MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2007 India National Olympiad
2007 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
5
1
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Inequality in an isosceles triangle
Let
A
B
C
ABC
A
BC
be a triangle in which AB\equal{}AC. Let
D
D
D
be the midpoint of
B
C
BC
BC
and
P
P
P
be a point on
A
D
AD
A
D
. Suppose
E
E
E
is the foot of perpendicular from
P
P
P
on
A
C
AC
A
C
. Define \frac{AP}{PD}\equal{}\frac{BP}{PE}\equal{}\lambda , \ \ \ \frac{BD}{AD}\equal{}m , \ \ \ z\equal{}m^2(1\plus{}\lambda) Prove that z^2 \minus{} (\lambda^3 \minus{} \lambda^2 \minus{} 2)z \plus{} 1 \equal{} 0 Hence show that
λ
≥
2
\lambda \ge 2
λ
≥
2
and \lambda \equal{} 2 if and only if
A
B
C
ABC
A
BC
is equilateral.
4
1
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How many permutations exist with exactly two inversions?
Let
σ
=
(
a
1
,
a
2
,
⋯
,
a
n
)
\sigma = (a_1, a_2, \cdots , a_n)
σ
=
(
a
1
,
a
2
,
⋯
,
a
n
)
be permutation of
(
1
,
2
,
⋯
,
n
)
(1, 2 ,\cdots, n)
(
1
,
2
,
⋯
,
n
)
. A pair
(
a
i
,
a
j
)
(a_i, a_j)
(
a
i
,
a
j
)
is said to correspond to an inversion of
σ
\sigma
σ
if
i
<
j
i<j
i
<
j
but
a
i
>
a
j
a_i>a_j
a
i
>
a
j
. How many permutations of
(
1
,
2
,
⋯
,
n
)
(1,2,\cdots,n)
(
1
,
2
,
⋯
,
n
)
,
n
≥
3
n \ge 3
n
≥
3
, have exactly two inversions?For example, In the permutation
(
2
,
4
,
5
,
3
,
1
)
(2,4,5,3,1)
(
2
,
4
,
5
,
3
,
1
)
, there are 6 inversions corresponding to the pairs
(
2
,
1
)
,
(
4
,
3
)
,
(
4
,
1
)
,
(
5
,
3
)
,
(
5
,
1
)
,
(
3
,
1
)
(2,1),(4,3),(4,1),(5,3),(5,1),(3,1)
(
2
,
1
)
,
(
4
,
3
)
,
(
4
,
1
)
,
(
5
,
3
)
,
(
5
,
1
)
,
(
3
,
1
)
.
3
1
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Prove that the roots of a quadratic are integers
Let
m
m
m
and
n
n
n
be positive integers such that x^2 \minus{} mx \plus{}n \equal{} 0 has real roots
α
\alpha
α
and
β
\beta
β
. Prove that
α
\alpha
α
and
β
\beta
β
are integers if and only if [m\alpha] \plus{} [m\beta] is the square of an integer. (Here
[
x
]
[x]
[
x
]
denotes the largest integer not exceeding
x
x
x
)
2
1
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Prove that 9n can be expressed in the required form...
Let
n
n
n
be a natural number such that n \equal{} a^2 \plus{} b^2 \plus{}c^2 for some natural numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
. Prove that 9n \equal{} (p_1a\plus{}q_1b\plus{}r_1c)^2 \plus{} (p_2a\plus{}q_2b\plus{}r_2c)^2 \plus{} (p_3a\plus{}q_3b\plus{}r_3c)^2 where
p
j
p_j
p
j
's ,
q
j
q_j
q
j
's ,
r
j
r_j
r
j
's are all nonzero integers. Further, if
3
3
3
does not divide at least one of
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
prove that
9
n
9n
9
n
can be expressed in the form x^2\plus{}y^2\plus{}z^2, where
x
,
y
,
z
x,y,z
x
,
y
,
z
are natural numbers none of which is divisible by
3
3
3
.
1
1
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Inequality in a right triangle
In a triangle
A
B
C
ABC
A
BC
right-angled at
C
C
C
, the median through
B
B
B
bisects the angle between
B
A
BA
B
A
and the bisector of
∠
B
\angle B
∠
B
. Prove that
5
2
<
A
B
B
C
<
3
\frac{5}{2} < \frac{AB}{BC} < 3
2
5
<
BC
A
B
<
3
6
1
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INMO 2007 Problem 6
If
x
x
x
,
y
y
y
,
z
z
z
are positive real numbers, prove that \left(x \plus{} y \plus{} z\right)^2 \left(yz \plus{} zx \plus{} xy\right)^2 \leq 3\left(y^2 \plus{} yz \plus{} z^2\right)\left(z^2 \plus{} zx \plus{} x^2\right)\left(x^2 \plus{} xy \plus{} y^2\right) .