MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2001 India National Olympiad
2001 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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All functions f with f(x +y) = f(x) f(y) f(xy)
Find all functions
f
:
R
→
R
f : \mathbb{R} \to\mathbb{R}
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
f
(
x
y
)
f(x +y) = f(x) f(y) f(xy)
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
f
(
x
y
)
for all
x
,
y
∈
R
.
x, y \in \mathbb{R}.
x
,
y
∈
R
.
5
1
Hide problems
Equilateral
A
B
C
ABC
A
BC
is a triangle.
M
M
M
is the midpoint of
B
C
BC
BC
.
∠
M
A
B
=
∠
C
\angle MAB = \angle C
∠
M
A
B
=
∠
C
, and
∠
M
A
C
=
1
5
∘
\angle MAC = 15^{\circ}
∠
M
A
C
=
1
5
∘
. Show that
∠
A
M
C
\angle AMC
∠
A
MC
is obtuse. If
O
O
O
is the circumcenter of
A
D
C
ADC
A
D
C
, show that
A
O
D
AOD
A
O
D
is equilateral.
4
1
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Yaphp
Show that given any nine integers, we can find four,
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that
a
+
b
−
c
−
d
a + b - c - d
a
+
b
−
c
−
d
is divisible by
20
20
20
. Show that this is not always true for eight integers.
3
1
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One on reals
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
a
b
c
=
1
abc= 1
ab
c
=
1
, Prove that
a
b
+
c
b
c
+
a
c
a
+
b
≤
1.
a^{b+c} b^{c+a} c^{a+b} \leq 1 .
a
b
+
c
b
c
+
a
c
a
+
b
≤
1.
2
1
Hide problems
Prove that there are infinte...
Show that the equation
x
2
+
y
2
+
z
2
=
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)
x
2
+
y
2
+
z
2
=
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
has infintely many solutions in integers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
1
1
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Triangles
Let
A
B
C
ABC
A
BC
be a triangle in which no angle is
9
0
∘
90^{\circ}
9
0
∘
. For any point
P
P
P
in the plane of the triangle, let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
denote the reflections of
P
P
P
in the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Prove that (i) If
P
P
P
is the incenter or an excentre of
A
B
C
ABC
A
BC
, then
P
P
P
is the circumenter of
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
; (ii) If
P
P
P
is the circumcentre of
A
B
C
ABC
A
BC
, then
P
P
P
is the orthocentre of
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
; (iii) If
P
P
P
is the orthocentre of
A
B
C
ABC
A
BC
, then
P
P
P
is either the incentre or an excentre of
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
.