MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1999 India National Olympiad
1999 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
Hide problems
Splitting integers
For which positive integer values of
n
n
n
can the set
{
1
,
2
,
3
,
…
,
4
n
}
\{ 1, 2, 3, \ldots, 4n \}
{
1
,
2
,
3
,
…
,
4
n
}
be split into
n
n
n
disjoint
4
4
4
-element subsets
{
a
,
b
,
c
,
d
}
\{ a,b,c,d \}
{
a
,
b
,
c
,
d
}
such that in each of these sets
a
=
b
+
c
+
d
3
a = \dfrac{b +c +d} {3}
a
=
3
b
+
c
+
d
.
5
1
Hide problems
Another quadratic.
Given any four distinct positive real numbers, show that one can choose three numbers
A
,
B
,
C
A,B,C
A
,
B
,
C
from among them, such that all three quadratic equations \begin{eqnarray*} Bx^2 + x + C &=& 0\\ Cx^2 + x + A &=& 0 \\ Ax^2 + x +B &=& 0 \end{eqnarray*} have only real roots, or all three equations have only imaginary roots.
4
1
Hide problems
Yet another circle!
Let
Γ
\Gamma
Γ
and
Γ
′
\Gamma'
Γ
′
be two concentric circles. Let
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
be any two equilateral triangles inscribed in
Γ
\Gamma
Γ
and
Γ
′
\Gamma'
Γ
′
respectively. If
P
P
P
and
P
′
P'
P
′
are any two points on
Γ
\Gamma
Γ
and
Γ
′
\Gamma'
Γ
′
respectively, show that
P
′
A
2
+
P
′
B
2
+
P
′
C
2
=
A
′
P
2
+
B
′
P
2
+
C
′
P
2
.
P'A^2 + P'B^2 + P'C^2 = A'P^2 + B'P^2 + C'P^2.
P
′
A
2
+
P
′
B
2
+
P
′
C
2
=
A
′
P
2
+
B
′
P
2
+
C
′
P
2
.
3
1
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One one polynomials
Show that there do not exist polynomials
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
each having integer coefficients and of degree greater than or equal to 1 such that
p
(
x
)
q
(
x
)
=
x
5
+
2
x
+
1.
p(x)q(x) = x^5 +2x +1 .
p
(
x
)
q
(
x
)
=
x
5
+
2
x
+
1.
2
1
Hide problems
Find length and breadth
In a village
1998
1998
1998
persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to
3996
3996
3996
feet. For this purpose, the field was divided into
1998
1998
1998
equal parts. If each part had an integer area, find the length and breadth of the field.
1
1
Hide problems
Find the perimeter
Let
A
B
C
ABC
A
BC
be an acute-angled triangle in which
D
,
E
,
F
D,E,F
D
,
E
,
F
are points on
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively such that
A
D
⊥
B
C
AD \perp BC
A
D
⊥
BC
;
A
E
=
B
C
AE = BC
A
E
=
BC
; and
C
F
CF
CF
bisects
∠
C
\angle C
∠
C
internally, Suppose
C
F
CF
CF
meets
A
D
AD
A
D
and
D
E
DE
D
E
in
M
M
M
and
N
N
N
respectively. If
F
M
FM
FM
=
2
= 2
=
2
,
M
N
=
1
MN =1
MN
=
1
,
N
C
=
3
NC=3
NC
=
3
, find the perimeter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
.