MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1992 India National Olympiad
1992 India National Olympiad
Part of
India National Olympiad
Subcontests
(10)
10
1
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Functional eqn.
Determine all functions
f
:
R
−
[
0
,
1
]
→
R
f : \mathbb{R} - [0,1] \to \mathbb{R}
f
:
R
−
[
0
,
1
]
→
R
such that
f
(
x
)
+
f
(
1
1
−
x
)
=
2
(
1
−
2
x
)
x
(
1
−
x
)
.
f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} .
f
(
x
)
+
f
(
1
−
x
1
)
=
x
(
1
−
x
)
2
(
1
−
2
x
)
.
9
1
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What is n???
Let
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
be an
n
n
n
-sided regular polygon. If
1
A
1
A
2
=
1
A
1
A
3
+
1
A
1
A
4
\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}
A
1
A
2
1
=
A
1
A
3
1
+
A
1
A
4
1
, find
n
n
n
.
8
1
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Perfecy square???
Determine all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers for which
2
m
+
3
n
2^{m} + 3^{n}
2
m
+
3
n
is a perfect square.
7
1
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Cheesboard!
Let
n
≥
3
n\geq 3
n
≥
3
be an integer. Find the number of ways in which one can place the numbers
1
,
2
,
3
,
…
,
n
2
1, 2, 3, \ldots, n^2
1
,
2
,
3
,
…
,
n
2
in the
n
2
n^2
n
2
squares of a
n
×
n
n \times n
n
×
n
chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.
6
1
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Integral functional eqn
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial in
x
x
x
with integer coefficients and suppose that for five distinct integers
a
1
,
…
,
a
5
a_1, \ldots, a_5
a
1
,
…
,
a
5
one has
f
(
a
1
)
=
f
(
a
2
)
=
…
=
f
(
a
5
)
=
2
f(a_1) = f(a_2) = \ldots = f(a_5) = 2
f
(
a
1
)
=
f
(
a
2
)
=
…
=
f
(
a
5
)
=
2
. Show that there does not exist an integer
b
b
b
such that
f
(
b
)
=
9
f(b) = 9
f
(
b
)
=
9
.
5
1
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Geometry - simple
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect at two distinct points
P
,
Q
P, Q
P
,
Q
in a plane. Let a line passing through
P
P
P
meet the circles
C
1
C_1
C
1
and
C
2
C_2
C
2
in
A
A
A
and
B
B
B
respectively. Let
Y
Y
Y
be the midpoint of
A
B
AB
A
B
and let
Q
Y
QY
Q
Y
meet the cirlces
C
1
C_1
C
1
and
C
2
C_2
C
2
in
X
X
X
and
Z
Z
Z
respectively. Show that
Y
Y
Y
is also the midpoint of
X
Z
XZ
XZ
.
4
1
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Find permutations
Find the number of permutations
(
p
1
,
p
2
,
p
3
,
p
4
,
p
5
,
p
6
)
( p_1, p_2, p_3 , p_4 , p_5 , p_6)
(
p
1
,
p
2
,
p
3
,
p
4
,
p
5
,
p
6
)
of
1
,
2
,
3
,
4
,
5
,
6
1, 2 ,3,4,5,6
1
,
2
,
3
,
4
,
5
,
6
such that for any
k
,
1
≤
k
≤
5
k, 1 \leq k \leq 5
k
,
1
≤
k
≤
5
,
(
p
1
,
…
,
p
k
)
(p_1, \ldots, p_k)
(
p
1
,
…
,
p
k
)
does not form a permutation of
1
,
2
,
…
,
k
1 , 2, \ldots, k
1
,
2
,
…
,
k
.
3
1
Hide problems
Find remainder
Find the remainder when
1
9
92
19^{92}
1
9
92
is divided by 92.
2
1
Hide problems
Inequality
If
x
,
y
,
z
∈
R
x , y, z \in \mathbb{R}
x
,
y
,
z
∈
R
such that
x
+
y
+
z
=
4
x+y +z =4
x
+
y
+
z
=
4
and
x
2
+
y
2
+
z
2
=
6
x^2 + y^2 +z^2 = 6
x
2
+
y
2
+
z
2
=
6
, then show that each of
x
,
y
,
z
x, y, z
x
,
y
,
z
lies in the closed interval
[
2
3
,
2
]
\left[ \dfrac{2}{3} , 2 \right]
[
3
2
,
2
]
. Can
x
x
x
attain the extreme value
2
3
\dfrac{2}{3}
3
2
or
2
2
2
?
1
1
Hide problems
Find relation in triangle
In a triangle
A
B
C
ABC
A
BC
,
∠
A
=
2
⋅
∠
B
\angle A = 2 \cdot \angle B
∠
A
=
2
⋅
∠
B
. Prove that
a
2
=
b
(
b
+
c
)
a^2 = b (b+c)
a
2
=
b
(
b
+
c
)
.