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Problems
Contests
National and Regional Contests
Albania Contests
Albania Round 2
2014 Albania Round 2
2014 Albania Round 2
Part of
Albania Round 2
Subcontests
(5)
5
1
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angles satisfying an equation
Prove that if the angles
α
\alpha
α
and
β
\beta
β
satisfy
sin
(
α
+
β
)
=
2
sin
α
\sin(\alpha + \beta) = 2 \sin \alpha
sin
(
α
+
β
)
=
2
sin
α
, Then
α
<
β
\alpha < \beta
α
<
β
4
1
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Number theory, Trigonometry with logarithms
Solve the equation,
sin
(
π
log
x
)
+
cos
(
π
log
x
)
=
1
\sin (\pi \log x) + \cos (\pi \log x) = 1
sin
(
π
lo
g
x
)
+
cos
(
π
lo
g
x
)
=
1
3
1
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Radii of incircles in right angled triangle.......easy geometry
In a right
Δ
A
B
C
\Delta ABC
Δ
A
BC
(
∠
C
=
9
0
∘
\angle C = 90^{\circ}
∠
C
=
9
0
∘
),
C
D
CD
C
D
is the height. Let
r
1
r_1
r
1
and
r
2
r_2
r
2
be the radii of inscribed circles of
Δ
A
C
D
\Delta ACD
Δ
A
C
D
and
Δ
D
C
B
\Delta DCB
Δ
D
CB
. Find the radius of inscribed circle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
2
1
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Sides of triangle in AP
Sides of a triangle form an arithmetic sequence with common difference
2
2
2
, and its area is
6
cm
2
6 \text{ cm }^2
6
cm
2
. Find its sides.
1
1
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Easy algebraic equation, Solve for x
Solve the equation,
x
+
5
+
16
−
x
2
=
x
2
−
25
\sqrt{x+5}+\sqrt{16-x^2}=x^2-25
x
+
5
+
16
−
x
2
=
x
2
−
25