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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2005 India Regional Mathematical Olympiad
2005 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Quadratic returns!
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three positive real numbers such that
a
+
b
+
c
=
1
a+ b +c =1
a
+
b
+
c
=
1
. Let
λ
=
m
i
n
{
a
3
+
a
2
b
c
,
b
3
+
b
2
a
c
,
c
3
+
a
b
c
2
}
\lambda = min \{ a^3 + a^2bc , b^3 + b^2 ac , c^3 + ab c^2 \}
λ
=
min
{
a
3
+
a
2
b
c
,
b
3
+
b
2
a
c
,
c
3
+
ab
c
2
}
Prove that the roots of
x
2
+
x
+
4
λ
=
0
x^2 + x + 4 \lambda = 0
x
2
+
x
+
4
λ
=
0
are real.
6
1
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Triples
Determine all triples of positive integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
a
≤
b
≤
c
a \leq b \leq c
a
≤
b
≤
c
and
a
+
b
+
c
+
a
b
+
b
c
+
c
a
=
a
b
c
+
1
a +b + c + ab+ bc +ca = abc +1
a
+
b
+
c
+
ab
+
b
c
+
c
a
=
ab
c
+
1
.
5
1
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Triangle
In a triangle ABC, D is midpoint of BC . If
∠
A
D
B
=
4
5
∘
\angle ADB = 45 ^{\circ}
∠
A
D
B
=
4
5
∘
and
∠
A
C
D
=
3
0
∘
\angle ACD = 30^{\circ}
∠
A
C
D
=
3
0
∘
, determine
∠
B
A
D
.
\angle BAD.
∠
B
A
D
.
4
1
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No. of nos
Find the number of 5-digit numbers that each contains the block '15' and is divisible by 15.
3
1
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Ineq 2
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive three real numbers such that
∣
a
−
b
∣
≥
c
,
∣
b
−
c
∣
≥
a
,
∣
c
−
a
∣
≥
b
| a-b | \geq c , | b-c | \geq a, | c-a | \geq b
∣
a
−
b
∣
≥
c
,
∣
b
−
c
∣
≥
a
,
∣
c
−
a
∣
≥
b
. Prove that one of
a
,
b
,
c
a,b,c
a
,
b
,
c
is equal to the sum of the other two.
2
1
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Divides
If
x
,
y
x,y
x
,
y
are integers and
17
17
17
divides both
x
2
−
2
x
y
+
y
2
−
5
x
+
7
y
x^2 -2xy + y^2 -5x + 7y
x
2
−
2
x
y
+
y
2
−
5
x
+
7
y
and
x
2
−
3
x
y
+
2
y
2
+
x
−
y
x^2 - 3xy + 2y^2 + x - y
x
2
−
3
x
y
+
2
y
2
+
x
−
y
, then prove that
17
17
17
divides
x
y
−
12
x
+
15
y
xy - 12x + 15y
x
y
−
12
x
+
15
y
.
1
1
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Rhombus
Let ABCD be a convex quadrilateral; P,Q, R,S are the midpoints of AB, BC, CD, DA respectively such that triangles AQR, CSP are equilateral. Prove that ABCD is a rhombus. Find its angles.