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Problems
Contests
International Contests
Danube Competition in Mathematics
2013 Danube Mathematical Competition
2013 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
2
2
Hide problems
a+b-c-d be a multiple of 2013, 6 numbers from a set of 63 < = 2012
Consider
64
64
64
distinct natural numbers, at most equal to
2012
2012
2012
. Show that it is possible to choose four of them, denoted as
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
a
+
b
−
c
−
d
a+b-c-d
a
+
b
−
c
−
d
to be a multiple of
2013
2013
2013
p divides a^2+ab+b^2 and a^n+b^n+c^n, but not a+b+c, prove n, p-1 not coprime
Let
a
,
b
,
c
,
n
a, b, c, n
a
,
b
,
c
,
n
be four integers, where n
≥
2
\ge 2
≥
2
, and let
p
p
p
be a prime dividing both
a
2
+
a
b
+
b
2
a^2+ab+b^2
a
2
+
ab
+
b
2
and
a
n
+
b
n
+
c
n
a^n+b^n+c^n
a
n
+
b
n
+
c
n
, but not
a
+
b
+
c
a+b+c
a
+
b
+
c
. for instance,
a
≡
b
≡
−
1
(
m
o
d
3
)
,
c
≡
1
(
m
o
d
3
)
,
n
a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n
a
≡
b
≡
−
1
(
m
o
d
3
)
,
c
≡
1
(
m
o
d
3
)
,
n
a positive even integer, and
p
=
3
p = 3
p
=
3
or
a
=
4
,
b
=
7
,
c
=
−
13
,
n
=
5
a = 4, b = 7, c = -13, n = 5
a
=
4
,
b
=
7
,
c
=
−
13
,
n
=
5
, and
p
=
31
p = 31
p
=
31
satisfy these conditions. Show that
n
n
n
and
p
−
1
p - 1
p
−
1
are not coprime.
3
2
Hide problems
diophantine 85^m-n^4=4
Determine the natural numbers
m
,
n
m,n
m
,
n
such as
8
5
m
−
n
4
=
4
85^m-n^4=4
8
5
m
−
n
4
=
4
r-chromatic simple graph with no cycle of < =6 edges
Show that, for every integer
r
≥
2
r \ge 2
r
≥
2
, there exists an
r
r
r
-chromatic simple graph (no loops, nor multiple edges) which has no cycle of less than
6
6
6
edges
1
2
Hide problems
Sum x_i = Sum 1/xi =0, from i=1 to n, find n
Determine the natural numbers
n
≥
2
n\ge 2
n
≥
2
for which exist
x
1
,
x
2
,
.
.
.
,
x
n
∈
R
∗
x_1,x_2,...,x_n \in R^*
x
1
,
x
2
,
...
,
x
n
∈
R
∗
, such that
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
=
0
x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0
x
1
+
x
2
+
...
+
x
n
=
x
1
1
+
x
2
1
+
...
+
x
n
1
=
0
if A,a,B,b,C, c concyclic then Pascal lines of AaBbCc, AbBcCa, AcBaCa concurrent
Given six points on a circle,
A
,
a
,
B
,
b
,
C
,
c
A, a, B, b, C, c
A
,
a
,
B
,
b
,
C
,
c
, show that the Pascal lines of the hexagrams
A
a
B
b
C
c
,
A
b
B
c
C
a
,
A
c
B
a
C
b
AaBbCc, AbBcCa, AcBaCb
A
a
B
b
C
c
,
A
b
B
c
C
a
,
A
c
B
a
C
b
are concurrent.
4
2
Hide problems
perpendicularity starting with a rectangle, danube junior 2013
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
A
B
≠
B
C
AB \ne BC
A
B
=
BC
and the center the point
O
O
O
. Perpendicular from
O
O
O
on
B
D
BD
B
D
intersects lines
A
B
AB
A
B
and
B
C
BC
BC
in points
E
E
E
and
F
F
F
respectively. Points
M
M
M
and
N
N
N
are midpoints of segments
[
C
D
]
[CD]
[
C
D
]
and
[
A
D
]
[AD]
[
A
D
]
respectively. Prove that
F
M
⊥
E
N
FM \perp EN
FM
⊥
EN
.
set {nx + S : n \in N}$ is finite, where nx + S ={nx + s : s \in S}
Show that there exists a proper non-empty subset
S
S
S
of the set of real numbers such that, for every real number
x
x
x
, the set
{
n
x
+
S
:
n
∈
N
}
\{nx + S : n \in N\}
{
n
x
+
S
:
n
∈
N
}
is finite, where
n
x
+
S
=
{
n
x
+
s
:
s
∈
S
}
nx + S =\{nx + s : s \in S\}
n
x
+
S
=
{
n
x
+
s
:
s
∈
S
}