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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1994 Iran MO (2nd round)
1994 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
1
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There exists P such that P|2^m -1 [Iran Second Round 1994]
Let
n
>
3
n >3
n
>
3
be an odd positive integer and
n
=
∏
i
=
1
k
p
i
α
i
n=\prod_{i=1}^k p_i^{\alpha_i}
n
=
∏
i
=
1
k
p
i
α
i
where
p
i
p_i
p
i
are primes and
α
i
\alpha_i
α
i
are positive integers. We know that
m
=
n
(
1
−
1
p
1
)
(
1
−
1
p
2
)
(
1
−
1
p
3
)
⋯
(
1
−
1
p
n
)
.
m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).
m
=
n
(
1
−
p
1
1
)
(
1
−
p
2
1
)
(
1
−
p
3
1
)
⋯
(
1
−
p
n
1
)
.
Prove that there exists a prime
P
P
P
such that
P
∣
2
m
−
1
P|2^m -1
P
∣
2
m
−
1
but
P
∤
n
.
P \nmid n.
P
∤
n
.
1
1
Hide problems
Prove that there exists a function [Iran Second Round 1994]
Let
a
1
a
2
a
3
…
a
n
‾
\overline{a_1a_2a_3\ldots a_n}
a
1
a
2
a
3
…
a
n
be the representation of a
n
−
n-
n
−
digits number in base
10.
10.
10.
Prove that there exists a one-to-one function like
f
:
{
0
,
1
,
2
,
3
,
…
,
9
}
→
{
0
,
1
,
2
,
3
,
…
,
9
}
f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}
f
:
{
0
,
1
,
2
,
3
,
…
,
9
}
→
{
0
,
1
,
2
,
3
,
…
,
9
}
such that
f
(
a
1
)
≠
0
f(a_1) \neq 0
f
(
a
1
)
=
0
and the number
f
(
a
1
)
f
(
a
2
)
f
(
a
3
)
…
f
(
a
n
)
‾
\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }
f
(
a
1
)
f
(
a
2
)
f
(
a
3
)
…
f
(
a
n
)
is divisible by
3.
3.
3.
2
2
Hide problems
Find the angle alpha [Iran Second Round 1994]
In the following diagram,
O
O
O
is the center of the circle. If three angles
α
,
β
\alpha, \beta
α
,
β
and
γ
\gamma
γ
be equal, find
α
.
\alpha.
α
.
[asy] unitsize(40); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0); draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){ return (0.6*cos(t)+0.64,0.6*sin(t)+2.24); } draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq); dot((0,0),ds); label("
O
O
O
", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("
A
A
A
", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("
B
B
B
", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("
C
C
C
", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("
E
E
E
", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("
D
D
D
", (-0.92,0.7), NE*lsf); label("
α
\alpha
α
", (0.48,1.38),NE*lsf); label("
β
\beta
β
", (-0.02,0.94),NE*lsf); label("
γ
\gamma
γ
", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle); [/asy]
Prove that the points are collinear [Iran Second Round 1994]
The incircle of triangle
A
B
C
ABC
A
BC
meet the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
and
B
C
BC
BC
in
M
,
N
M,N
M
,
N
and
P
P
P
, respectively. Prove that the orthocenter of triangle
M
N
P
,
MNP,
MNP
,
the incenter and the circumcenter of triangle
A
B
C
ABC
A
BC
are collinear.[asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1); draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6)); dot((8,17.58),ds); label("
A
A
A
", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("
B
B
B
", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("
C
C
C
", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("
O
O
O
", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("
M
M
M
", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("
N
N
N
", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("
P
P
P
", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("
D
D
D
", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("
F
F
F
", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("
G
G
G
", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("
H
H
H
", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle); [/asy]