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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1996 India Regional Mathematical Olympiad
1996 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Show that a contains a square
If
A
A
A
is a fifty element subset of the set
1
,
2
,
…
100
1,2,\ldots 100
1
,
2
,
…
100
such that no two numbers from
A
A
A
add up to
100
100
100
, show that
A
A
A
contains a square.
6
1
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Show that there exist..
Given any positive integer
n
n
n
, show that there are two positive rational numbers
a
a
a
and
b
b
b
,
a
≠
b
a \not= b
a
=
b
, which are not integers and which are such that
a
−
b
,
a
2
−
b
2
,
…
a
n
−
b
n
a - b, a^2 - b^2 , \ldots a^n - b^n
a
−
b
,
a
2
−
b
2
,
…
a
n
−
b
n
are all integers.
5
1
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A triangle ineq
Let
A
B
C
ABC
A
BC
be a triangle and
h
a
h_a
h
a
be the altitude through
A
A
A
. Prove that
(
b
+
c
)
2
≥
a
2
+
4
h
a
2
.
(b+c)^2 \geq a^2 + 4h_a ^2 .
(
b
+
c
)
2
≥
a
2
+
4
h
a
2
.
4
1
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Find numbers
Suppose
N
N
N
is an
n
n
n
digit positive integer such that (a) all its digits are distinct; (b) the sum of any three consecutive digits is divisible by
5
5
5
. Prove that
n
≤
6
n \leq 6
n
≤
6
. Further, show that starting with any digit, one can find a six digit number with these properties.
3
1
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Solve for real x and y
Solve for real numbers
x
x
x
and
y
y
y
, \begin{eqnarray*} \\ xy^2 &=& 15x^2 + 17xy +15y^2 ; \\ \\ x^2y &=& 20x^2 + 3y^2. \end{eqnarray*}
2
1
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Find all triples
Find all triples
a
,
b
,
c
a,b,c
a
,
b
,
c
of positive integers such that
(
1
+
1
a
)
(
1
+
1
b
)
(
1
+
1
c
)
=
3.
( 1 + \frac{1}{a} ) ( 1 + \frac{1}{b}) ( 1 + \frac{1}{c} ) = 3.
(
1
+
a
1
)
(
1
+
b
1
)
(
1
+
c
1
)
=
3.
1
1
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Find the circumradius
The sides of a triangle are three consecutive integers and its inradius is
4
4
4
. Find the circumradius.