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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1992 India Regional Mathematical Olympiad
1992 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(8)
8
1
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One on octagon
The cyclic octagon
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
has sides
a
,
a
,
a
,
a
,
b
,
b
,
b
,
b
a,a,a,a,b,b,b,b
a
,
a
,
a
,
a
,
b
,
b
,
b
,
b
respectively. Find the radius of the circle that circumscribes
A
B
C
D
E
F
G
H
.
ABCDEFGH.
A
BC
D
EFG
H
.
7
1
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Solve the system
Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter
A
A
A
.
6
1
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A good inequality
Prove that
1
<
1
1001
+
1
1002
+
1
1003
+
⋯
+
1
3001
<
1
1
3
.
1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}.
1
<
1001
1
+
1002
1
+
1003
1
+
⋯
+
3001
1
<
1
3
1
.
5
1
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Another simple one on quads
A
B
C
D
ABCD
A
BC
D
is a quadrilateral and
P
,
Q
P,Q
P
,
Q
are the midpoints of
C
D
,
A
B
,
A
P
,
D
Q
CD, AB, AP, DQ
C
D
,
A
B
,
A
P
,
D
Q
meet at
X
X
X
and
B
P
,
C
Q
BP, CQ
BP
,
CQ
meet at
Y
Y
Y
. Prove that
A
[
A
D
X
]
+
A
[
B
C
Y
]
=
A
[
P
X
O
Y
]
A[ADX]+A[BCY] = A[PXOY]
A
[
A
D
X
]
+
A
[
BC
Y
]
=
A
[
PXO
Y
]
.
4
1
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A simple one on cyclic quads
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral with
A
C
⊥
B
D
AC \perp BD
A
C
⊥
B
D
;
A
C
AC
A
C
meets
B
D
BD
B
D
at
E
E
E
. Prove that
E
A
2
+
E
B
2
+
E
C
2
+
E
D
2
=
4
R
2
EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2
E
A
2
+
E
B
2
+
E
C
2
+
E
D
2
=
4
R
2
where
R
R
R
is the radius of the circumscribing circle.
3
1
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Determine the largest prime s.t.
Determine the largest
3
3
3
digit prime number that is a factor of
(
2000
1000
)
{2000 \choose 1000}
(
1000
2000
)
.
2
1
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Yet another square
If
1
a
+
1
b
=
1
c
\frac{1}{a} + \frac{1}{b} = \frac{1}{c}
a
1
+
b
1
=
c
1
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integers with no common factor, prove that
(
a
+
b
)
(a +b)
(
a
+
b
)
is a square.
1
1
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Determine the set
Determine the set of integers
n
n
n
for which
n
2
+
19
n
+
92
n^2+19n+92
n
2
+
19
n
+
92
is a square.