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Problems
Contests
International Contests
Danube Competition in Mathematics
2012 Danube Mathematical Competition
2012 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
2
2
Hide problems
randomly remove 2012 numbers from decimal number 1/p, after the comma
Consider the natural number prime
p
,
p
>
5
p, p> 5
p
,
p
>
5
. From the decimal number
1
p
\frac1p
p
1
, randomly remove
2012
2012
2012
numbers, after the comma. Show that the remaining number can be represented as
a
b
\frac{a}{b}
b
a
, where
a
a
a
and
b
b
b
are coprime numbers , and
b
b
b
is multiple of
p
p
p
.
orthocenter of a triangle is circumenter of another when triangles are similar
Let
A
B
C
ABC
A
BC
be an acute triangle and let
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
be points on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively. Show that the triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
are similar (
∠
A
=
∠
A
1
,
∠
B
=
∠
B
1
,
∠
C
=
∠
C
1
\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1
∠
A
=
∠
A
1
,
∠
B
=
∠
B
1
,
∠
C
=
∠
C
1
) if and only if the orthocentre of the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and the circumcentre of the triangle
A
B
C
ABC
A
BC
coincide.
4
2
Hide problems
7 elements of set {1,2,3, ...,26}, exist 2 with the same sum
Let
A
A
A
be a subset with seven elements of the set
{
1
,
2
,
3
,
.
.
.
,
26
}
\{1,2,3, ...,26\}
{
1
,
2
,
3
,
...
,
26
}
. Show that there are two distinct elements of
A
A
A
, having the same sum of their elements.
partition set of n elements in m sets with same sum of elements, criterion
Given a positive integer
n
n
n
, show that the set
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
can be partitioned into
m
m
m
sets, each with the same sum, if and only if m is a divisor of
n
(
n
+
1
)
2
\frac{n(n + 1)}{2}
2
n
(
n
+
1
)
which does not exceed
n
+
1
2
\frac{n + 1}{2}
2
n
+
1
.
1
2
Hide problems
ab+1,bc+1, cd+1, da+1 simultaneously perfect square
a) Exist
a
,
b
,
c
,
∈
N
a, b, c, \in N
a
,
b
,
c
,
∈
N
, such that the numbers
a
b
+
1
,
b
c
+
1
ab+1,bc+1
ab
+
1
,
b
c
+
1
and
c
a
+
1
ca+1
c
a
+
1
are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two)
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
, so that the numbers
a
b
+
1
,
b
c
+
1
,
c
d
+
1
ab+1,bc+1, cd+1
ab
+
1
,
b
c
+
1
,
c
d
+
1
and
d
a
+
1
da+1
d
a
+
1
are simultaneously perfect squares.
max no of lattice points that covers a square of side n +1/(2n+1)
Given a positive integer
n
n
n
, determine the maximum number of lattice points in the plane a square of side length
n
+
1
2
n
+
1
n +\frac{1}{2n+1}
n
+
2
n
+
1
1
may cover.
3
2
Hide problems
equal angles in a right triangle, danube junior geometry 2012
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
9
0
o
\angle BAC = 90^o
∠
B
A
C
=
9
0
o
. Angle bisector of the
∠
C
B
A
\angle CBA
∠
CB
A
intersects the segment
(
A
B
)
(AB)
(
A
B
)
at point
E
E
E
. If there exists
D
∈
(
C
E
)
D \in (CE)
D
∈
(
CE
)
so that
∠
D
A
C
=
∠
B
D
E
=
x
o
\angle DAC = \angle BDE =x^o
∠
D
A
C
=
∠
B
D
E
=
x
o
, calculate
x
x
x
.
Danube 2012 #3
Let
p
p
p
and
q
,
p
<
q
,
q, p < q,
q
,
p
<
q
,
be two primes such that
1
+
p
+
p
2
+
.
.
.
+
p
m
1 + p + p^2+...+p^m
1
+
p
+
p
2
+
...
+
p
m
is a power of
q
q
q
for some positive integer
m
m
m
, and
1
+
q
+
q
2
+
.
.
.
+
q
n
1 + q + q^2+...+q^n
1
+
q
+
q
2
+
...
+
q
n
is a power of
p
p
p
for some positive integer
n
n
n
. Show that
p
=
2
p = 2
p
=
2
and
q
=
2
t
−
1
q = 2^t-1
q
=
2
t
−
1
where
t
t
t
is prime.