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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1961 Poland - Second Round
1961 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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tape with covers max area of a triangle
A tape with width
d
<
A
B
d < AB
d
<
A
B
and edges perpendicular to
A
B
AB
A
B
moves in the plane of the acute-angled triangle
A
B
C
ABC
A
BC
. At what position of the tape will it cover the largest part of the triangle?
5
1
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a + b + c> 0, ab + b, c + ca > 0, abc > 0
Prove that if the real numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy the inequalities
a
+
b
+
c
>
0
,
a + b + c> 0,
a
+
b
+
c
>
0
,
a
b
+
b
c
+
c
a
>
0
ab + bc + ca > 0
ab
+
b
c
+
c
a
>
0
a
b
c
>
0
abc > 0
ab
c
>
0
then
a
>
0
,
b
>
0
,
c
>
0
a > 0, b > 0, c > 0
a
>
0
,
b
>
0
,
c
>
0
.
4
1
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last four digits of 5^{5555}
Find the last four digits of
5
5555
5^{5555}
5
5555
.
3
1
Hide problems
1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z = ...
Prove that for any angles
x
,
y
,
z
x,y,z
x
,
y
,
z
holds the equality
1
−
cos
2
x
−
cos
2
y
−
y
−
cos
2
z
+
2
cos
x
cos
y
cos
z
=
4
sin
x
+
y
+
z
2
sin
x
+
y
−
z
2
sin
x
−
y
+
z
2
sin
−
x
−
y
+
z
2
.
1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}.
1
−
cos
2
x
−
cos
2
y
−
y
−
cos
2
z
+
2
cos
x
cos
y
cos
z
=
4
sin
2
x
+
y
+
z
sin
2
x
+
y
−
z
sin
2
x
−
y
+
z
sin
2
−
x
−
y
+
z
.
2
1
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concurrent altitudes in tetrahedron
Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.
1
1
Hide problems
2^n = not sum of consecutives
Prove that no number of the form
2
n
2^n
2
n
, where
n
n
n
is a natural number, is the sum of two or more consecutive natural numbers.