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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
1997 Iran MO (3rd Round)
1997 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(6)
5
1
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D, E, I are collinear <=> P, Q, O are collinear
In an acute triangle
A
B
C
ABC
A
BC
let
A
D
AD
A
D
and
B
E
BE
BE
be altitudes, and
A
P
AP
A
P
and
B
Q
BQ
BQ
be bisectors. Let
I
I
I
and
O
O
O
be centers of incircle and circumcircle, respectively. Prove that the points
D
,
E
D, E
D
,
E
, and
I
I
I
are collinear if and only if the points
P
,
Q
P, Q
P
,
Q
, and
O
O
O
are collinear.
1
3
Hide problems
All functions f such that f(1-x)=1-f(f(x))
Find all strictly ascending functions
f
f
f
such that for all
x
∈
R
x\in \mathbb R
x
∈
R
,
f
(
1
−
x
)
=
1
−
f
(
f
(
x
)
)
.
f(1-x)=1-f(f(x)).
f
(
1
−
x
)
=
1
−
f
(
f
(
x
))
.
P(a)P(b)=-(a-b)^2 => Prove that P(a)+P(b)=0
Let
P
P
P
be a polynomial with integer coefficients. There exist integers
a
a
a
and
b
b
b
such that
P
(
a
)
⋅
P
(
b
)
=
−
(
a
−
b
)
2
P(a) \cdot P(b)=-(a-b)^2
P
(
a
)
⋅
P
(
b
)
=
−
(
a
−
b
)
2
. Prove that
P
(
a
)
+
P
(
b
)
=
0
P(a)+P(b)=0
P
(
a
)
+
P
(
b
)
=
0
.
Prove that a=x and b=x^x - [Iran Third Round 1998]
Suppose that
a
,
b
,
x
a, b, x
a
,
b
,
x
are positive integers such that
x
a
+
b
=
a
b
b
x^{a+b}=a^bb
x
a
+
b
=
a
b
b
Prove that
a
=
x
a=x
a
=
x
and
b
=
x
x
b=x^x
b
=
x
x
.
4
1
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\sqrt {x-1}+...
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers greater than
1
1
1
such that
1
x
+
1
y
+
1
z
=
2
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2
x
1
+
y
1
+
z
1
=
2
. Prove that
x
−
1
+
y
−
1
+
z
−
1
≤
x
+
y
+
z
.
\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.
x
−
1
+
y
−
1
+
z
−
1
≤
x
+
y
+
z
.
3
3
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Coloring the points with rational coordinates
Let
d
d
d
be a real number such that
d
2
=
r
2
+
s
2
d^2=r^2+s^2
d
2
=
r
2
+
s
2
, where
r
r
r
and
s
s
s
are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance
d
d
d
have different colors.
An interesting problem about weighing coins
There are
30
30
30
bags and there are
100
100
100
similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most
999
999
999
grams in each weighing. Using this scale, we want to find the weight of coins of each bag.(a) Show that this operation is possible by
10
10
10
times of weighing, and(b) It's not possible by
9
9
9
times of weighing.
Rational
Let
S
=
{
x
0
,
x
1
,
…
,
x
n
}
S = \{x_0, x_1,\dots , x_n\}
S
=
{
x
0
,
x
1
,
…
,
x
n
}
be a finite set of numbers in the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
with
x
0
=
0
x_0 = 0
x
0
=
0
and
x
1
=
1
x_1 = 1
x
1
=
1
. We consider pairwise distances between numbers in
S
S
S
. If every distance that appears, except the distance
1
1
1
, occurs at least twice, prove that all the
x
i
x_i
x
i
are rational.
6
1
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Coordinate
Let
P
\mathcal P
P
be the set of all points in
R
n
\mathbb R^n
R
n
with rational coordinates. For the points
A
,
B
∈
l
P
A,B \in \mathcal l{P}
A
,
B
∈
l
P
, one can move from
A
A
A
to
B
B
B
if the distance
A
B
AB
A
B
is
1
1
1
. Prove that every point in
l
P
\mathcal l{ P}
l
P
can be reached from any other point in
P
\mathcal{P}
P
by a finite sequence of moves if and only if
n
≥
5
n \geq 5
n
≥
5
.
2
3
Hide problems
Nice Geometric inequality on angles sines and side lengths
Show that for any arbitrary triangle
A
B
C
ABC
A
BC
, we have
sin
(
A
2
)
⋅
sin
(
B
2
)
⋅
sin
(
C
2
)
≤
a
b
c
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
.
\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.
sin
(
2
A
)
⋅
sin
(
2
B
)
⋅
sin
(
2
C
)
≤
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
ab
c
.
two traingles and 6 points of intersection
Let
A
B
C
ABC
A
BC
and
X
Y
Z
XYZ
X
Y
Z
be two triangles. Define
A
1
=
B
C
∩
Z
X
,
A
2
=
B
C
∩
X
Y
,
A_1=BC\cap ZX, A_2=BC\cap XY,
A
1
=
BC
∩
ZX
,
A
2
=
BC
∩
X
Y
,
B
1
=
C
A
∩
X
Y
,
B
2
=
C
A
∩
Y
Z
,
B_1=CA\cap XY, B_2=CA\cap YZ,
B
1
=
C
A
∩
X
Y
,
B
2
=
C
A
∩
Y
Z
,
C
1
=
A
B
∩
Y
Z
,
C
2
=
A
B
∩
Z
X
.
C_1=AB\cap YZ, C_2=AB\cap ZX.
C
1
=
A
B
∩
Y
Z
,
C
2
=
A
B
∩
ZX
.
Hereby, the abbreviation
g
∩
h
g\cap h
g
∩
h
means the point of intersection of two lines
g
g
g
and
h
h
h
.Prove that
C
1
C
2
A
B
=
A
1
A
2
B
C
=
B
1
B
2
C
A
\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}
A
B
C
1
C
2
=
BC
A
1
A
2
=
C
A
B
1
B
2
holds if and only if
A
1
C
2
X
Z
=
C
1
B
2
Z
Y
=
B
1
A
2
Y
X
\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}
XZ
A
1
C
2
=
Z
Y
C
1
B
2
=
Y
X
B
1
A
2
.
Circle pass through mid of bc
In an acute triangle
A
B
C
ABC
A
BC
, points
D
,
E
,
F
D,E,F
D
,
E
,
F
are the feet of the altitudes from
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively. A line through
D
D
D
parallel to
E
F
EF
EF
meets
A
C
AC
A
C
at
Q
Q
Q
and
A
B
AB
A
B
at
R
R
R
. Lines
B
C
BC
BC
and
E
F
EF
EF
intersect at
P
P
P
. Prove that the circumcircle of triangle
P
Q
R
PQR
PQR
passes through the midpoint of
B
C
BC
BC
.