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Problems
Contests
International Contests
Danube Competition in Mathematics
2015 Danube Mathematical Competition
2015 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(5)
4
2
Hide problems
prove that the rectangle is a square, double angle is given
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
A
B
≥
B
C
AB\ge BC
A
B
≥
BC
Point
M
M
M
is located on the side
(
A
D
)
(AD)
(
A
D
)
, and the perpendicular bisector of
[
M
C
]
[MC]
[
MC
]
intersects the line
B
C
BC
BC
at the point
N
N
N
. Let
Q
=
M
N
∪
A
B
{Q} =MN\cup AB
Q
=
MN
∪
A
B
. Knowing that
∠
M
Q
A
=
2
⋅
∠
B
C
Q
\angle MQA= 2\cdot \angle BCQ
∠
MQ
A
=
2
⋅
∠
BCQ
, show that the quadrilateral
A
B
C
D
ABCD
A
BC
D
is a square.
Nice problem
Given an integer
n
≥
2
n \ge 2
n
≥
2
,determine the numbers that written in the form
a
1
a_1
a
1
a
2
a_2
a
2
+
+
+
a
2
a_2
a
2
a
3
a_3
a
3
+
+
+
.
.
.
...
...
a
k
−
1
a_{k-1}
a
k
−
1
a
k
a_k
a
k
, where
k
k
k
is an integer greater than or equal to 2, and
a
1
a_1
a
1
,...
a
k
a_k
a
k
are positive integers with sum
n
n
n
.
5
1
Hide problems
A lantern and $2n$ batteries...
A lantern needs exactly
2
2
2
charged batteries in order to work.We have available
n
n
n
charged batteries and
n
n
n
uncharged batteries,
n
≥
4
n\ge 4
n
≥
4
(all batteries look the same). A try consists in introducing two batteries in the lantern and verifying if the lantern works.Prove that we can find a pair of charged batteries in at most
n
+
2
n+2
n
+
2
tries.
2
2
Hide problems
5 subsets with some propriety...
Consider the set
A
=
{
1
,
2
,
.
.
.
,
120
}
A=\{1,2,...,120\}
A
=
{
1
,
2
,
...
,
120
}
and
M
M
M
a subset of
A
A
A
such that
∣
M
∣
=
30
|M|=30
∣
M
∣
=
30
.Prove that there are
5
5
5
different subsets of
M
M
M
,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.
a simple finite graph problem
Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even
1
2
Hide problems
Trunk numbers
Consider a positive integer
n
=
a
1
a
2
.
.
.
a
k
‾
,
k
≥
2
n=\overline{a_1a_2...a_k},k\ge 2
n
=
a
1
a
2
...
a
k
,
k
≥
2
.A trunk of
n
n
n
is a number of the form
a
1
a
2
.
.
.
a
t
‾
,
1
≤
t
≤
k
−
1
\overline{a_1a_2...a_t},1\le t\le k-1
a
1
a
2
...
a
t
,
1
≤
t
≤
k
−
1
.(For example,the number
23
23
23
is a trunk of
2351
2351
2351
.) By
T
(
n
)
T(n)
T
(
n
)
we denote the sum of all trunk of
n
n
n
and let
S
(
n
)
=
a
1
+
a
2
+
.
.
.
+
a
k
S(n)=a_1+a_2+...+a_k
S
(
n
)
=
a
1
+
a
2
+
...
+
a
k
.Prove that
n
=
S
(
n
)
+
9
⋅
T
(
n
)
n=S(n)+9\cdot T(n)
n
=
S
(
n
)
+
9
⋅
T
(
n
)
.
cyclic ABCD, diagonals intersection, 2 incenters, isosceles wanted
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrangle, let the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
cross at
O
O
O
, and let
I
I
I
and
J
J
J
be the incentres of the triangles
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
, respectively. The line
I
J
IJ
I
J
crosses the segments
O
A
OA
O
A
and
O
B
OB
OB
at
M
M
M
and
N
N
N
, respectively. Prove that the triangle
O
M
N
OMN
OMN
is isosceles.
3
2
Hide problems
Danube Math Competition,juniors problem 3
Solve in N
a
2
=
2
b
3
c
+
1
a^2 = 2^b3^c + 1
a
2
=
2
b
3
c
+
1
.
P48. Romania Danube Mathematical Competition
Determine all positive integers
n
n
n
such that all positive integers less than or equal to
n
n
n
and relatively prime to
n
n
n
are pairwise coprime.