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National and Regional Contests
India Contests
India National Olympiad
1986 India National Olympiad
1986 India National Olympiad
Part of
India National Olympiad
Subcontests
(9)
9
1
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Indian Mathematical Olympiad 1986 - Problem 9
Show that among all quadrilaterals of a given perimeter the square has the largest area.
8
1
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Indian Mathematical Olympiad 1986 - Problem 8
Suppose
A
1
,
…
,
A
6
A_1,\dots, A_6
A
1
,
…
,
A
6
are six sets each with four elements and
B
1
,
…
,
B
n
B_1,\dots,B_n
B
1
,
…
,
B
n
are
n
n
n
sets each with two elements, Let S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n. Given that each elements of
S
S
S
belogs to exactly four of the
A
A
A
's and to exactly three of the
B
B
B
's, find
n
n
n
.
7
1
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Indian Mathematical Olympiad 1986 - Problem 7
If
a
a
a
,
b
b
b
,
x
x
x
,
y
y
y
are integers greater than 1 such that
a
a
a
and
b
b
b
have no common factor except 1 and x^a \equal{} y^b show that x \equal{} n^b, y \equal{} n^a for some integer
n
n
n
greater than 1.
6
1
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Indian Mathematical Olympiad 1986 - Problem 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
5
1
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Indian Mathematical Olympiad 1986 - Problem 5
If
P
(
x
)
P(x)
P
(
x
)
is a polynomial with integer coefficients and
a
a
a
,
b
b
b
,
c
c
c
, three distinct integers, then show that it is impossible to have P(a)\equal{}b, P(b)\equal{}c, P(c)\equal{}a.
4
1
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Indian Mathematical Olympiad 1986 - Problem 4
Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.
3
1
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Indian Mathematical Olympiad 1986 - Problem 3
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}
2
1
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Indian Mathematical Olympiad 1986 - Problem 2
Solve \left\{ \begin{array}{l} \log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\ \log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\ \log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\ \end{array} \right.
1
1
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Indian Mathematical Olympiad 1986 - Problem 1
A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?