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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1999 India Regional Mathematical Olympiad
1999 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Find the number of quadratics
Find the number of quadratic polynomials
a
x
2
+
b
x
+
c
ax^2 + bx +c
a
x
2
+
b
x
+
c
which satisfy the following: (a)
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct; (b)
a
,
b
,
c
∈
{
1
,
2
,
3
,
⋯
1999
}
a,b,c \in \{ 1,2,3,\cdots 1999 \}
a
,
b
,
c
∈
{
1
,
2
,
3
,
⋯
1999
}
; (c)
x
+
1
x+1
x
+
1
divides
a
x
2
+
b
x
+
c
ax^2 + bx+c
a
x
2
+
b
x
+
c
.
6
1
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An equation - good
Find all solutions in integers
m
,
n
m,n
m
,
n
of the equation
(
m
−
n
)
2
=
4
m
n
m
+
n
−
1
.
(m-n)^2 = \frac{4mn}{ m+n-1}.
(
m
−
n
)
2
=
m
+
n
−
1
4
mn
.
5
1
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Well known triangle ineq
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are sides of a triangle, prove that
a
c
+
a
−
b
+
b
a
+
b
−
c
+
c
b
+
c
−
a
≥
3.
\frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3.
c
+
a
−
b
a
+
a
+
b
−
c
b
+
b
+
c
−
a
c
≥
3.
4
1
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A simple cubic
If
p
,
q
,
r
p,q,r
p
,
q
,
r
are the roots of the cubic equation
x
3
−
3
p
x
2
+
3
q
2
x
−
r
3
=
0
x^3 - 3px^2 + 3q^2 x - r^3 = 0
x
3
−
3
p
x
2
+
3
q
2
x
−
r
3
=
0
, then show that
p
=
q
=
r
p = q =r
p
=
q
=
r
.
3
1
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A simple geometry
Let
A
B
C
D
ABCD
A
BC
D
be a square and
M
,
N
M,N
M
,
N
points on sides
A
B
,
B
C
AB, BC
A
B
,
BC
respectively such that
∠
M
D
N
=
4
5
∘
\angle MDN = 45^{\circ}
∠
M
D
N
=
4
5
∘
. If
R
R
R
is the midpoint of
M
N
MN
MN
show that
R
P
=
R
Q
RP =RQ
RP
=
RQ
where
P
,
Q
P,Q
P
,
Q
are points of intersection of
A
C
AC
A
C
with the lines
M
D
,
N
D
MD, ND
M
D
,
N
D
.
2
1
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No of divisors
Find the number of positive integers which divide
1
0
999
10^{999}
1
0
999
but not
1
0
998
10^{998}
1
0
998
.
1
1
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A simple on on in circles
Prove that the inradius of a right angled triangle with integer sides is an integer.