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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1987 Iran MO (2nd round)
1987 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
Hide problems
Prove that 5 S{A'B'C'D'} =S{ABCD} - [Iran Second Round 1987]
In the following diagram, let
A
B
C
D
ABCD
A
BC
D
be a square and let
M
,
N
,
P
M,N,P
M
,
N
,
P
and
Q
Q
Q
be the midpoints of its sides. Prove that
S
A
′
B
′
C
′
D
′
=
1
5
S
A
B
C
D
.
S_{A'B'C'D'} = \frac 15 S_{ABCD}.
S
A
′
B
′
C
′
D
′
=
5
1
S
A
BC
D
.
[asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("
A
A
A
", (0.07,4.12), NE*lsf); dot((0,0),ds); label("
D
D
D
", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("
C
C
C
", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("
B
B
B
", (4.08,4.12), NE*lsf); dot((2,4),ds); label("
M
M
M
", (2.08,4.12), NE*lsf); dot((4,2),ds); label("
N
N
N
", (4.2,1.98), NE*lsf); dot((2,0),ds); label("
P
P
P
", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("
Q
Q
Q
", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("
A
′
A'
A
′
", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("
B
′
B'
B
′
", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("
C
′
C'
C
′
", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("
D
′
D'
D
′
", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy][
S
X
S_{X}
S
X
denotes area of the
X
.
X.
X
.
]
Minimum and maximum number of lines-[Iran Second Round 1987]
Let
L
1
,
L
2
,
L
3
,
L
4
L_1, L_2, L_3, L_4
L
1
,
L
2
,
L
3
,
L
4
be four lines in the space such that no three of them are in the same plane. Let
L
1
,
L
2
L_1, L_2
L
1
,
L
2
intersect in
A
A
A
,
L
2
,
L
3
L_2,L_3
L
2
,
L
3
intersect in
B
B
B
and
L
3
,
L
4
L_3, L_4
L
3
,
L
4
intersect in
C
.
C.
C
.
Find minimum and maximum number of lines in the space that intersect
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
and
L
4
.
L_4.
L
4
.
Justify your answer.
2
2
Hide problems
The function g is bounded - [Iran Second Round 1987]
Let
f
f
f
be a real function defined in the interval
[
0
,
+
∞
)
[0, +\infty )
[
0
,
+
∞
)
and suppose that there exist two functions
f
′
,
f
′
′
f', f''
f
′
,
f
′′
in the interval
[
0
,
+
∞
)
[0, +\infty )
[
0
,
+
∞
)
such that
f
′
′
(
x
)
=
1
x
2
+
f
′
(
x
)
2
+
1
and
f
(
0
)
=
f
′
(
0
)
=
0.
f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.
f
′′
(
x
)
=
x
2
+
f
′
(
x
)
2
+
1
1
and
f
(
0
)
=
f
′
(
0
)
=
0.
Let
g
g
g
be a function for which
g
(
0
)
=
0
and
g
(
x
)
=
f
(
x
)
x
.
g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.
g
(
0
)
=
0
and
g
(
x
)
=
x
f
(
x
)
.
Prove that
g
g
g
is bounded.
Find all functions - [Iran Second Round 1987]
Find all continuous functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
such that f(x^2-y^2)=f(x)^2 + f(y)^2, \forall x,y \in \mathbb R.
1
2
Hide problems
System of Equations - [Iran Second Round 1987]
Solve the following system of equations in positive integers
{
a
3
−
b
3
−
c
3
=
3
a
b
c
a
2
=
2
(
b
+
c
)
\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.
⎩
⎨
⎧
a
3
−
b
3
−
c
3
=
3
ab
c
a
2
=
2
(
b
+
c
)
Calculate the product of sines - [Iran Second Round 1987]
Calculate the product:
A
=
sin
1
∘
×
sin
2
∘
×
sin
3
∘
×
⋯
×
sin
8
9
∘
A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ
A
=
sin
1
∘
×
sin
2
∘
×
sin
3
∘
×
⋯
×
sin
8
9
∘