MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2000 India National Olympiad
2000 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
Hide problems
Find no of triangles
For any natural numbers
n
n
n
, (
n
≥
3
n \geq 3
n
≥
3
), let
f
(
n
)
f(n)
f
(
n
)
denote the number of congruent integer-sided triangles with perimeter
n
n
n
. Show that (i)
f
(
1999
)
>
f
(
1996
)
f(1999) > f (1996)
f
(
1999
)
>
f
(
1996
)
; (ii)
f
(
2000
)
=
f
(
1997
)
f(2000) = f(1997)
f
(
2000
)
=
f
(
1997
)
.
5
1
Hide problems
Solution of a cubic
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three real numbers such that
1
≥
a
≥
b
≥
c
≥
0
1 \geq a \geq b \geq c \geq 0
1
≥
a
≥
b
≥
c
≥
0
. prove that if
λ
\lambda
λ
is a root of the cubic equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
(real or complex), then
∣
λ
∣
≤
1.
| \lambda | \leq 1.
∣
λ
∣
≤
1.
4
1
Hide problems
One on quads
In a convex quadrilateral
P
Q
R
S
PQRS
PQRS
,
P
Q
=
R
S
PQ =RS
PQ
=
RS
,
(
3
+
1
)
Q
R
=
S
P
(\sqrt{3} +1 )QR = SP
(
3
+
1
)
QR
=
SP
and
∠
R
S
P
−
∠
S
Q
P
=
3
0
∘
\angle RSP - \angle SQP = 30^{\circ}
∠
RSP
−
∠
SQP
=
3
0
∘
. Prove that
∠
P
Q
R
−
∠
Q
R
S
=
9
0
∘
.
\angle PQR - \angle QRS = 90^{\circ}.
∠
PQR
−
∠
QRS
=
9
0
∘
.
3
1
Hide problems
Prove that they are equal
If
a
,
b
,
c
,
x
a,b,c,x
a
,
b
,
c
,
x
are real numbers such that
a
b
c
≠
0
abc \not= 0
ab
c
=
0
and
x
b
+
(
1
−
x
)
c
a
=
x
c
+
(
1
−
x
)
a
b
=
x
a
+
(
1
−
x
)
b
c
,
\frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c},
a
x
b
+
(
1
−
x
)
c
=
b
x
c
+
(
1
−
x
)
a
=
c
x
a
+
(
1
−
x
)
b
,
then prove that
a
=
b
=
c
a = b = c
a
=
b
=
c
.
2
1
Hide problems
Solve in integers
Solve for integers
x
,
y
,
z
x,y,z
x
,
y
,
z
:
{
x
+
y
=
1
−
z
x
3
+
y
3
=
1
−
z
2
.
\{ \begin{array}{ccc} x + y &=& 1 - z \\ x^3 + y^3 &=& 1 - z^2 . \end{array}
{
x
+
y
x
3
+
y
3
=
=
1
−
z
1
−
z
2
.
1
1
Hide problems
One on incircle: PQ bisects AB and AC
The incircle of
A
B
C
ABC
A
BC
touches
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
K
K
K
,
L
L
L
,
M
M
M
respectively. The line through
A
A
A
parallel to
L
K
LK
L
K
meets
M
K
MK
M
K
at
P
P
P
, and the line through
A
A
A
parallel to
M
K
MK
M
K
meets
L
K
LK
L
K
at
Q
Q
Q
. Show that the line
P
Q
PQ
PQ
bisects
A
B
AB
A
B
and bisects
A
C
AC
A
C
.