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International Contests
Romanian Masters of Mathematics Collection
2008 Romanian Master of Mathematics
2008 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(4)
4
1
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(n+1)^2 points inside a n x n square. Looks classical.
Consider a square of sidelength
n
n
n
and (n\plus{}1)^2 interior points. Prove that we can choose
3
3
3
of these points so that they determine a triangle (eventually degenerated) of area at most
1
2
\frac12
2
1
.
3
1
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Every positive integer divides some number of a sequence
Let
a
>
1
a>1
a
>
1
be a positive integer. Prove that every non-zero positive integer
N
N
N
has a multiple in the sequence
(
a
n
)
n
≥
1
(a_n)_{n\ge1}
(
a
n
)
n
≥
1
, a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor.
2
1
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Bijective function on Z is sum of 2 bijective functions on Z
Prove that every bijective function
f
:
Z
→
Z
f: \mathbb{Z}\rightarrow\mathbb{Z}
f
:
Z
→
Z
can be written in the way f\equal{}u\plus{}v where
u
,
v
:
Z
→
Z
u,v: \mathbb{Z}\rightarrow\mathbb{Z}
u
,
v
:
Z
→
Z
are bijective functions.
1
1
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Equilateral triangle and an interior point. Seems classical.
Let
A
B
C
ABC
A
BC
be an equilateral triangle and
P
P
P
in its interior. The distances from
P
P
P
to the triangle's sides are denoted by
a
2
,
b
2
,
c
2
a^2, b^2,c^2
a
2
,
b
2
,
c
2
respectively, where
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
. Find the locus of the points
P
P
P
for which
a
,
b
,
c
a,b,c
a
,
b
,
c
can be the sides of a non-degenerate triangle.