MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
1995 India Regional Mathematical Olympiad
1995 India Regional Mathematical Olympiad
Part of
Regional Mathematical Olympiad
Subcontests
(7)
7
1
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Another trig ineq
Show that for any real number
x
x
x
:
x
2
sin
x
+
x
cos
x
+
x
2
+
1
2
>
0.
x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0 .
x
2
sin
x
+
x
cos
x
+
x
2
+
2
1
>
0.
6
1
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A 21-gon
Let
A
1
A
2
A
3
…
A
21
A_1A_2A_3 \ldots A_{21}
A
1
A
2
A
3
…
A
21
be a 21-sided regular polygon inscribed in a circle with centre
O
O
O
. How many triangles
A
i
A
j
A
k
A_iA_jA_k
A
i
A
j
A
k
,
1
≤
i
<
j
<
k
≤
21
1 \leq i < j < k \leq 21
1
≤
i
<
j
<
k
≤
21
, contain the centre point
O
O
O
in their interior?
5
1
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A triangle ineq - well known
Show that for any triangle
A
B
C
ABC
A
BC
, the following inequality is true:
a
2
+
b
2
+
c
2
>
3
m
a
x
{
∣
a
2
−
b
2
∣
,
∣
b
2
−
c
2
∣
,
∣
c
2
−
a
2
∣
}
.
a^2 + b^2 +c^2 > \sqrt{3} max \{ |a^2 - b^2|, |b^2 -c^2|, |c^2 -a^2| \} .
a
2
+
b
2
+
c
2
>
3
ma
x
{
∣
a
2
−
b
2
∣
,
∣
b
2
−
c
2
∣
,
∣
c
2
−
a
2
∣
}
.
4
1
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An integer root
Show that the quadratic equation
x
2
+
7
x
−
14
(
q
2
+
1
)
=
0
x^2 + 7x - 14 (q^2 +1) =0
x
2
+
7
x
−
14
(
q
2
+
1
)
=
0
, where
q
q
q
is an integer, has no integer root.
3
1
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Php
Prove that among any
18
18
18
consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.
2
1
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Good integers
Call a positive integer
n
n
n
good if there are
n
n
n
integers, positive or negative, and not necessarily distinct, such that their sum and products are both equal to
n
n
n
. Show that the integers of the form
4
k
+
1
4k+1
4
k
+
1
and
4
l
4l
4
l
are good.
1
1
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Perpendicularity
In triangle
A
B
C
ABC
A
BC
,
K
K
K
and
L
L
L
are points on the side
B
C
BC
BC
(
K
K
K
being closer to
B
B
B
than
L
L
L
) such that
B
C
⋅
K
L
=
B
K
⋅
C
L
BC \cdot KL = BK \cdot CL
BC
⋅
K
L
=
B
K
⋅
C
L
and
A
L
AL
A
L
bisects
∠
K
A
C
\angle KAC
∠
K
A
C
. Show that
A
L
⊥
A
B
.
AL \perp AB.
A
L
⊥
A
B
.