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National and Regional Contests
Poland Contests
Poland - Second Round
1997 Poland - Second Round
1997 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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max distance of 8 points inside a unit cube is 1
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than
1
1
1
.
5
1
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k white and m black dice, equal sums remainders mod 7, probability
We have thrown
k
k
k
white dice and
m
m
m
black dice. Find the probability that the remainder modulo
7
7
7
of the sum of the numbers on the white dice is equal to the remainder modulo
7
7
7
of the sum of the numbers on the black dice.
3
1
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n points, 2 segments with a common endpoint are of different colors, min colours
Let be given
n
n
n
points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible
1
1
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Nice algebra
For the real number
a
a
a
find the number of solutions
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of a system of the equations:
{
x
+
y
2
+
z
2
=
a
x
2
+
y
+
z
2
=
a
x
2
+
y
2
+
z
=
a
\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.
⎩
⎨
⎧
x
+
y
2
+
z
2
=
a
x
2
+
y
+
z
2
=
a
x
2
+
y
2
+
z
=
a
4
1
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set of integers
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.
2
1
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P inside ABC such that 3 < ABP = 3 < ACP = < ABC + < ACB
Let P be a point inside triangle ABC such that 3