MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2002 India National Olympiad
2002 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Show that $ b_1 + b_2 + ... + b_n$ =
The numbers
1
,
2
,
3
1, 2, 3
1
,
2
,
3
,
…
\ldots
…
,
n
2
n^2
n
2
are arranged in an
n
×
n
n\times n
n
×
n
array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let
a
i
j
a_{ij}
a
ij
be the number in position
i
,
j
i, j
i
,
j
. Let
b
j
b_j
b
j
be the number of possible values for
a
j
j
a_{jj}
a
jj
. Show that
b
1
+
b
2
+
⋯
+
b
n
=
n
(
n
2
−
3
n
+
5
)
3
.
b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} .
b
1
+
b
2
+
⋯
+
b
n
=
3
n
(
n
2
−
3
n
+
5
)
.
5
1
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An arithmetic progression
Do there exist distinct positive integers
a
a
a
,
b
b
b
,
c
c
c
such that
a
a
a
,
b
b
b
,
c
c
c
,
−
a
+
b
+
c
-a+b+c
−
a
+
b
+
c
,
a
−
b
+
c
a-b+c
a
−
b
+
c
,
a
+
b
−
c
a+b-c
a
+
b
−
c
,
a
+
b
+
c
a+b+c
a
+
b
+
c
form an arithmetic progression (in some order).
4
1
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2002 points
Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?
3
1
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Inegalité indian
If
x
x
x
,
y
y
y
are positive reals such that
x
+
y
=
2
x + y = 2
x
+
y
=
2
show that
x
3
y
3
(
x
3
+
y
3
)
≤
2
x^3y^3(x^3+ y^3) \leq 2
x
3
y
3
(
x
3
+
y
3
)
≤
2
.
2
1
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The smallest positive value
Find the smallest positive value taken by
a
3
+
b
3
+
c
3
−
3
a
b
c
a^3 + b^3 + c^3 - 3abc
a
3
+
b
3
+
c
3
−
3
ab
c
for positive integers
a
a
a
,
b
b
b
,
c
c
c
. Find all
a
a
a
,
b
b
b
,
c
c
c
which give the smallest value
1
1
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Hexagon
For a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
in which each pair of opposite sides is unequal, consider the following statements.(
a
1
a_1
a
1
)
A
B
AB
A
B
is parallel to
D
E
DE
D
E
. (
a
2
a_2
a
2
) AE \equal{} BD.(
b
1
b_1
b
1
)
B
C
BC
BC
is parallel to
E
F
EF
EF
. (
b
2
b_2
b
2
) BF \equal{} CE.(
c
1
c_1
c
1
)
C
D
CD
C
D
is parallel to
F
A
FA
F
A
. (
c
2
c_2
c
2
) CA \equal{} DF.
(
a
)
(a)
(
a
)
Show that if all six of these statements are true then the hexagon is cyclic.
(
b
)
(b)
(
b
)
Prove that, in fact, five of the six statements suffice.