, find the angle
∠
A
\angle A
∠
A
.
9
1
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Concurrent
I
a
I_a
I
a
is the excenter of the triangle
A
B
C
ABC
A
BC
with respect to
A
A
A
, and
A
I
a
AI_a
A
I
a
intersects the circumcircle of
A
B
C
ABC
A
BC
at
T
T
T
. Let
X
X
X
be a point on
T
I
a
TI_a
T
I
a
such that XI_a^2\equal{}XA.XT. Draw a perpendicular line from
X
X
X
to
B
C
BC
BC
so that it intersects
B
C
BC
BC
in
A
′
A'
A
′
. Define
B
′
B'
B
′
and
C
′
C'
C
′
in the same way. Prove that
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
are concurrent.
8
1
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Perfect square preserving polynomial
Find all polynomials
p
p
p
of one variable with integer coefficients such that if
a
a
a
and
b
b
b
are natural numbers such that a \plus{} b is a perfect square, then p\left(a\right) \plus{} p\left(b\right) is also a perfect square.
7
1
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Intersection of elements of a family
Let
S
S
S
be a set with
n
n
n
elements, and
F
F
F
be a family of subsets of
S
S
S
with 2^{n\minus{}1} elements, such that for each
A
,
B
,
C
∈
F
A,B,C\in F
A
,
B
,
C
∈
F
,
A
∩
B
∩
C
A\cap B\cap C
A
∩
B
∩
C
is not empty. Prove that the intersection of all of the elements of
F
F
F
is not empty.
6
1
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A tournament
Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.
5
1
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sqrt{a^3 + a} + sqrt{b^3+b} + sqrt{c^3+c} \geq 2 sqrt{a+b+c}
Let
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
and
a
b
+
b
c
+
c
a
=
1
ab+bc+ca = 1
ab
+
b
c
+
c
a
=
1
. Prove that:
a
3
+
a
+
b
3
+
b
+
c
3
+
c
≥
2
a
+
b
+
c
.
\sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}.
a
3
+
a
+
b
3
+
b
+
c
3
+
c
≥
2
a
+
b
+
c
.
4
1
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Minimum of a function
Let
P
1
,
P
2
,
P
3
,
P
4
P_1,P_2,P_3,P_4
P
1
,
P
2
,
P
3
,
P
4
be points on the unit sphere. Prove that \sum_{i\neq j}\frac1{|P_i\minus{}P_j|} takes its minimum value if and only if these four points are vertices of a regular pyramid.
2
1
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Concurrency
Suppose that
I
I
I
is incenter of triangle
A
B
C
ABC
A
BC
and
l
′
l'
l
′
is a line tangent to the incircle. Let
l
l
l
be another line such that intersects
A
B
,
A
C
,
B
C
AB,AC,BC
A
B
,
A
C
,
BC
respectively at
C
′
,
B
′
,
A
′
C',B',A'
C
′
,
B
′
,
A
′
. We draw a tangent from
A
′
A'
A
′
to the incircle other than
B
C
BC
BC
, and this line intersects with
l
′
l'
l
′
at
A
1
A_1
A
1
.
B
1
,
C
1
B_1,C_1
B
1
,
C
1
are similarly defined. Prove that
A
A
1
,
B
B
1
,
C
C
1
AA_1,BB_1,CC_1
A
A
1
,
B
B
1
,
C
C
1
are concurrent.
3
1
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A tree with k edges
Suppose that
T
T
T
is a tree with
k
k
k
edges. Prove that the
k
k
k
-dimensional cube can be partitioned to graphs isomorphic to
T
T
T
.
1
1
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f(xf(y)) + y + f(x) = f(x + f(y)) + yf(x)
Find all functions
f
:
R
⟶
R
f: \mathbb R\longrightarrow \mathbb R
f
:
R
⟶
R
such that for each
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
: f(xf(y)) \plus{} y \plus{} f(x) \equal{} f(x \plus{} f(y)) \plus{} yf(x)