MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1996 India National Olympiad
1996 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Arrays
There is a
2
n
×
2
n
2n \times 2n
2
n
×
2
n
array (matrix) consisting of
0
′
s
0's
0
′
s
and
1
′
s
1's
1
′
s
and there are exactly
3
n
3n
3
n
zeroes. Show that it is possible to remove all the zeroes by deleting some
n
n
n
rows and some
n
n
n
columns.
5
1
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Sequnece 1
Define a sequence
(
a
n
)
n
≥
1
(a_n)_{n \geq 1}
(
a
n
)
n
≥
1
by
a
1
=
1
a_1 =1
a
1
=
1
and
a
2
=
2
a_2 =2
a
2
=
2
and
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
a_{n+2} = 2 a_{n+1} - a_n + 2
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
for
n
≥
1
n \geq 1
n
≥
1
. prove that for any
m
m
m
,
a
m
a
m
+
1
a_m a_{m+1}
a
m
a
m
+
1
is also a term in this sequence.
4
1
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Find the triples
Let
X
X
X
be a set containing
n
n
n
elements. Find the number of ordered triples
(
A
,
B
,
C
)
(A,B, C)
(
A
,
B
,
C
)
of subsets of
X
X
X
such that
A
A
A
is a subset of
B
B
B
and
B
B
B
is a proper subset of
C
C
C
.
3
1
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Yet another system!
Solve the following system for real
a
,
b
,
c
,
d
,
e
a , b, c, d, e
a
,
b
,
c
,
d
,
e
:
{
3
a
=
(
b
+
c
+
d
)
3
3
b
=
(
c
+
d
+
e
)
3
3
c
=
(
d
+
e
+
a
)
3
3
d
=
(
e
+
a
+
b
)
3
3
e
=
(
a
+
b
+
c
)
3
.
\left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right.
⎩
⎨
⎧
3
a
3
b
3
c
3
d
3
e
=
=
=
=
=
(
b
+
c
+
d
)
3
(
c
+
d
+
e
)
3
(
d
+
e
+
a
)
3
(
e
+
a
+
b
)
3
(
a
+
b
+
c
)
3
.
2
1
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Two circles ...
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two concentric circles in the plane with radii
R
R
R
and
3
R
3R
3
R
respectively. Show that the orthocenter of any triangle inscribed in circle
C
1
C_1
C
1
lies in the interior of circle
C
2
C_2
C
2
. Conversely, show that every point in the interior of
C
2
C_2
C
2
is the orthocenter of some triangle inscribed in
C
1
C_1
C
1
.
1
1
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Division
a) Given any positive integer
n
n
n
, show that there exist distint positive integers
x
x
x
and
y
y
y
such that
x
+
j
x + j
x
+
j
divides
y
+
j
y + j
y
+
j
for
j
=
1
,
2
,
3
,
…
,
n
j = 1 , 2, 3, \ldots, n
j
=
1
,
2
,
3
,
…
,
n
; b) If for some positive integers
x
x
x
and
y
y
y
,
x
+
j
x+j
x
+
j
divides
y
+
j
y+j
y
+
j
for all positive integers
j
j
j
, prove that
x
=
y
x = y
x
=
y
.