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India Contests
Postal Coaching
2012 Postal Coaching
2012 Postal Coaching
Part of
Postal Coaching
Subcontests
(5)
5
1
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Triangle Geometry
In triangle
A
B
C
ABC
A
BC
, \angle BAC = 94^{\circ},\ \angle ACB = 39^{\circ}. Prove that
B
C
2
=
A
C
2
+
A
C
⋅
A
B
BC^2 = AC^2 + AC\cdot AB
B
C
2
=
A
C
2
+
A
C
⋅
A
B
.
4
1
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Combinatorial Geometry show distances equal
Choose arbitrarily
n
n
n
vertices of a regular
2
n
−
2n-
2
n
−
gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.
3
1
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Show identity involving logarithmic and power functions
Given an integer
n
≥
2
n\ge 2
n
≥
2
, prove that
⌊
n
⌋
+
⌊
n
3
⌋
+
⋯
+
⌊
n
n
⌋
=
⌊
log
2
n
⌋
+
⌊
log
3
n
⌋
+
⋯
+
⌊
log
n
n
⌋
\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor
⌊
n
⌋
+
⌊
3
n
⌋
+
⋯
+
⌊
n
n
⌋
=
⌊
lo
g
2
n
⌋
+
⌊
lo
g
3
n
⌋
+
⋯
+
⌊
lo
g
n
n
⌋
.[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)
2
1
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Show that an expression does not divide another
Let a_1, a_2,\cdots ,a_n be positive integers and let
a
a
a
be an integer greater than
1
1
1
and divisible by the product
a
1
a
2
⋯
a
n
a_1a_2\cdots a_n
a
1
a
2
⋯
a
n
. Prove that
a
n
+
1
+
a
−
1
a^{n+1} + a-1
a
n
+
1
+
a
−
1
is not divisible by the product (a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1).
1
1
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Geometric inequality - angle bisector configuration
Given a triangle
A
B
C
ABC
A
BC
, the internal bisectors through
A
A
A
and
B
B
B
meet the opposite sides in
D
D
D
and
E
E
E
, respectively. Prove that DE \le (3 - 2\sqrt2)(AB + BC + CA) and determine the cases of equality.