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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2002 India Regional Mathematical Olympiad
2002 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Find all integers
Find all integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that (i)
1
≤
a
≤
b
≤
c
≤
d
1 \leq a \leq b \leq c \leq d
1
≤
a
≤
b
≤
c
≤
d
; (ii)
a
b
+
c
d
=
a
+
b
+
c
+
d
+
3
ab + cd = a +b +c +d + 3
ab
+
c
d
=
a
+
b
+
c
+
d
+
3
.
6
1
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Simple inequality
Prove that for any natural number
n
>
1
n > 1
n
>
1
,
1
2
<
1
n
2
+
1
+
2
n
2
+
2
+
…
+
n
n
2
+
n
<
1
2
+
1
2
n
.
\frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}.
2
1
<
n
2
+
1
1
+
n
2
+
2
2
+
…
+
n
2
+
n
n
<
2
1
+
2
n
1
.
5
1
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A cyclic quad
The circumference of a circle is divided into eight arcs by a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by
p
,
q
,
r
,
s
p,q,r,s
p
,
q
,
r
,
s
in counter-clockwise direction. Suppose
p
+
r
=
q
+
s
p+r = q+s
p
+
r
=
q
+
s
. Prove that
A
B
C
D
ABCD
A
BC
D
is cyclic.
4
1
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Partitioning
Suppose the integers
1
,
2
,
…
10
1,2,\ldots 10
1
,
2
,
…
10
are split into two disjoint collections
a
1
,
a
2
,
…
a
5
a_1,a_2, \ldots a_5
a
1
,
a
2
,
…
a
5
and
b
1
,
…
b
5
b_1 , \ldots b_5
b
1
,
…
b
5
such that
a
1
<
a
2
<
a
3
<
a
4
<
a
5
,
b
1
<
b
2
<
b
3
<
b
4
<
b
5
a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5
a
1
<
a
2
<
a
3
<
a
4
<
a
5
,
b
1
<
b
2
<
b
3
<
b
4
<
b
5
(i) Show that the larger number in any pair
{
a
j
,
b
j
}
\{ a_j, b_j \}
{
a
j
,
b
j
}
,
1
≤
j
≤
5
1 \leq j \leq 5
1
≤
j
≤
5
is at least
6
6
6
. (ii) Show that
∑
i
=
1
5
∣
a
i
−
b
i
∣
\sum_{i=1} ^{5} | a_i - b_i|
∑
i
=
1
5
∣
a
i
−
b
i
∣
= 25 for every such partition.
3
1
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Divides
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers such that
a
a
a
divides
b
2
b^2
b
2
,
b
b
b
divides
c
2
c^2
c
2
and
c
c
c
divides
a
2
a^2
a
2
. Prove that
a
b
c
abc
ab
c
divides
(
a
+
b
+
c
)
7
(a + b +c)^7
(
a
+
b
+
c
)
7
.
2
1
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Solve the eqn
Solve for real
x
x
x
:
(
x
2
+
x
−
2
)
3
+
(
2
x
2
−
x
−
1
)
3
=
27
(
x
2
−
1
)
3
.
(x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3.
(
x
2
+
x
−
2
)
3
+
(
2
x
2
−
x
−
1
)
3
=
27
(
x
2
−
1
)
3
.
1
1
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Altitudes
In an acute triangle
A
B
C
ABC
A
BC
points
D
,
E
,
F
D,E,F
D
,
E
,
F
are located on the sides
B
C
,
C
A
,
A
B
BC,CA, AB
BC
,
C
A
,
A
B
such that
C
D
C
E
=
C
A
C
B
,
A
E
A
F
=
A
B
A
C
,
B
F
F
D
=
B
C
B
A
\frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA}
CE
C
D
=
CB
C
A
,
A
F
A
E
=
A
C
A
B
,
F
D
BF
=
B
A
BC
Prove that
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
are altitudes of triangle
A
B
C
ABC
A
BC
.