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Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1983 Iran MO (2nd round)
1983 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(7)
7
1
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Find the sum - [Iran Second Round 1983]
Find the sum
∑
i
=
1
∞
n
2
n
.
\sum_{i=1}^{\infty} \frac{n}{2^n}.
∑
i
=
1
∞
2
n
n
.
6
1
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Limit of the function - [Iran Second Round 1983]
Suppose that f(x)=\{\begin{array}{cc}n,& \qquad n \in \mathbb N , x= \frac 1n\\ \text{} \\x, & \mbox{otherwise}\end{array}i) In which points, the function has a limit?ii) Prove that there does not exist limit of
f
f
f
in the point
x
=
0.
x=0.
x
=
0.
5
1
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Sum of arctangents - [Iran Second Round 1983]
Find the value of
S
n
=
arctan
1
2
+
arctan
1
8
+
arctan
1
18
+
⋯
+
arctan
1
2
n
2
.
S_n= \arctan \frac 12 + \arctan \frac 18+ \arctan \frac {1}{18} + \cdots + \arctan \frac {1}{2n^2}.
S
n
=
arctan
2
1
+
arctan
8
1
+
arctan
18
1
+
⋯
+
arctan
2
n
2
1
.
Also find
lim
n
→
∞
S
n
.
\lim_{n \to \infty} S_n.
lim
n
→
∞
S
n
.
4
1
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Sum of squares of lengths is fixed -[Iran Second Round 1983]
The point
M
M
M
moves such that the sum of squares of the lengths from
M
M
M
to faces of a cube, is fixed. Find the locus of
M
.
M.
M
.
3
1
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Find the matrix A - [Iran Second Round 1983]
Find a matrix
A
(
2
×
2
)
A_{(2 \times 2)}
A
(
2
×
2
)
for which
[
2
1
3
2
]
A
[
3
2
4
3
]
=
[
1
2
2
1
]
.
\begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.
[
2
3
1
2
]
A
[
3
4
2
3
]
=
[
1
2
2
1
]
.
2
1
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The number is irrational - [Iran Second Round 1983]
Prove that the number
x
=
1
+
2
x = \sqrt{1 + \sqrt 2}
x
=
1
+
2
is irrational.
1
1
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Prove that f is bijective - [Iran Second Round 1983]
Let
f
,
g
:
R
→
R
f, g : \mathbb R \to \mathbb R
f
,
g
:
R
→
R
be two functions such that
g
∘
f
:
R
→
R
g\circ f : \mathbb R \to \mathbb R
g
∘
f
:
R
→
R
is an injective function. Prove that
f
f
f
is also injective.