MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1995 India National Olympiad
1995 India National Olympiad
Part of
India National Olympiad
Subcontests
(6)
6
1
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Find primes
Find all primes
p
p
p
for which the quotient
2
p
−
1
−
1
p
\dfrac{2^{p-1} - 1 }{p}
p
2
p
−
1
−
1
is a square.
5
1
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Solve summation
Let
n
≥
2
n \geq 2
n
≥
2
. Let
a
1
,
a
2
,
a
3
,
…
a
n
a_1 , a_2 , a_3 , \ldots a_n
a
1
,
a
2
,
a
3
,
…
a
n
be
n
n
n
real numbers all less than
1
1
1
and such that
∣
a
k
−
a
k
+
1
∣
<
1
|a_k - a_{k+1} | < 1
∣
a
k
−
a
k
+
1
∣
<
1
for
1
≤
k
≤
n
−
1
1 \leq k \leq n-1
1
≤
k
≤
n
−
1
. Show that
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
…
+
a
n
−
1
a
n
+
a
n
a
1
<
2
n
−
1.
\dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 .
a
2
a
1
+
a
3
a
2
+
a
4
a
3
+
…
+
a
n
a
n
−
1
+
a
1
a
n
<
2
n
−
1.
4
1
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Find the ratio of radii
Let
A
B
C
ABC
A
BC
be a triangle and a circle
Γ
′
\Gamma'
Γ
′
be drawn lying outside the triangle, touching its incircle
Γ
\Gamma
Γ
externally, and also the two sides
A
B
AB
A
B
and
A
C
AC
A
C
. Show that the ratio of the radii of the circles
Γ
′
\Gamma'
Γ
′
and
Γ
\Gamma
Γ
is equal to
tan
2
(
π
−
A
4
)
.
\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.
tan
2
(
4
π
−
A
)
.
3
1
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No. of subsets
Show that the number of
3
−
3-
3
−
element subsets
{
a
,
b
,
c
}
\{ a , b, c \}
{
a
,
b
,
c
}
of
{
1
,
2
,
3
,
…
,
63
}
\{ 1 , 2, 3, \ldots, 63 \}
{
1
,
2
,
3
,
…
,
63
}
with
a
+
b
+
c
<
95
a+b +c < 95
a
+
b
+
c
<
95
is less than the number of those with
a
+
b
+
c
≥
95.
a + b +c \geq 95.
a
+
b
+
c
≥
95.
2
1
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Quadratic eqns
Show that there are infintely many pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.
1
1
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Triangle property [A = 30° yields AT = 2 * BC]
In an acute angled triangle
A
B
C
ABC
A
BC
,
∠
A
=
3
0
∘
\angle A = 30^{\circ}
∠
A
=
3
0
∘
,
H
H
H
is the orthocenter, and
M
M
M
is the midpoint of
B
C
BC
BC
. On the line
H
M
HM
H
M
, take a point
T
T
T
such that
H
M
=
M
T
HM = MT
H
M
=
MT
. Show that
A
T
=
2
B
C
AT = 2 BC
A
T
=
2
BC
.