MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2000 India Regional Mathematical Olympiad
2000 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Equation roots
Find all real values of
a
a
a
such that
x
4
−
2
a
x
2
+
x
+
a
2
−
a
=
0
x^4 - 2ax^2 + x + a^2 -a = 0
x
4
−
2
a
x
2
+
x
+
a
2
−
a
=
0
has all its roots real.
6
1
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Least positive integer
(i) Consider two positive integers
a
a
a
and
b
b
b
which are such that
a
a
b
b
a^a b^b
a
a
b
b
is divisible by
2000
2000
2000
. What is the least possible value of
a
b
ab
ab
? (ii) Consider two positive integers
a
a
a
and
b
b
b
which are such that
a
b
b
a
a^b b^a
a
b
b
a
is divisible by
2000
2000
2000
. What is the least possible value of
a
b
ab
ab
?
5
1
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Parallelism
The internal bisector of angle
A
A
A
in a triangle
A
B
C
ABC
A
BC
with
A
C
>
A
B
AC > AB
A
C
>
A
B
meets the circumcircle
Γ
\Gamma
Γ
of the triangle in
D
D
D
. Join
D
D
D
to the center
O
O
O
of the circle
Γ
\Gamma
Γ
and suppose that
D
O
DO
D
O
meets
A
C
AC
A
C
in
E
E
E
, possibly when extended. Given that
B
E
BE
BE
is perpendicular to
A
D
AD
A
D
, show that
A
O
AO
A
O
is parallel to
B
D
BD
B
D
.
4
1
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Find the 2000th number
All the
7
7
7
digit numbers containing each of the digits
1
,
2
,
3
,
4
,
5
,
6
,
7
1,2,3,4,5,6,7
1
,
2
,
3
,
4
,
5
,
6
,
7
exactly once , and not divisible by
5
5
5
are arranged in increasing order. Find the
200
t
h
200th
200
t
h
number in the list.
3
1
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An inequality - good
Suppose
{
x
n
}
n
≥
1
\{ x_n \}_{n\geq 1}
{
x
n
}
n
≥
1
is a sequence of positive real numbers such that
x
1
≥
x
2
≥
x
3
…
≥
x
n
…
x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots
x
1
≥
x
2
≥
x
3
…
≥
x
n
…
, and for all
n
n
n
x
1
1
+
x
4
2
+
x
9
3
+
…
+
x
n
2
n
≤
1.
\frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 .
1
x
1
+
2
x
4
+
3
x
9
+
…
+
n
x
n
2
≤
1.
Show that for all
k
k
k
x
1
1
+
x
2
2
+
…
+
x
k
k
≤
3.
\frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3.
1
x
1
+
2
x
2
+
…
+
k
x
k
≤
3.
2
1
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Solve the eqn
Solve the equation
y
3
=
x
3
+
8
x
2
−
6
x
+
8
y^3 = x^3 + 8x^2 - 6x +8
y
3
=
x
3
+
8
x
2
−
6
x
+
8
, for positive integers
x
x
x
and
y
y
y
.
1
1
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Prove an eq triangle
Let
A
C
AC
A
C
be a line segment in the plane and
B
B
B
a points between
A
A
A
and
C
C
C
. Construct isosceles triangles
P
A
B
PAB
P
A
B
and
Q
A
C
QAC
Q
A
C
on one side of the segment
A
C
AC
A
C
such that
∠
A
P
B
=
∠
B
Q
C
=
12
0
∘
\angle APB = \angle BQC = 120^{\circ}
∠
A
PB
=
∠
BQC
=
12
0
∘
and an isosceles triangle
R
A
C
RAC
R
A
C
on the other side of
A
C
AC
A
C
such that
∠
A
R
C
=
12
0
∘
.
\angle ARC = 120^{\circ}.
∠
A
RC
=
12
0
∘
.
Show that
P
Q
R
PQR
PQR
is an equilateral triangle.