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Contests
International Contests
Danube Competition in Mathematics
2016 Danube Mathematical Olympiad
2016 Danube Mathematical Olympiad
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
2
Hide problems
Tiling A Special Board
A unit square is removed from the corner of an
n
×
n
n\times n
n
×
n
grid where
n
≥
2
n \geq 2
n
≥
2
. Prove that the remainder can be covered by copies of the "L-shapes" consisting of
3
3
3
or
5
5
5
unit square, as depicted in the figure below. Every square must be covered once and the L-shapes must not go over the bounds of the grid. [asy] import geometry;draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0));draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy]Estonian Olympiad, 2009
Problem 4, Number Theory
4.Prove that there exist only finitely many positive integers n such that
(
n
1
+
1
)
(
n
2
+
2
)
.
.
.
(
n
n
+
n
)
(\frac{n}{1}+1)(\frac{n}{2}+2)...(\frac{n}{n}+n)
(
1
n
+
1
)
(
2
n
+
2
)
...
(
n
n
+
n
)
is an integer.
3
2
Hide problems
Smallest Value Of An Angle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
,
AB < AC,
A
B
<
A
C
,
I
I
I
its incenter, and
M
M
M
the midpoint of the side
B
C
BC
BC
. If
I
A
=
I
M
,
IA=IM,
I
A
=
I
M
,
determine the smallest possible value of the angle
A
I
M
AIM
A
I
M
.
Problem 3, Inequality with Integral
3. Let n > 1 be an integer and
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, . . . , a_n
a
1
,
a
2
,
...
,
a
n
be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that
∑
a
k
1
+
(
a
0
+
a
1
+
.
.
.
+
a
k
−
1
)
2
≥
c
\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c
∑
1
+
(
a
0
+
a
1
+
...
+
a
k
−
1
)
2
a
k
≥
c
, where
a
0
=
0
a_0 = 0
a
0
=
0
. b) Show that ’the best’ value of c is at least
π
4
\frac{\pi}{4}
4
π
.
1
2
Hide problems
Can The Sum Be 2016?
Let
S
=
x
1
x
2
+
x
3
x
4
+
⋯
+
x
2015
x
2016
,
S=x_1x_2+x_3x_4+\cdots+x_{2015}x_{2016},
S
=
x
1
x
2
+
x
3
x
4
+
⋯
+
x
2015
x
2016
,
where
x
1
,
x
2
,
…
,
x
2016
∈
{
3
−
2
,
3
+
2
}
.
x_1,x_2,\ldots,x_{2016}\in\{\sqrt{3}-\sqrt{2},\sqrt{3}+\sqrt{2}\}.
x
1
,
x
2
,
…
,
x
2016
∈
{
3
−
2
,
3
+
2
}
.
Can
S
S
S
be equal to
2016
?
2016?
2016
?
Cristian Lazăr
Problem 1, Geometric Inequality
1.Let
A
B
C
ABC
A
BC
be a triangle,
D
D
D
the foot of the altitude from
A
A
A
and
M
M
M
the midpoint of the side
B
C
BC
BC
. Let
S
S
S
be a point on the closed segment
D
M
DM
D
M
and let
P
,
Q
P, Q
P
,
Q
the projections of
S
S
S
on the lines
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Prove that the length of the segment
P
Q
PQ
PQ
does not exceed one quarter the perimeter of the triangle
A
B
C
ABC
A
BC
.
2
2
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Divisors Yield Primes
Determine all positive integers
n
>
1
n>1
n
>
1
such that for any divisor
d
d
d
of
n
,
n,
n
,
the numbers
d
2
−
d
+
1
d^2-d+1
d
2
−
d
+
1
and
d
2
+
d
+
1
d^2+d+1
d
2
+
d
+
1
are prime.Lucian Petrescu
Combinatorics
A bank has a set S of codes formed only with 0 and 1,each one with length n.Two codes are 'friends' if they are different on only one position.We know that each code has exactly k 'friends'.Prove that: 1)S has an even number of elements 2)S contains at least
2
k
2^k
2
k
codes