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Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1989 Iran MO (2nd round)
1989 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
Hide problems
Prove that the limit is 2 - [Iran Second Round 1989]
Let
{
a
n
}
n
≥
1
\{a_n\}_{n \geq 1}
{
a
n
}
n
≥
1
be a sequence in which
a
1
=
1
a_1=1
a
1
=
1
and
a
2
=
2
a_2=2
a
2
=
2
and
a
n
+
1
=
1
+
a
1
a
2
a
3
⋯
a
n
−
1
+
(
a
1
a
2
a
3
⋯
a
n
−
1
)
2
∀
n
≥
2.
a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.
a
n
+
1
=
1
+
a
1
a
2
a
3
⋯
a
n
−
1
+
(
a
1
a
2
a
3
⋯
a
n
−
1
)
2
∀
n
≥
2.
Prove that
lim
n
→
∞
(
1
a
1
+
1
a
2
+
1
a
3
+
⋯
+
1
a
n
)
=
2
\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2
n
→
∞
lim
(
a
1
1
+
a
2
1
+
a
3
1
+
⋯
+
a
n
1
)
=
2
Faithful Lines - [Iran Second Round 1989]
A line
d
d
d
is called faithful to triangle
A
B
C
ABC
A
BC
if
d
d
d
be in plane of triangle
A
B
C
ABC
A
BC
and the reflections of
d
d
d
over the sides of
A
B
C
ABC
A
BC
be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one faithful line to both of them, or there exist infinitely faithful lines to them.
1
2
Hide problems
Floor Problem and the equation - Iran 1989 Second Round
(a) Let
n
n
n
be a positive integer, prove that
n
+
1
−
n
<
1
2
n
\sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}
n
+
1
−
n
<
2
n
1
(b) Find a positive integer
n
n
n
for which
⌊
1
+
1
2
+
1
3
+
1
4
+
⋯
+
1
n
⌋
=
12
\bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12
⌊
1
+
2
1
+
3
1
+
4
1
+
⋯
+
n
1
⌋
=
12
The old problem of tournament - [Iran Second Round 1989]
In a sport competition,
m
m
m
teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team
x
x
x
such that for each team
y
,
y,
y
,
either
x
x
x
wins
y
y
y
or there exists a team
z
z
z
for which
x
x
x
wins
z
z
z
and
z
z
z
wins
y
.
y.
y
.
[i.e. prove that in every tournament there exists a king.]
2
2
Hide problems
Plane passes through a fixed point [Iran Second Round 1989]
A sphere
S
S
S
with center
O
O
O
and radius
R
R
R
is given. Let
P
P
P
be a fixed point on this sphere. Points
A
,
B
,
C
A,B,C
A
,
B
,
C
move on the sphere
S
S
S
such that we have
∠
A
P
B
=
∠
B
P
C
=
∠
C
P
A
=
9
0
∘
.
\angle APB = \angle BPC = \angle CPA = 90^\circ.
∠
A
PB
=
∠
BPC
=
∠
CP
A
=
9
0
∘
.
Prove that the plane of triangle
A
B
C
ABC
A
BC
passes through a fixed point.
Root of a polynomial
Let
n
n
n
be a positive integer. Prove that the polynomial
P
(
x
)
=
x
n
n
!
+
x
n
−
1
(
n
−
1
)
!
+
.
.
.
+
x
+
1
P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1
P
(
x
)
=
n
!
x
n
+
(
n
−
1
)!
x
n
−
1
+
...
+
x
+
1
Does not have any rational root.