MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1991 India National Olympiad
1991 India National Olympiad
Part of
India National Olympiad
Subcontests
(10)
10
1
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Find s(n)
For any positive integer
n
n
n
, let
s
(
n
)
s(n)
s
(
n
)
denote the number of ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers for which
1
x
+
1
y
=
1
n
\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}
x
1
+
y
1
=
n
1
. Determine the set of positive integers for which
s
(
n
)
=
5
s(n) = 5
s
(
n
)
=
5
9
1
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Prove the similarity
Triangle
A
B
C
ABC
A
BC
has an incenter
I
I
I
l its incircle touches the side
B
C
BC
BC
at
T
T
T
. The line through
T
T
T
parallel to
I
A
IA
I
A
meets the incircle again at
S
S
S
and the tangent to the incircle at
S
S
S
meets
A
B
,
A
C
AB , AC
A
B
,
A
C
at points
C
′
,
B
′
C' , B'
C
′
,
B
′
respectively. Prove that triangle
A
B
′
C
′
AB'C'
A
B
′
C
′
is similar to triangle
A
B
C
ABC
A
BC
.
8
1
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Prove the weight!
There are
10
10
10
objects of total weight
20
20
20
, each of the weights being a positive integers. Given that none of the weights exceeds
10
10
10
, prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance.
7
1
Hide problems
Solve this system
Solve the following system for real
x
,
y
,
z
x,y,z
x
,
y
,
z
{
x
+
y
−
z
=
4
x
2
−
y
2
+
z
2
=
−
4
x
y
z
=
6.
\{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array}
{
x
+
y
−
z
x
2
−
y
2
+
z
2
x
yz
=
=
=
4
−
4
6.
6
1
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Does it divide???
(i) Determine the set of all positive integers
n
n
n
for which
3
n
+
1
3^{n+1}
3
n
+
1
divides
2
3
n
+
1
2^{3^n} + 1
2
3
n
+
1
; (ii) Prove that
3
n
+
2
3^{n+2}
3
n
+
2
does not divide
2
3
n
+
1
2^{3^n} + 1
2
3
n
+
1
for any positive integer
n
n
n
.
5
1
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Find the angle
Triangle
A
B
C
ABC
A
BC
has an incenter
I
I
I
. Let points
X
X
X
,
Y
Y
Y
be located on the line segments
A
B
AB
A
B
,
A
C
AC
A
C
respectively, so that
B
X
⋅
A
B
=
I
B
2
BX \cdot AB = IB^2
BX
⋅
A
B
=
I
B
2
and
C
Y
⋅
A
C
=
I
C
2
CY \cdot AC = IC^2
C
Y
⋅
A
C
=
I
C
2
. Given that the points
X
,
I
,
Y
X, I, Y
X
,
I
,
Y
lie on a straight line, find the possible values of the measure of angle
A
A
A
.
4
1
Hide problems
Simple inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers with
0
<
a
<
1
0 < a< 1
0
<
a
<
1
,
0
<
b
<
1
0 < b < 1
0
<
b
<
1
,
0
<
c
<
1
0 < c < 1
0
<
c
<
1
, and
a
+
b
+
c
=
2
a+b + c = 2
a
+
b
+
c
=
2
. Prove that
a
1
−
a
⋅
b
1
−
b
⋅
c
1
−
c
≥
8
\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8
1
−
a
a
⋅
1
−
b
b
⋅
1
−
c
c
≥
8
.
3
1
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Simple trignometry
Given a triangle
A
B
C
ABC
A
BC
let \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) \\ y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) \\ z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that
x
+
y
+
z
+
x
y
z
=
0
x+ y + z + xyz = 0
x
+
y
+
z
+
x
yz
=
0
.
2
1
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Prove an area
Given an acute-angled triangle
A
B
C
ABC
A
BC
, let points
A
′
,
B
′
,
C
′
A' , B' , C'
A
′
,
B
′
,
C
′
be located as follows:
A
′
A'
A
′
is the point where altitude from
A
A
A
on
B
C
BC
BC
meets the outwards-facing semicircle on
B
C
BC
BC
as diameter. Points
B
′
,
C
′
B', C'
B
′
,
C
′
are located similarly. Prove that
A
[
B
C
A
′
]
2
+
A
[
C
A
B
′
]
2
+
A
[
A
B
C
′
]
2
=
A
[
A
B
C
]
2
A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2
A
[
BC
A
′
]
2
+
A
[
C
A
B
′
]
2
+
A
[
A
B
C
′
]
2
=
A
[
A
BC
]
2
where
A
[
A
B
C
]
A[ABC]
A
[
A
BC
]
is the area of triangle
A
B
C
ABC
A
BC
.
1
1
Hide problems
Find the no of Integers
Find the number of positive integers
n
n
n
for which (i)
n
≤
1991
n \leq 1991
n
≤
1991
; (ii) 6 is a factor of
(
n
2
+
3
n
+
2
)
(n^2 + 3n +2)
(
n
2
+
3
n
+
2
)
.