MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1998 India Regional Mathematical Olympiad
1998 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
1
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A set
Given the
7
7
7
-element set
A
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
}
A = \{ a ,b,c,d,e,f,g \}
A
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
}
, find a collection
T
T
T
of
3
3
3
-element subsets of
A
A
A
such that each pair of elements from
A
A
A
occurs exactly once on one of the subsets of
T
T
T
.
5
1
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Find numbers whose sum
Find the minimum possible least common multiple of twenty natural numbers whose sum is
801
801
801
.
4
1
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An eq triangle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
B
A
C
=
3
0
∘
\angle BAC = 30^{\circ}
∠
B
A
C
=
3
0
∘
, Let
A
′
A'
A
′
be the reflection of
A
A
A
in the line
B
C
BC
BC
;
B
′
B'
B
′
be the reflection of
B
B
B
in the line
C
A
CA
C
A
;
C
′
C'
C
′
be the reflection of
C
C
C
in line
A
B
AB
A
B
, Show that
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is an equilateral triangle.
3
1
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An ineq - simple
Prove that for every natural number
n
>
1
n > 1
n
>
1
1
n
+
1
(
1
+
1
3
+
1
5
+
…
+
1
2
n
−
1
)
>
1
n
(
1
2
+
1
4
+
…
+
1
2
n
)
.
\frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) .
n
+
1
1
(
1
+
3
1
+
5
1
+
…
+
2
n
−
1
1
)
>
n
1
(
2
1
+
4
1
+
…
+
2
n
1
)
.
2
1
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Yet another divisibility
Let
n
n
n
be a positive integer and
p
1
,
p
2
,
p
3
,
…
p
n
p_1, p_2, p_3, \ldots p_n
p
1
,
p
2
,
p
3
,
…
p
n
be
n
n
n
prime numbers all larger than
5
5
5
such that
6
6
6
divides
p
1
2
+
p
2
2
+
p
3
2
+
⋯
p
n
2
p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2
p
1
2
+
p
2
2
+
p
3
2
+
⋯
p
n
2
. prove that
6
6
6
divides
n
n
n
.
1
1
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Find an angle
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral in which
∠
B
A
C
=
5
0
∘
,
∠
C
A
D
=
6
0
∘
\angle BAC = 50^{\circ}, \angle CAD = 60^{\circ}
∠
B
A
C
=
5
0
∘
,
∠
C
A
D
=
6
0
∘
and
∠
B
D
C
=
2
5
∘
\angle BDC = 25^{\circ}
∠
B
D
C
=
2
5
∘
. If
E
E
E
is the point of intersection of
A
C
AC
A
C
and
B
D
BD
B
D
, find
∠
A
E
B
\angle AEB
∠
A
EB
.