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Problems
Contests
International Contests
Danube Competition in Mathematics
2017 Danube Mathematical Olympiad
2017 Danube Mathematical Olympiad
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
2
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Diophantine Equation
Determine all triples of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
4
+
y
4
=
2
z
2
x^4+y^4 =2z^2
x
4
+
y
4
=
2
z
2
and
x
x
x
and
y
y
y
are relatively prime.
Problem 4
Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any
n
∗
n
n*n
n
∗
n
square is less or equal than
1
1
1
. Prove that the absolute value of the sum of the numbers from any
m
∗
n
m*n
m
∗
n
rectangular is less or equal than
4
4
4
.
3
2
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Perpendicular Contains Midpoint
Consider an acute triangle
A
B
C
ABC
A
BC
in which
A
1
,
B
1
,
A_1, B_1,
A
1
,
B
1
,
and
C
1
C_1
C
1
are the feet of the altitudes from
A
,
B
,
A, B,
A
,
B
,
and
C
,
C,
C
,
respectively, and
H
H
H
is the orthocenter. The perpendiculars from
H
H
H
onto
A
1
C
1
A_1C_1
A
1
C
1
and
A
1
B
1
A_1B_1
A
1
B
1
intersect lines
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
,
Q,
Q
,
respectively. Prove that the line perpendicular to
B
1
C
1
B_1C_1
B
1
C
1
that passes through
A
A
A
also contains the midpoint of the line segment
P
Q
PQ
PQ
.
Problem 3
Let
O
,
H
O,H
O
,
H
be the circumcenter and the orthocenter of triangle
A
B
C
ABC
A
BC
. Let
F
F
F
be the foot of the perpendicular from C onto AB, and
M
M
M
the midpoint of
C
H
CH
C
H
. Let N be the foot of the perpendicular from C onto the parallel through H at
O
M
OM
OM
. Let
D
D
D
be on
A
B
AB
A
B
such that
C
A
=
C
D
CA=CD
C
A
=
C
D
. Let
B
N
BN
BN
intersect
C
D
CD
C
D
at
P
P
P
. Let
P
H
PH
P
H
intersect
C
A
CA
C
A
at
Q
Q
Q
. Prove that
Q
F
⊥
O
F
QF\perp OF
QF
⊥
OF
.
2
2
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Placing Numbers On A Board
Let
n
≥
3
n\geq 3
n
≥
3
be a positive integer. Consider an
n
×
n
n\times n
n
×
n
square. In each cell of the square, one of the numbers from the set
M
=
{
1
,
2
,
…
,
2
n
−
1
}
M=\{1,2,\ldots,2n-1\}
M
=
{
1
,
2
,
…
,
2
n
−
1
}
is to be written. One such filling is called good if, for every index
1
≤
i
≤
n
,
1\leq i\leq n,
1
≤
i
≤
n
,
row no.
i
i
i
and column no.
i
,
i,
i
,
together, contain all the elements of
M
M
M
.[*]Prove that there exists
n
≥
3
n\geq 3
n
≥
3
for which a good filling exists. [*]Prove that for
n
=
2017
n=2017
n
=
2017
there is no good filling of the
n
×
n
n\times n
n
×
n
square.
Problem 2
Let n be a positive interger. Let n real numbers be wrote on a paper. We call a "transformation" :choosing 2 numbers
a
,
b
a,b
a
,
b
and replace both of them with
a
∗
b
a*b
a
∗
b
. Find all n for which after a finite number of transformations and any n real numbers, we can have the same number written n times on the paper.
1
2
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Smallest Sum Of Digits
What is the smallest value that the sum of the digits of the number
3
n
2
+
n
+
1
,
3n^2+n+1,
3
n
2
+
n
+
1
,
n
∈
N
n\in\mathbb{N}
n
∈
N
can take?
Problem 1-Polynomial
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that
a
2
+
b
2
−
c
2
a^2+b^2-c^2
a
2
+
b
2
−
c
2
divides
P
(
a
)
+
P
(
b
)
−
P
(
c
)
P(a)+P(b)-P(c)
P
(
a
)
+
P
(
b
)
−
P
(
c
)
, for all integers
a
,
b
,
c
a,b,c
a
,
b
,
c
.