MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1984 Iran MO (2nd round)
1984 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(8)
8
1
Hide problems
Operation is associative - [Iran Second Round 1984]
Define the operation
⨁
\bigoplus
⨁
on the set of real numbers such that
x
⨁
y
=
x
+
y
−
x
y
∀
x
,
y
∈
R
.
x \bigoplus y = x+y-xy \qquad \forall x,y \in \mathbb R.
x
⨁
y
=
x
+
y
−
x
y
∀
x
,
y
∈
R
.
Prove that this operation is associative.
7
1
Hide problems
Find the locus of the points M - [Iran Second Round 1984]
Let
B
B
B
and
C
C
C
be two fixed point on the plane
P
.
P.
P
.
Find the locus of the points
M
M
M
on the plane
P
P
P
for which
M
B
2
+
k
M
C
2
=
a
2
.
MB^2 + kMC^2 = a^2.
M
B
2
+
k
M
C
2
=
a
2
.
(
k
k
k
and
a
a
a
are two given numbers and
k
>
0.
k>0.
k
>
0.
)
6
1
Hide problems
Length of common perpendicular - [Iran Second Round 1984]
Let
D
D
D
and
D
′
D'
D
′
be two lines with the equations \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \text{and} \frac{x+1}{2} = \frac{y+2}{4} = \frac{z-1}{3}. Find the length of their common perpendicular.
5
1
Hide problems
Find limit of S_n - [Iran Second Round 1984]
Suppose that
S
n
=
5
9
×
14
20
×
27
35
×
⋯
×
2
n
2
−
n
−
1
2
n
2
+
n
−
1
S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}
S
n
=
9
5
×
20
14
×
35
27
×
⋯
×
2
n
2
+
n
−
1
2
n
2
−
n
−
1
Find
lim
n
→
∞
S
n
.
\lim_{n \to \infty} S_n.
lim
n
→
∞
S
n
.
4
1
Hide problems
Number of terms after expanding - [Iran Second Round 1984]
Find number of terms when we expand
(
a
+
b
+
c
)
99
(a+b+c)^{99}
(
a
+
b
+
c
)
99
(in the general case).
3
1
Hide problems
The function has derivative - [Iran Second Round 1984]
Let
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
be a function such that
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
∀
x
,
y
∈
R
f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
∀
x
,
y
∈
R
Suppose that
f
(
0
)
≠
0
f(0) \neq 0
f
(
0
)
=
0
and
f
(
0
)
f(0)
f
(
0
)
exists and it is finite
(
f
(
0
)
≠
∞
)
(f(0) \neq \infty)
(
f
(
0
)
=
∞
)
. Prove that
f
f
f
has derivative in each point
x
∈
R
.
x \in \mathbb R.
x
∈
R
.
2
1
Hide problems
Find the period - [Iran Second Round 1984]
Consider the function
f
(
x
)
=
sin
(
π
2
⌊
x
⌋
)
.
f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).
f
(
x
)
=
sin
(
2
π
⌊
x
⌋
)
.
Find the period of
f
f
f
and sketch diagram of
f
f
f
in one period. Also prove that
lim
x
→
1
f
(
x
)
\lim_{x \to 1} f(x)
lim
x
→
1
f
(
x
)
does not exist.
1
1
Hide problems
Prove that limits are equal - [Iran Second Round 1984]
Let
f
f
f
and
g
g
g
be two functions such that f(x)=\frac{1}{\lfloor | x | \rfloor}, g(x)=\frac{1}{|\lfloor x \rfloor |}. Find the domains of
f
f
f
and
g
g
g
and then prove that
lim
x
→
−
1
+
f
(
x
)
=
lim
x
→
1
−
g
(
x
)
.
\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).
x
→
−
1
+
lim
f
(
x
)
=
x
→
1
−
lim
g
(
x
)
.