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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1994 India Regional Mathematical Olympiad
1994 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(8)
8
1
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An inequality
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
a
+
b
+
c
=
1
a+b+c = 1
a
+
b
+
c
=
1
, prove that
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
≥
8
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
.
(1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) .
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
≥
8
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
.
7
1
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Find all rationals s.t..
Find the number of rationals
m
n
\frac{m}{n}
n
m
such that (i)
0
<
m
n
<
1
0 < \frac{m}{n} < 1
0
<
n
m
<
1
;(ii)
m
m
m
and
n
n
n
are relatively prime;(iii)
m
n
=
25
!
mn = 25!
mn
=
25
!
.
6
1
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Prove a parallelogram
Let
A
C
AC
A
C
and
B
D
BD
B
D
be two chords of a circle with center
O
O
O
such that they intersect at right angles inside the circle at the point
M
M
M
. Suppose
K
K
K
and
L
L
L
are midpoints of the chords
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Prove that
O
K
M
L
OKML
O
K
M
L
is a parallelogram.
5
1
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Gcd
Let
A
A
A
be a set of
16
16
16
positive integers with the property that the product of any two distinct members of
A
A
A
will not exceed 1994. Show that there are numbers
a
a
a
and
b
b
b
in the set
A
A
A
such that the gcd of
a
a
a
and
b
b
b
is greater than 1.
4
1
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Solve the system
Solve the system of equations for real
x
x
x
and
y
y
y
: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
3
1
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Find all numbers
Find all 6-digit numbers
a
1
a
2
a
3
a
4
a
5
a
6
a_1a_2a_3a_4a_5a_6
a
1
a
2
a
3
a
4
a
5
a
6
formed by using the digits
1
,
2
,
3
,
4
,
5
,
6
1,2,3,4,5,6
1
,
2
,
3
,
4
,
5
,
6
once each such that the number
a
1
a
2
a
2
…
a
k
a_1a_2a_2\ldots a_k
a
1
a
2
a
2
…
a
k
is divisible by
k
k
k
for
1
≤
k
≤
6
1 \leq k \leq 6
1
≤
k
≤
6
.
2
1
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Find the sides
In a triangle
A
B
C
ABC
A
BC
, the incircle touches the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. If the radius if the incircle is
4
4
4
units and if
B
D
,
C
E
,
A
F
BD, CE , AF
B
D
,
CE
,
A
F
are consecutive integers, find the sides of the triangle
A
B
C
ABC
A
BC
.
1
1
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A leaf from a book
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is
15000
15000
15000
. What are the page numbers on the torn leaf?