MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1994 India National Olympiad
1994 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Another func eq!!
Find all real-valued functions
f
f
f
on the reals such that
f
(
−
x
)
=
−
f
(
x
)
f(-x) = -f(x)
f
(
−
x
)
=
−
f
(
x
)
,
f
(
x
+
1
)
=
f
(
x
)
+
1
f(x+1) = f(x) + 1
f
(
x
+
1
)
=
f
(
x
)
+
1
for all
x
x
x
, and
f
(
1
x
)
=
f
(
x
)
x
2
f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}
f
(
x
1
)
=
x
2
f
(
x
)
for
x
≠
0
x \not = 0
x
=
0
.
5
1
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Find another area!
A circle passes through the vertex of a rectangle
A
B
C
D
ABCD
A
BC
D
and touches its sides
A
B
AB
A
B
and
A
D
AD
A
D
at
M
M
M
and
N
N
N
respectively. If the distance from
C
C
C
to the line segment
M
N
MN
MN
is equal to
5
5
5
units, find the area of rectangle
A
B
C
D
ABCD
A
BC
D
.
4
1
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Find the no. of triangles
Find the number of nondegenerate triangles whose vertices lie in the set of points
(
s
,
t
)
(s,t)
(
s
,
t
)
in the plane such that
0
≤
s
≤
4
0 \leq s \leq 4
0
≤
s
≤
4
,
0
≤
t
≤
4
0 \leq t \leq 4
0
≤
t
≤
4
,
s
s
s
and
t
t
t
are integers.
3
1
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Yet another Pigeon Hole!
In any set of
181
181
181
square integers, prove that one can always find a subset of
19
19
19
numbers, sum of whose elements is divisible by
19
19
19
.
2
1
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One related to polynomials
If
x
5
−
x
3
+
x
=
a
,
x^5 - x ^3 + x = a,
x
5
−
x
3
+
x
=
a
,
prove that
x
6
≥
2
a
−
1
x^6 \geq 2a - 1
x
6
≥
2
a
−
1
.
1
1
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Prove another similarity!
Let
G
G
G
be the centroid of the triangle
A
B
C
ABC
A
BC
in which the angle at
C
C
C
is obtuse and
A
D
AD
A
D
and
C
F
CF
CF
be the medians from
A
A
A
and
C
C
C
respectively onto the sides
B
C
BC
BC
and
A
B
AB
A
B
. If the points
B
,
D
,
G
\ B,\ D, \ G
B
,
D
,
G
and
F
\ F
F
are concyclic, show that
A
C
B
C
≥
2
\dfrac{AC}{BC} \geq \sqrt{2}
BC
A
C
≥
2
. If further
P
P
P
is a point on the line
B
G
BG
BG
extended such that
A
G
C
P
AGCP
A
GCP
is a parallelogram, show that triangle
A
B
C
ABC
A
BC
and
G
A
P
GAP
G
A
P
are similar.