MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2003 India Regional Mathematical Olympiad
2003 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
Hide problems
A set with prod
Consider the set
X
X
X
=
{
1
,
2
…
10
}
\{ 1,2 \ldots 10 \}
{
1
,
2
…
10
}
. Find two disjoint nonempty sunsets
A
A
A
and
B
B
B
of
X
X
X
such that a)
A
∪
B
=
X
A \cup B = X
A
∪
B
=
X
; b)
∏
x
∈
A
x
\prod_{x\in A}x
∏
x
∈
A
x
is divisible by
∏
x
∈
B
x
\prod_{x\in B}x
∏
x
∈
B
x
, where
∏
x
∈
C
x
\prod_{x\in C}x
∏
x
∈
C
x
is the product of all numbers in
C
C
C
; c)
∏
x
∈
A
x
∏
x
∈
B
x
\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}
x
∈
B
∏
x
x
∈
A
∏
x
is as small as possible.
6
1
Hide problems
An equation -good
Find all real numbers
a
a
a
for which the equation
x
2
a
−
2
x
+
1
=
3
∣
x
∣
x^2a- 2x + 1 = 3 |x|
x
2
a
−
2
x
+
1
=
3∣
x
∣
has exactly three distinct real solutions in
x
x
x
.
5
1
Hide problems
Perp distances
Suppose
P
P
P
is an interior point of a triangle
A
B
C
ABC
A
BC
such that the ratios
d
(
A
,
B
C
)
d
(
P
,
B
C
)
,
d
(
B
,
C
A
)
d
(
P
,
C
A
)
,
d
(
C
,
A
B
)
d
(
P
,
A
B
)
\frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)} , \frac{d(C,AB)}{d(P,AB)}
d
(
P
,
BC
)
d
(
A
,
BC
)
,
d
(
P
,
C
A
)
d
(
B
,
C
A
)
,
d
(
P
,
A
B
)
d
(
C
,
A
B
)
are all equal. Find the common value of these ratios.
d
(
X
,
Y
Z
)
d(X,YZ)
d
(
X
,
Y
Z
)
represents the perpendicular distance fro
X
X
X
to the line
Y
Z
YZ
Y
Z
.
4
1
Hide problems
Find the no of triples
Find the number of ordered triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of non-negative integers satisfying (i)
x
≤
y
≤
z
x \leq y \leq z
x
≤
y
≤
z
(ii)
x
+
y
+
z
≤
100.
x + y + z \leq 100.
x
+
y
+
z
≤
100.
3
1
Hide problems
Ineq - simple
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three positive real numbers such that
a
+
b
+
c
=
1
a + b +c =1
a
+
b
+
c
=
1
. prove that among the three numbers
a
−
a
b
,
b
−
b
c
,
c
−
c
a
a-ab, b - bc, c-ca
a
−
ab
,
b
−
b
c
,
c
−
c
a
there is one which is at most
1
4
\frac{1}{4}
4
1
and there is one which is at least
2
9
\frac{2}{9}
9
2
.
2
1
Hide problems
Gint and c
If
n
n
n
is an integer greater than
7
7
7
, prove that
(
n
7
)
−
[
n
7
]
{n \choose 7} - \left[ \frac{n}{7} \right]
(
7
n
)
−
[
7
n
]
is divisible by
7
7
7
.
1
1
Hide problems
Prove an angle
Let
A
B
C
ABC
A
BC
be a triangle in which
A
B
=
A
C
AB =AC
A
B
=
A
C
and
∠
C
A
B
=
9
0
∘
\angle CAB = 90^{\circ}
∠
C
A
B
=
9
0
∘
. Suppose that
M
M
M
and
N
N
N
are points on the hypotenuse
B
C
BC
BC
such that
B
M
2
+
C
N
2
=
M
N
2
BM^2 + CN^2 = MN^2
B
M
2
+
C
N
2
=
M
N
2
. Prove that
∠
M
A
N
=
4
5
∘
\angle MAN = 45^{\circ}
∠
M
A
N
=
4
5
∘
.