MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1986 Iran MO (2nd round)
1986 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(4)
5
1
Hide problems
Dividing - [Iran Second Round 1986]
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.]
3
2
Hide problems
Smallest positive integer - [Iran Second Round 1986]
Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be
3
2
\frac 32
2
3
times of the original number.
The arctangent function - [Iran Second Round 1986]
Prove that
arctan
1
2
+
arctan
1
3
=
π
4
.
\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.
arctan
2
1
+
arctan
3
1
=
4
π
.
2
2
Hide problems
Functions f, g - [Iran Second Round 1986]
(a) Sketch the diagram of the function
f
f
f
if f(x)=4x(1-|x|) , |x| \leq 1.(b) Does there exist derivative of
f
f
f
in the point
x
=
0
?
x=0 \ ?
x
=
0
?
(c) Let
g
g
g
be a function such that
g
(
x
)
=
{
f
(
x
)
x
e
m
s
p
;
:
x
≠
0
4
e
m
s
p
;
:
x
=
0
g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x}   : x \neq 0\\ \text{ } \\ 4 \ \ \ \   : x=0\end{array}\right.
g
(
x
)
=
⎩
⎨
⎧
x
f
(
x
)
4
e
m
s
p
;
:
x
=
0
e
m
s
p
;
:
x
=
0
Is the function
g
g
g
continuous in the point
x
=
0
?
x=0 \ ?
x
=
0
?
(d) Sketch the diagram of
g
.
g.
g
.
Find the area in terms of a, b - [Iran Second Round 1986]
In a trapezoid
A
B
C
D
ABCD
A
BC
D
, the legs
A
B
AB
A
B
and
C
D
CD
C
D
meet in
M
M
M
and the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
meet in
N
.
N.
N
.
Let
A
C
=
a
AC=a
A
C
=
a
and
B
C
=
b
.
BC=b.
BC
=
b
.
Find the area of triangles
A
M
D
AMD
A
M
D
and
A
N
D
AND
A
N
D
in terms of
a
a
a
and
b
.
b.
b
.
1
2
Hide problems
Find the limit of the function - [Iran Second Round 1986]
Let
f
f
f
be a function such that
f
(
x
)
=
(
x
2
−
2
x
+
1
)
sin
1
x
−
1
sin
π
x
.
f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.
f
(
x
)
=
sin
π
x
(
x
2
−
2
x
+
1
)
sin
x
−
1
1
.
Find the limit of
f
f
f
in the point
x
0
=
1.
x_0=1.
x
0
=
1.
Midpoint is a Fixed Point - [Iran Second Round 1986]
O
O
O
is a point in the plane. Let
O
′
O'
O
′
be an arbitrary point on the axis
O
x
Ox
O
x
of the plane and let
M
M
M
be an arbitrary point. Rotate
M
M
M
,
9
0
∘
90^\circ
9
0
∘
clockwise around
O
O
O
to get the point
M
′
M'
M
′
and rotate
M
M
M
,
9
0
∘
90^\circ
9
0
∘
anticlockwise around
O
′
O'
O
′
to get the point
M
′
′
.
M''.
M
′′
.
Prove that the midpoint of the segment
M
M
′
′
MM''
M
M
′′
is a fixed point.