MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1997 India National Olympiad
1997 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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A cubic eqn
Suppose
a
a
a
and
b
b
b
are two positive real numbers such that the roots of the cubic equation
x
3
−
a
x
+
b
=
0
x^3 - ax + b = 0
x
3
−
a
x
+
b
=
0
are all real. If
α
\alpha
α
is a root of this cubic with minimal absolute value, prove that
b
a
<
α
<
3
b
2
a
.
\dfrac{b}{a} < \alpha < \dfrac{3b}{2a}.
a
b
<
α
<
2
a
3
b
.
5
1
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Yet another array!
Find the number of
4
×
4
4 \times 4
4
×
4
array whose entries are from the set
{
0
,
1
,
2
,
3
}
\{ 0 , 1, 2, 3 \}
{
0
,
1
,
2
,
3
}
and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by
4
4
4
.
4
1
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Division of square
In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr
p
p
p
, show that
14
10
<
p
<
15
10
\dfrac{14}{10} < p < \dfrac{15}{10}
10
14
<
p
<
10
15
.
3
1
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Simple one!
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are three real numbers and
a
+
1
b
=
b
+
1
c
=
c
+
1
a
=
t
a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t
a
+
b
1
=
b
+
c
1
=
c
+
a
1
=
t
for some real number
t
t
t
, prove that
a
b
c
+
t
=
0.
abc + t = 0 .
ab
c
+
t
=
0.
2
1
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Prove that there are no integers...
Show that there do not exist positive integers
m
m
m
and
n
n
n
such that
m
n
+
n
+
1
m
=
4.
\dfrac{m}{n} + \dfrac{n+1}{m} = 4 .
n
m
+
m
n
+
1
=
4.
1
1
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Parallelogram problem
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Suppose a line passing through
C
C
C
and lying outside the parallelogram meets
A
B
AB
A
B
and
A
D
AD
A
D
produced at
E
E
E
and
F
F
F
respectively. Show that
A
C
2
+
C
E
⋅
C
F
=
A
B
⋅
A
E
+
A
D
⋅
A
F
.
AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF .
A
C
2
+
CE
⋅
CF
=
A
B
⋅
A
E
+
A
D
⋅
A
F
.