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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1997 India Regional Mathematical Olympiad
1997 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
1
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Find the number of unordered pairs
Find the number of unordered pairs
{
A
,
B
}
\{ A,B \}
{
A
,
B
}
of subsets of an n-element set
X
X
X
that satisfies the following: (a)
A
≠
B
A \not= B
A
=
B
(b)
A
∪
B
=
X
A \cup B = X
A
∪
B
=
X
5
1
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These are lengths of sides
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be three distinct real positive numbers, Determine whether or not the three real numbers
∣
x
y
−
y
x
∣
,
∣
y
z
−
z
y
∣
,
∣
z
x
−
x
z
∣
\left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right|
y
x
−
x
y
,
z
y
−
y
z
,
x
z
−
z
x
can be the lengths of the sides of a triangle.
4
1
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Quad ineq
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
, it is given that
A
B
AB
A
B
is parallel to
C
D
CD
C
D
and the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular to each other. Show that (a)
A
D
⋅
B
C
≥
A
B
⋅
C
D
AD \cdot BC \geq AB \cdot CD
A
D
⋅
BC
≥
A
B
⋅
C
D
(b)
A
D
+
B
C
≥
A
B
+
C
D
.
AD + BC \geq AB + CD.
A
D
+
BC
≥
A
B
+
C
D
.
3
1
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Gint
Solve for real
x
x
x
:
1
[
x
]
+
1
[
2
x
]
=
x
−
[
x
]
+
1
3
.
\frac{1}{[x]} + \frac{1}{[2x]} = x - [x] + \frac{1}{3}.
[
x
]
1
+
[
2
x
]
1
=
x
−
[
x
]
+
3
1
.
2
1
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A sequence
For each positive integer
n
n
n
, define
a
n
=
20
+
n
2
a_n = 20 + n^2
a
n
=
20
+
n
2
and
d
n
=
g
c
d
(
a
n
,
a
n
+
1
)
d_n = gcd(a_n, a_{n+1})
d
n
=
g
c
d
(
a
n
,
a
n
+
1
)
. Find the set of all values that are taken by
d
n
d_n
d
n
and show by examples that each of these values is attained.
1
1
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Areas
Let
P
P
P
be an interior point of a triangle
A
B
C
ABC
A
BC
and let
B
P
BP
BP
and
C
P
CP
CP
meet
A
C
AC
A
C
and
A
B
AB
A
B
in
E
E
E
and
F
F
F
respectively. IF
S
B
P
F
=
4
S_{BPF} = 4
S
BPF
=
4
,
S
B
P
C
=
8
S_{BPC} = 8
S
BPC
=
8
and
S
C
P
E
=
13
S_{CPE} = 13
S
CPE
=
13
, find
S
A
F
P
E
.
S_{AFPE}.
S
A
FPE
.