MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2016 Serbia National Math Olympiad
2016 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
4
1
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Easy geometry
Let
A
B
C
ABC
A
BC
be a triangle, and
I
I
I
the incenter,
M
M
M
midpoint of
B
C
BC
BC
,
D
D
D
the touch point of incircle and
B
C
BC
BC
. Prove that perpendiculars from
M
,
D
,
A
M, D, A
M
,
D
,
A
to
A
I
,
I
M
,
B
C
AI, IM, BC
A
I
,
I
M
,
BC
respectively are concurrent
6
1
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A lot of squares. Can all of them be different?
Let
a
1
,
a
2
,
…
,
a
2
2016
a_1, a_2, \dots, a_{2^{2016}}
a
1
,
a
2
,
…
,
a
2
2016
be positive integers not bigger than
2016
2016
2016
. We know that for each
n
≤
2
2016
n \leq 2^{2016}
n
≤
2
2016
,
a
1
a
2
…
a
n
+
1
a_1a_2 \dots a_{n} +1
a
1
a
2
…
a
n
+
1
is a perfect square. Prove that for some
i
i
i
,
a
i
=
1
a_i=1
a
i
=
1
.
5
1
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Combinatorics with subsets,SMO 2016P5
There are
2
n
−
1
2n-1
2
n
−
1
twoelement subsets of set
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
. Prove that one can choose
n
n
n
out of these such that their union contains no more than
2
3
n
+
1
\frac{2}{3}n+1
3
2
n
+
1
elements.
3
1
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Geometry, SMO 2016, not easy
Let
A
B
C
ABC
A
BC
be a triangle and
O
O
O
its circumcentre. A line tangent to the circumcircle of the triangle
B
O
C
BOC
BOC
intersects sides
A
B
AB
A
B
at
D
D
D
and
A
C
AC
A
C
at
E
E
E
. Let
A
′
A'
A
′
be the image of
A
A
A
under
D
E
DE
D
E
. Prove that the circumcircle of the triangle
A
′
D
E
A'DE
A
′
D
E
is tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
.
2
1
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Recursively defined function
Let
n
n
n
be a positive integer. Let
f
f
f
be a function from nonnegative integers to themselves. Let
f
(
0
,
i
)
=
f
(
i
,
0
)
=
0
f (0,i)=f (i,0)=0
f
(
0
,
i
)
=
f
(
i
,
0
)
=
0
,
f
(
1
,
1
)
=
n
f (1, 1)=n
f
(
1
,
1
)
=
n
, and
f
(
i
,
j
)
=
[
f
(
i
−
1
,
j
)
2
]
+
[
f
(
i
,
j
−
1
)
2
]
f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}]
f
(
i
,
j
)
=
[
2
f
(
i
−
1
,
j
)
]
+
[
2
f
(
i
,
j
−
1
)
]
for positive integers
i
,
j
i, j
i
,
j
such that
i
∗
j
>
1
i*j>1
i
∗
j
>
1
. Find the number of pairs
(
i
,
j
)
(i,j)
(
i
,
j
)
such that
f
(
i
,
j
)
f (i, j)
f
(
i
,
j
)
is an odd number.(
[
x
]
[x]
[
x
]
is the floor function).
1
1
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Exist big m such that...
Let
n
>
1
n>1
n
>
1
be an integer. Prove that there exist
m
>
n
n
m>n^n
m
>
n
n
such that
n
m
−
m
n
m
+
n
\frac {n^m-m^n}{m+n}
m
+
n
n
m
−
m
n
is a positive integer.