MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2001 India Regional Mathematical Olympiad
2001 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Product of integers
Prove that the product of the first
1000
1000
1000
positive even integers differs from the product of the first
1000
1000
1000
positive odd integers by a multiple of
2001
2001
2001
.
6
1
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Ineq triangle
If
x
,
y
,
z
x,y,z
x
,
y
,
z
are sides of a triangle, prove that
∣
x
2
(
y
−
z
)
+
y
2
(
z
−
x
)
+
z
2
(
x
−
y
)
∣
<
x
y
z
.
| x^2(y-z) + y^2(z-x) + z^2(x-y) | < xyz.
∣
x
2
(
y
−
z
)
+
y
2
(
z
−
x
)
+
z
2
(
x
−
y
)
∣
<
x
yz
.
5
1
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Prove an angle
In a triangle
A
B
C
ABC
A
BC
,
D
D
D
is a point on
B
C
BC
BC
such that
A
D
AD
A
D
is the internal bisector of
∠
A
\angle A
∠
A
. Suppose
∠
B
=
2
∠
C
\angle B = 2 \angle C
∠
B
=
2∠
C
and
C
D
=
A
B
CD =AB
C
D
=
A
B
. prove that
∠
A
=
7
2
∘
\angle A = 72^{\circ}
∠
A
=
7
2
∘
.
4
1
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An array
Consider an
n
×
n
n \times n
n
×
n
array of numbers
a
i
j
a_{ij}
a
ij
(standard notation). Suppose each row consists of the
n
n
n
numbers
1
,
2
,
…
n
1,2,\ldots n
1
,
2
,
…
n
in some order and
a
i
j
=
a
j
i
a_{ij} = a_{ji}
a
ij
=
a
ji
for
i
,
j
=
1
,
2
,
…
n
i , j = 1,2, \ldots n
i
,
j
=
1
,
2
,
…
n
. If
n
n
n
is odd, prove that the numbers
a
11
,
a
22
,
…
a
n
n
a_{11}, a_{22} , \ldots a_{nn}
a
11
,
a
22
,
…
a
nn
are
1
,
2
,
3
,
…
n
1,2,3, \ldots n
1
,
2
,
3
,
…
n
in some order.
3
1
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A gint equation
Find the number of positive integers
x
x
x
such that
[
x
99
]
=
[
x
101
]
.
\left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] .
[
99
x
]
=
[
101
x
]
.
2
1
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Find all primes
Find all primes
p
p
p
and
q
q
q
such that
p
2
+
7
p
q
+
q
2
p^2 + 7pq + q^2
p
2
+
7
pq
+
q
2
is a perfect square.
1
1
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Another parallel
Let
B
E
BE
BE
and
C
F
CF
CF
be the altitudes of an acute triangle
A
B
C
ABC
A
BC
with
E
E
E
on
A
C
AC
A
C
and
F
F
F
on
A
B
AB
A
B
. Let
O
O
O
be the point of intersection of
B
E
BE
BE
and
C
F
CF
CF
. Take any line
K
L
KL
K
L
through
O
O
O
with
K
K
K
on
A
B
AB
A
B
and
L
L
L
on
A
C
AC
A
C
. Suppose
M
M
M
and
N
N
N
are located on
B
E
BE
BE
and
C
F
CF
CF
respectively. such that
K
M
KM
K
M
is perpendicular to
B
E
BE
BE
and
L
N
LN
L
N
is perpendicular to
C
F
CF
CF
. Prove that
F
M
FM
FM
is parallel to
E
N
EN
EN
.