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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2008 India Regional Mathematical Olympiad
2008 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(6)
6
2
Hide problems
Number of triangles with integer sides and other conditions
Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter
2008
2008
2008
. [16 points out of 100 for the 6 problems]
Trig bash
Let
B
C
D
K
BCDK
BC
DK
be a convex quadrilateral such that
B
C
=
B
K
BC=BK
BC
=
B
K
and
D
C
=
D
K
DC=DK
D
C
=
DK
.
A
A
A
and
E
E
E
are points such that
A
B
C
D
E
ABCDE
A
BC
D
E
is a convex pentagon such that
A
B
=
B
C
AB=BC
A
B
=
BC
and
D
E
=
D
C
DE=DC
D
E
=
D
C
and
K
K
K
lies in the interior of the pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
. If
∠
A
B
C
=
12
0
∘
\angle ABC=120^{\circ}
∠
A
BC
=
12
0
∘
and
∠
C
D
E
=
6
0
∘
\angle CDE=60^{\circ}
∠
C
D
E
=
6
0
∘
and
B
D
=
2
BD=2
B
D
=
2
then determine area of the pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
.
5
2
Hide problems
Find all H.P.s with conditions
Three nonzero real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
are said to be in harmonic progression if \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}. Find all three term harmonic progressions
a
,
b
,
c
a,b,c
a
,
b
,
c
of strictly increasing positive integers in which a \equal{} 20 and
b
b
b
divides
c
c
c
. [17 points out of 100 for the 6 problems]
How many good integers are there?
Let
N
N
N
be a ten digit positive integer divisible by
7
7
7
. Suppose the first and the last digit of
N
N
N
are interchanged and the resulting number (not necessarily ten digit) is also divisible by
7
7
7
then we say that
N
N
N
is a good integer. How many ten digit good integers are there?
4
2
Hide problems
Counting 6 digit numbers with conditions
Find the number of all
6
6
6
-digit natural numbers such that the sum of their digits is
10
10
10
and each of the digits
0
,
1
,
2
,
3
0,1,2,3
0
,
1
,
2
,
3
occurs at least once in them. [14 points out of 100 for the 6 problems]
Number Theory
Determine all the natural numbers
n
n
n
such that
21
21
21
divides
2
2
n
+
2
n
+
1.
2^{2^{n}}+2^n+1.
2
2
n
+
2
n
+
1.
3
2
Hide problems
inequality on coefficients of cubic
Suppose
a
a
a
and
b
b
b
are real numbers such that the roots of the cubic equation ax^3\minus{}x^2\plus{}bx\minus{}1 are positive real numbers. Prove that:
(
i
)
0
<
3
a
b
≤
1
and
(
i
)
b
≥
3
(i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3}
(
i
)
0
<
3
ab
≤
1
and
(
i
)
b
≥
3
[19 points out of 100 for the 6 problems]
An Inequality
Prove that for every positive integer
n
n
n
and a non-negative real number
a
a
a
, the following inequality holds:
n
(
n
+
1
)
a
+
2
n
⩾
4
a
(
1
+
2
+
⋯
+
n
)
.
n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).
n
(
n
+
1
)
a
+
2
n
⩾
4
a
(
1
+
2
+
⋯
+
n
)
.
2
2
Hide problems
Existence of infinite sequences
Prove that there exist two infinite sequences
{
a
n
}
n
≥
1
\{a_n\}_{n\ge 1}
{
a
n
}
n
≥
1
and
{
b
n
}
n
≥
1
\{b_n\}_{n\ge 1}
{
b
n
}
n
≥
1
of positive integers such that the following conditions hold simultaneously:
(
i
)
(i)
(
i
)
0
<
a
1
<
a
2
<
a
3
<
⋯
0 < a_1 < a_2 < a_3 < \cdots
0
<
a
1
<
a
2
<
a
3
<
⋯
;
(
i
i
)
(ii)
(
ii
)
a
n
<
b
n
<
a
n
2
a_n < b_n < a_n^2
a
n
<
b
n
<
a
n
2
, for all
n
≥
1
n\ge 1
n
≥
1
;
(
i
i
i
)
(iii)
(
iii
)
a_n \minus{} 1 divides b_n \minus{} 1, for all
n
≥
1
n\ge 1
n
≥
1
(
i
v
)
(iv)
(
i
v
)
a_n^2 \minus{} 1 divides b_n^2 \minus{} 1, for all
n
≥
1
n\ge 1
n
≥
1
[19 points out of 100 for the 6 problems]
A system of equations
Solve the system of equation
x
+
y
+
z
=
2
;
x+y+z=2;
x
+
y
+
z
=
2
;
(
x
+
y
)
(
y
+
z
)
+
(
y
+
z
)
(
z
+
x
)
+
(
z
+
x
)
(
x
+
y
)
=
1
;
(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;
(
x
+
y
)
(
y
+
z
)
+
(
y
+
z
)
(
z
+
x
)
+
(
z
+
x
)
(
x
+
y
)
=
1
;
x
2
(
y
+
z
)
+
y
2
(
z
+
x
)
+
z
2
(
x
+
y
)
=
−
6.
x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.
x
2
(
y
+
z
)
+
y
2
(
z
+
x
)
+
z
2
(
x
+
y
)
=
−
6.
1
2
Hide problems
Segment equal to circumradius
Let
A
B
C
ABC
A
BC
be an acute angled triangle; let
D
,
F
D,F
D
,
F
be the midpoints of
B
C
,
A
B
BC,AB
BC
,
A
B
respectively. Let the perpendicular from
F
F
F
to
A
C
AC
A
C
and the perpendicular from
B
B
B
ti
B
C
BC
BC
meet in
N
N
N
: Prove that
N
D
ND
N
D
is the circumradius of
A
B
C
ABC
A
BC
. [15 points out of 100 for the 6 problems]
Trig bash or angle chase
On a semicircle with diameter
A
B
AB
A
B
and centre
S
S
S
, points
C
C
C
and
D
D
D
are given such that point
C
C
C
belongs to arc
A
D
AD
A
D
. Suppose
∠
C
S
D
=
12
0
∘
\angle CSD = 120^\circ
∠
CS
D
=
12
0
∘
. Let
E
E
E
be the point of intersection of the straight lines
A
C
AC
A
C
and
B
D
BD
B
D
and
F
F
F
the point of intersection of the straight lines
A
D
AD
A
D
and
B
C
BC
BC
. Prove that
E
F
=
3
A
B
EF=\sqrt{3}AB
EF
=
3
A
B
.