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National and Regional Contests
Iran Contests
Iran Team Selection Test
2009 Iran Team Selection Test
2009 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(12)
12
1
Hide problems
Iran TST 2009-Day4-P3
T
T
T
is a subset of
1
,
2
,
.
.
.
,
n
{1,2,...,n}
1
,
2
,
...
,
n
which has this property : for all distinct
i
,
j
∈
T
i,j \in T
i
,
j
∈
T
,
2
j
2j
2
j
is not divisible by
i
i
i
. Prove that :
∣
T
∣
≤
4
9
n
+
log
2
n
+
2
|T| \leq \frac {4}{9}n + \log_2 n + 2
∣
T
∣
≤
9
4
n
+
lo
g
2
n
+
2
11
1
Hide problems
Iran TST 2009-Day4-P2
Let
n
n
n
be a positive integer. Prove that
3
5
2
n
−
1
2
n
+
2
≡
(
−
5
)
3
2
n
−
1
2
n
+
2
(
m
o
d
2
n
+
4
)
.
3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}.
3
2
n
+
2
5
2
n
−
1
≡
(
−
5
)
2
n
+
2
3
2
n
−
1
(
mod
2
n
+
4
)
.
10
1
Hide problems
Iran TST 2009-Day4-P1
Let
A
B
C
ABC
A
BC
be a triangle and
A
B
≠
A
C
AB\ne AC
A
B
=
A
C
.
D
D
D
is a point on
B
C
BC
BC
such that BA \equal{} BD and
B
B
B
is between
C
C
C
and
D
D
D
. Let
I
c
I_{c}
I
c
be center of the circle which touches
A
B
AB
A
B
and the extensions of
A
C
AC
A
C
and
B
C
BC
BC
.
C
I
c
CI_{c}
C
I
c
intersect the circumcircle of
A
B
C
ABC
A
BC
again at
T
T
T
. If \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4} then find
∠
A
\angle A
∠
A
7
1
Hide problems
Iran TST 2009-Day3-P1
Suppose three direction on the plane . We draw
11
11
11
lines in each direction . Find maximum number of the points on the plane which are on three lines .
9
1
Hide problems
Iran TST 2009-Day3-P3
In triangle
A
B
C
ABC
A
BC
,
D
D
D
,
E
E
E
and
F
F
F
are the points of tangency of incircle with the center of
I
I
I
to
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively. Let
M
M
M
be the foot of the perpendicular from
D
D
D
to
E
F
EF
EF
.
P
P
P
is on
D
M
DM
D
M
such that
D
P
=
M
P
DP = MP
D
P
=
MP
. If
H
H
H
is the orthocenter of
B
I
C
BIC
B
I
C
, prove that
P
H
PH
P
H
bisects
E
F
EF
EF
.
8
1
Hide problems
Iran TST 2009-Day3-P2
Find all polynomials
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
such that for all reals
x
x
x
and
y
y
y
,
P
(
x
2
,
y
2
)
=
P
(
(
x
+
y
)
2
2
,
(
x
−
y
)
2
2
)
.
P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).
P
(
x
2
,
y
2
)
=
P
(
2
(
x
+
y
)
2
,
2
(
x
−
y
)
2
)
.
6
1
Hide problems
Iran TST 2009-Day2-P3
We have a closed path on a vertices of a
n
n
n
×
n
n
n
square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .
5
1
Hide problems
Iran TST 2009-Day2-P2
A
B
C
ABC
A
BC
is a triangle and
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
are three altitudes of this triangle . Let
P
P
P
be the feet of perpendicular from
C
′
C'
C
′
to
A
′
B
′
A'B'
A
′
B
′
, and
Q
Q
Q
is a point on
A
′
B
′
A'B'
A
′
B
′
such that QA \equal{} QB . Prove that : \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C
4
1
Hide problems
Iran TST 2009-Day2-P1
Find all polynomials
f
f
f
with integer coefficient such that, for every prime
p
p
p
and natural numbers
u
u
u
and
v
v
v
with the condition:
p
∣
u
v
−
1
p \mid uv - 1
p
∣
uv
−
1
we always have
p
∣
f
(
u
)
f
(
v
)
−
1
p \mid f(u)f(v) - 1
p
∣
f
(
u
)
f
(
v
)
−
1
.
1
1
Hide problems
Iran TST 2009-Day1-P1
Let
A
B
C
ABC
A
BC
be a triangle and
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
lie on
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively such that the incenter of
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
and
A
B
C
ABC
A
BC
are coincide and the inradius of
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is half of inradius of
A
B
C
ABC
A
BC
. Prove that
A
B
C
ABC
A
BC
is equilateral .
3
1
Hide problems
Iran TST 2009-Day1-P3
Suppose that
a
a
a
,
b
b
b
,
c
c
c
be three positive real numbers such that a\plus{}b\plus{}c\equal{}3 . Prove that : \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}
2
1
Hide problems
Iran TST 2009-Day1-P2
Let
a
a
a
be a fix natural number . Prove that the set of prime divisors of 2^{2^{n}} \plus{} a for n \equal{} 1,2,\cdots is infinite