MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2005 Iran Team Selection Test
2005 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(3)
3
2
Hide problems
Persian Gulf
Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.
function
Suppose
S
=
{
1
,
2
,
…
,
n
}
S= \{1,2,\dots,n\}
S
=
{
1
,
2
,
…
,
n
}
and
n
≥
3
n \geq 3
n
≥
3
. There is
f
:
S
k
⟼
S
f:S^k \longmapsto S
f
:
S
k
⟼
S
that if
a
,
b
∈
S
k
a,b \in S^k
a
,
b
∈
S
k
and
a
a
a
and
b
b
b
differ in all of elements then
f
(
a
)
≠
f
(
b
)
f(a) \neq f(b)
f
(
a
)
=
f
(
b
)
. Prove that
f
f
f
is a function of one of its elements.
2
2
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PT is perpendicular to XY
Assume
A
B
C
ABC
A
BC
is an isosceles triangle that
A
B
=
A
C
AB=AC
A
B
=
A
C
Suppose
P
P
P
is a point on extension of side
B
C
BC
BC
.
X
X
X
and
Y
Y
Y
are points on
A
B
AB
A
B
and
A
C
AC
A
C
that:
P
X
∣
∣
A
C
,
P
Y
∣
∣
A
B
PX || AC \ , \ PY ||AB
PX
∣∣
A
C
,
P
Y
∣∣
A
B
Also
T
T
T
is midpoint of arc
B
C
BC
BC
. Prove that
P
T
⊥
X
Y
PT \perp XY
PT
⊥
X
Y
O is constant
Suppose there are
n
n
n
distinct points on plane. There is circle with radius
r
r
r
and center
O
O
O
on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move
O
O
O
to the baricenter of the point in the circle. Prove that location of
O
O
O
is constant after some steps.
1
2
Hide problems
a_1,...a_n
Suppose that
a
1
a_1
a
1
,
a
2
a_2
a
2
, ...,
a
n
a_n
a
n
are positive real numbers such that
a
1
≤
a
2
≤
⋯
≤
a
n
a_1 \leq a_2 \leq \dots \leq a_n
a
1
≤
a
2
≤
⋯
≤
a
n
. Let {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. Suppose that, for some
i
i
i
, we know
a
i
≤
m
a_i \leq m
a
i
≤
m
. Prove that: n \minus{} i \geq n \left(m \minus{} a_i\right)^2
f:N -->N
Find all
f
:
N
⟼
N
f : N \longmapsto N
f
:
N
⟼
N
that there exist
k
∈
N
k \in N
k
∈
N
and a prime
p
p
p
that:
∀
n
≥
k
f
(
n
+
p
)
=
f
(
n
)
\forall n \geq k \ f(n+p)=f(n)
∀
n
≥
k
f
(
n
+
p
)
=
f
(
n
)
and also if
m
∣
n
m \mid n
m
∣
n
then
f
(
m
+
1
)
∣
f
(
n
)
+
1
f(m+1) \mid f(n)+1
f
(
m
+
1
)
∣
f
(
n
)
+
1