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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1949 Kurschak Competition
1949 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
Hide problems
sin x + 1/2 sin 2x + 1/3 sin 3x > 0$
Prove that
sin
x
+
1
2
sin
2
x
+
1
3
sin
3
x
>
0
\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0
sin
x
+
2
1
sin
2
x
+
3
1
sin
3
x
>
0
for
0
<
x
<
18
0
o
0 < x < 180^o
0
<
x
<
18
0
o
.
3
1
Hide problems
positive integers cannot be represented as a sum of consecutive integers
Which positive integers cannot be represented as a sum of (two or more) consecutive integers?
2
1
Hide problems
reflection of P in the line QR lies on the circumcircle of the triangle
P
P
P
is a point on the base of an isosceles triangle. Lines parallel to the sides through
P
P
P
meet the sides at
Q
Q
Q
and
R
R
R
. Show that the reflection of
P
P
P
in the line
Q
R
QR
QR
lies on the circumcircle of the triangle.