MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2006 India National Olympiad
2006 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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If n is an integer greater than 4011^2 then ...
(a) Prove that if
n
n
n
is a integer such that
n
≥
401
1
2
n \geq 4011^2
n
≥
401
1
2
then there exists an integer
l
l
l
such that
n
<
l
2
<
(
1
+
1
2005
)
n
.
n < l^2 < (1 + \frac{1}{{2005}})n .
n
<
l
2
<
(
1
+
2005
1
)
n
.
(b) Find the smallest positive integer
M
M
M
for which whenever an integer
n
n
n
is such that
n
≥
M
n \geq M
n
≥
M
then there exists an integer
l
l
l
such that
n
<
l
2
<
(
1
+
1
2005
)
n
.
n < l^2 < (1 + \frac{1}{{2005}})n .
n
<
l
2
<
(
1
+
2005
1
)
n
.
5
1
Hide problems
In a cyclic quadrilateral ABCD, <ABC=120 degrees
In a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
B
=
a
AB=a
A
B
=
a
,
B
C
=
b
BC=b
BC
=
b
,
C
D
=
c
CD=c
C
D
=
c
,
∠
A
B
C
=
12
0
∘
\angle ABC = 120^\circ
∠
A
BC
=
12
0
∘
and
∠
A
B
D
=
3
0
∘
\angle ABD = 30^\circ
∠
A
B
D
=
3
0
∘
. Prove that (1)
c
≥
a
+
b
c \ge a + b
c
≥
a
+
b
; (2)
∣
c
+
a
−
c
+
b
∣
=
c
−
a
−
b
|\sqrt{c + a} - \sqrt{c + b} | = \sqrt{c - a - b}
∣
c
+
a
−
c
+
b
∣
=
c
−
a
−
b
.
1
1
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Non equilateral triangle $abc$ the sides $a,b,c$ form an ap
In a non equilateral triangle
A
B
C
ABC
A
BC
the sides
a
,
b
,
c
a,b,c
a
,
b
,
c
form an arithmetic progression. Let
I
I
I
be the incentre and
O
O
O
the circumcentre of the triangle
A
B
C
ABC
A
BC
. Prove that (1)
I
O
IO
I
O
is perpendicular to
B
I
BI
B
I
; (2) If
B
I
BI
B
I
meets
A
C
AC
A
C
in
K
K
K
, and
D
D
D
,
E
E
E
are the midpoints of
B
C
BC
BC
,
B
A
BA
B
A
respectively then
I
I
I
is the circumcentre of triangle
D
K
E
DKE
DK
E
.
4
1
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46 squares are colored in red
Some 46 squares are randomly chosen from a
9
×
9
9 \times 9
9
×
9
chess board and colored in red. Show that there exists a
2
×
2
2\times 2
2
×
2
block of 4 squares of which at least three are colored in red.
3
1
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f(a,b,c) = (a+b+c, ab+bc+ca, abc)
Let
X
=
Z
3
X=\mathbb{Z}^3
X
=
Z
3
denote the set of all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of integers. Define
f
:
X
→
X
f: X \to X
f
:
X
→
X
by
f
(
a
,
b
,
c
)
=
(
a
+
b
+
c
,
a
b
+
b
c
+
c
a
,
a
b
c
)
.
f(a,b,c) = (a+b+c, ab+bc+ca, abc) .
f
(
a
,
b
,
c
)
=
(
a
+
b
+
c
,
ab
+
b
c
+
c
a
,
ab
c
)
.
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
f
(
f
(
a
,
b
,
c
)
)
=
(
a
,
b
,
c
)
.
f(f(a,b,c)) = (a,b,c) .
f
(
f
(
a
,
b
,
c
))
=
(
a
,
b
,
c
)
.
2
1
Hide problems
n = \frac{1}{2}(a + b - 1)(a + b - 2) + a
Prove that for every positive integer
n
n
n
there exists a unique ordered pair
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers such that
n
=
1
2
(
a
+
b
−
1
)
(
a
+
b
−
2
)
+
a
.
n = \frac{1}{2}(a + b - 1)(a + b - 2) + a .
n
=
2
1
(
a
+
b
−
1
)
(
a
+
b
−
2
)
+
a
.