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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1972 Poland - Second Round
1972 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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|f'(x) - f'(y)| <= 4|x-y| , differentiable
Prove that there exists a function
f
f
f
defined and differentiable in the set of all real numbers, satisfying the conditions
∣
f
′
(
x
)
−
f
′
(
y
)
∣
≤
4
∣
x
−
y
∣
|f'(x) - f'(y)| \leq 4|x-y|
∣
f
′
(
x
)
−
f
′
(
y
)
∣
≤
4∣
x
−
y
∣
.
5
1
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concurrent perpendiculars in cyclic quad
Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.
4
1
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n^3 unit cubes partiton a cube
A cube with edge length
n
n
n
is divided into
n
3
n^3
n
3
unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?
3
1
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diameter of circumcircle <= product of sides, lattice points
The coordinates of the triangle's vertices in the Cartesian system
X
O
Y
XOY
XO
Y
are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.
2
1
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120 unit squares in 20x25 rectangle, circle of diameter 1
In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.
1
1
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ax^2 + bx + c = a(x - x_2)(x - x_3), 3x3 system
Prove that there are no real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
,
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
such that for every real number
x
x
x
a
x
2
+
b
x
+
c
=
a
(
x
−
x
2
)
(
x
−
x
3
)
ax^2 + bx + c = a(x - x_2)(x - x_3)
a
x
2
+
b
x
+
c
=
a
(
x
−
x
2
)
(
x
−
x
3
)
b
x
2
+
c
x
+
a
=
b
(
x
−
x
3
)
(
x
−
x
1
)
bx^2 + cx + a = b(x - x_3) (x - x_1)
b
x
2
+
c
x
+
a
=
b
(
x
−
x
3
)
(
x
−
x
1
)
c
x
2
+
a
x
+
b
=
c
(
x
−
x
1
)
(
x
−
x
2
)
cx^2 + ax + b = c(x - x_1) (x - x_2)
c
x
2
+
a
x
+
b
=
c
(
x
−
x
1
)
(
x
−
x
2
)
and
x
1
≠
x
2
x_1 \neq x_2
x
1
=
x
2
,
x
2
≠
x
3
x_2 \neq x_3
x
2
=
x
3
,
x
3
≠
x
1
x_3 \neq x_1
x
3
=
x
1
,
a
b
c
≠
0
abc \neq 0
ab
c
=
0
.