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Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad Second Round
2017 Polish Junior Math Olympiad Second Round
Part of
2017 Polish Junior Math Olympiad
Subcontests
(5)
1.
1
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2017 Polish Junior Math Olympiad Second Round P1
In each square of a
4
×
4
4\times 4
4
×
4
board, we are to write an integer in such a way that the sums of the numbers in each column and in each row are nonnegative integral powers of
2
2
2
. Is it possible to do this in such a way that every two of these eight sums are different? Justify your answer.
2.
1
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2017 Polish Junior Math Olympiad Second Round P2
Prove that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum of the lengths of the sides of this trapezoid.
3.
1
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2017 Polish Junior Math Olympiad Second Round P3
Let
a
a
a
,
b
b
b
, and
d
d
d
be positive integers. It is known that
a
+
b
a+b
a
+
b
is divisible by
d
d
d
and
a
⋅
b
a\cdot b
a
⋅
b
is divisible by
d
2
d^2
d
2
. Prove that both
a
a
a
and
b
b
b
are divisible by
d
d
d
.
4.
1
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2017 Polish Junior Math Olympiad Second Round P4
Do numbers
x
1
,
x
2
,
…
,
x
99
x_1, x_2, \ldots, x_{99}
x
1
,
x
2
,
…
,
x
99
exist, where each of them is equal to
2
+
1
\sqrt{2}+1
2
+
1
or
2
−
1
\sqrt{2}-1
2
−
1
, and satisfy the equation
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
…
+
x
98
x
99
+
x
99
x
1
=
199
?
x_1x_2+x_2x_3+x_3x_4+\ldots+x_{98}x_{99}+x_{99}x_1=199\,?
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
…
+
x
98
x
99
+
x
99
x
1
=
199
?
Justify your answer.
5.
1
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2017 Polish Junior Math Olympiad Second Round P5
Does there exist a convex polyhedron in which each internal angle of each of its faces is either a right angle or an obtuse angle, and which has exactly
100
100
100
edges? Justify your answer.