MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2006 India Regional Mathematical Olympiad
2006 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
Hide problems
Problem 7 of RMO 2006 (Regional Mathematical Olympiad-India)
Let
X
X
X
be the set of all positive integers greater than or equal to
8
8
8
and let
f
:
X
→
X
f: X\rightarrow X
f
:
X
→
X
be a function such that f(x\plus{}y)\equal{}f(xy) for all
x
≥
4
,
y
≥
4.
x\ge 4, y\ge 4 .
x
≥
4
,
y
≥
4.
if f(8)\equal{}9, determine
f
(
9
)
.
f(9) .
f
(
9
)
.
5
1
Hide problems
Problem 5 of RMO 2006 (Regional Mathematical Olympiad-India)
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral in which
A
B
AB
A
B
is parallel to
C
D
CD
C
D
and perpendicular to AD; AB \equal{} 3CD; and the area of the quadrilateral is
4
4
4
. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.
3
1
Hide problems
Problem 3 of RMO 2006 (Regional Mathematical Olympiad-India)
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are three positive real numbers, prove that \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3
1
1
Hide problems
Problem 1 of RMO 2006 (Regional Mathematical Olympiad-India)
Let
A
B
C
ABC
A
BC
be an acute-angled triangle and let
D
,
E
,
F
D,E,F
D
,
E
,
F
be the feet of perpendiculars from
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively to
B
C
,
C
A
,
A
B
.
BC,CA,AB .
BC
,
C
A
,
A
B
.
Let the perpendiculars from
F
F
F
to
C
B
,
C
A
,
A
D
,
B
E
CB,CA,AD,BE
CB
,
C
A
,
A
D
,
BE
meet them in
P
,
Q
,
M
,
N
P,Q,M,N
P
,
Q
,
M
,
N
respectively. Prove that the points
P
,
Q
,
M
,
N
P,Q,M,N
P
,
Q
,
M
,
N
are collinear.
4
1
Hide problems
Problem 4 of RMO 2006 (Regional Mathematical Olympiad-India)
A
6
×
6
6\times 6
6
×
6
square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
6
1
Hide problems
Problem 6 of RMO 2006 (Regional Mathematical Olympiad-India)
Prove that there are infinitely many positive integers
n
n
n
such that n(n\plus{}1) can be represented as a sum of two positive squares in at least two different ways. (Here a^{2}\plus{}b^{2} and b^{2}\plus{}a^{2} are considered as the same representation.)
2
1
Hide problems
a exhaustive question
If
a
a
a
and
b
b
b
are natural numbers such that a\plus{}13b is divisible by
11
11
11
and a\plus{}11b is divisible by
13
13
13
, then find the least possible value of a\plus{}b.