MathDB
Problems
Contests
International Contests
Benelux
2014 Benelux
2014 Benelux
Part of
Benelux
Subcontests
(4)
4
1
Hide problems
Arbitrary point P inside square ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a square. Consider a variable point
P
P
P
inside the square for which
∠
B
A
P
≥
6
0
∘
.
\angle BAP \ge 60^\circ.
∠
B
A
P
≥
6
0
∘
.
Let
Q
Q
Q
be the intersection of the line
A
D
AD
A
D
and the perpendicular to
B
P
BP
BP
in
P
P
P
. Let
R
R
R
be the intersection of the line
B
Q
BQ
BQ
and the perpendicular to
B
P
BP
BP
from
C
C
C
. [*] (a) Prove that
∣
B
P
∣
≥
∣
B
R
∣
|BP|\ge |BR|
∣
BP
∣
≥
∣
BR
∣
[*] (b) For which point(s)
P
P
P
does the inequality in (a) become an equality?
3
1
Hide problems
2k-l and 2l-k divide n
For all integers
n
≥
2
n\ge 2
n
≥
2
with the following property: [*] for each pair of positive divisors
k
,
ℓ
<
n
k,~\ell <n
k
,
ℓ
<
n
, at least one of the numbers
2
k
−
ℓ
2k-\ell
2
k
−
ℓ
and
2
ℓ
−
k
2\ell-k
2
ℓ
−
k
is a (not necessarily positive) divisor of
n
n
n
as well.
2
1
Hide problems
Red and blue chips
Let
k
≥
1
k\ge 1
k
≥
1
be a positive integer.We consider
4
k
4k
4
k
chips,
2
k
2k
2
k
of which are red and
2
k
2k
2
k
of which are blue. A sequence of those
4
k
4k
4
k
chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from
r
b
b
‾
b
r
r
r
‾
b
r\underline{bb}br\underline{rr}b
r
bb
b
r
rr
b
to
r
r
r
‾
b
r
b
b
‾
b
r\underline{rr}br\underline{bb}b
r
rr
b
r
bb
b
where
r
r
r
denotes a red chip and
b
b
b
denotes a blue chip.Determine the smallest number
n
n
n
(as a function of
k
k
k
) such that starting from any initial sequence of the
4
k
4k
4
k
chips, we need at most
n
n
n
moves to reach the state in which the first
2
k
2k
2
k
chips are red.
1
1
Hide problems
Minimum of Floor Functions
Find the smallest possible value of the expression
⌊
a
+
b
+
c
d
⌋
+
⌊
b
+
c
+
d
a
⌋
+
⌊
c
+
d
+
a
b
⌋
+
⌊
d
+
a
+
b
c
⌋
\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor
⌊
d
a
+
b
+
c
⌋
+
⌊
a
b
+
c
+
d
⌋
+
⌊
b
c
+
d
+
a
⌋
+
⌊
c
d
+
a
+
b
⌋
in which
a
,
b
,
c
a,~ b,~ c
a
,
b
,
c
, and
d
d
d
vary over the set of positive integers.(Here
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the biggest integer which is smaller than or equal to
x
x
x
.)