MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1998 India National Olympiad
1998 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Choose number such that...
It is desired to choose
n
n
n
integers from the collection of
2
n
2n
2
n
integers, namely,
0
,
0
,
1
,
1
,
2
,
2
,
…
,
n
−
1
,
n
−
1
0,0,1,1,2,2,\ldots,n-1,n-1
0
,
0
,
1
,
1
,
2
,
2
,
…
,
n
−
1
,
n
−
1
such that the average of these
n
n
n
chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer
n
n
n
and find this minimum value for each
n
n
n
.
5
1
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Another quadratic!
Suppose
a
,
b
,
c
a,b,c
a
,
b
,
c
are three rela numbers such that the quadratic equation
x
2
−
(
a
+
b
+
c
)
x
+
(
a
b
+
b
c
+
c
a
)
=
0
x^2 - (a +b +c )x + (ab +bc +ca) = 0
x
2
−
(
a
+
b
+
c
)
x
+
(
ab
+
b
c
+
c
a
)
=
0
has roots of the form
α
+
i
β
\alpha + i \beta
α
+
i
β
where
α
>
0
\alpha > 0
α
>
0
and
β
≠
0
\beta \not= 0
β
=
0
are real numbers. Show that (i) The numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
are all positive. (ii) The numbers
a
,
b
,
c
\sqrt{a}, \sqrt{b} , \sqrt{c}
a
,
b
,
c
form the sides of a triangle.
4
1
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Inequality => square
Suppose
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral inscribed in a circle of radius one unit. If
A
B
⋅
B
C
⋅
C
D
⋅
D
A
≥
4
AB \cdot BC \cdot CD \cdot DA \geq 4
A
B
⋅
BC
⋅
C
D
⋅
D
A
≥
4
, prove that
A
B
C
D
ABCD
A
BC
D
is a square.
3
1
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Divisiblity by 5
Let
p
,
q
,
r
,
s
p , q, r , s
p
,
q
,
r
,
s
be four integers such that
s
s
s
is not divisible by
5
5
5
. If there is an integer
a
a
a
such that
p
a
3
+
q
a
2
+
r
a
+
s
pa^3 + qa^2+ ra +s
p
a
3
+
q
a
2
+
r
a
+
s
is divisible be 5, prove that there is an integer
b
b
b
such that
s
b
3
+
r
b
2
+
q
b
+
p
sb^3 + rb^2 + qb + p
s
b
3
+
r
b
2
+
q
b
+
p
is also divisible by 5.
2
1
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Rationals/irrationals
Let
a
a
a
and
b
b
b
be two positive rational numbers such that
a
3
+
b
3
\sqrt[3] {a} + \sqrt[3]{b}
3
a
+
3
b
is also a rational number. Prove that
a
3
\sqrt[3]{a}
3
a
and
b
3
\sqrt[3] {b}
3
b
themselves are rational numbers.
1
1
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Yet another circle!
In a circle
C
1
C_1
C
1
with centre
O
O
O
, let
A
B
AB
A
B
be a chord that is not a diameter. Let
M
M
M
be the midpoint of this chord
A
B
AB
A
B
. Take a point
T
T
T
on the circle
C
2
C_2
C
2
with
O
M
OM
OM
as diameter. Let the tangent to
C
2
C_2
C
2
at
T
T
T
meet
C
1
C_1
C
1
at
P
P
P
. Show that
P
A
2
+
P
B
2
=
4
⋅
P
T
2
PA^2 + PB^2 = 4 \cdot PT^2
P
A
2
+
P
B
2
=
4
⋅
P
T
2
.