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Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2017 Romanian Master of Mathematics
2017 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
4
1
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Graphs of the Quadratic Functions
In the Cartesian plane, let
G
1
G_1
G
1
and
G
2
G_2
G
2
be the graphs of the quadratic functions
f
1
(
x
)
=
p
1
x
2
+
q
1
x
+
r
1
f_1(x) = p_1x^2 + q_1x + r_1
f
1
(
x
)
=
p
1
x
2
+
q
1
x
+
r
1
and
f
2
(
x
)
=
p
2
x
2
+
q
2
x
+
r
2
f_2(x) = p_2x^2 + q_2x + r_2
f
2
(
x
)
=
p
2
x
2
+
q
2
x
+
r
2
, where
p
1
>
0
>
p
2
p_1 > 0 > p_2
p
1
>
0
>
p
2
. The graphs
G
1
G_1
G
1
and
G
2
G_2
G
2
cross at distinct points
A
A
A
and
B
B
B
. The four tangents to
G
1
G_1
G
1
and
G
2
G_2
G
2
at
A
A
A
and
B
B
B
form a convex quadrilateral which has an inscribed circle. Prove that the graphs
G
1
G_1
G
1
and
G
2
G_2
G
2
have the same axis of symmetry.
6
1
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Cyclic Quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be any convex quadrilateral and let
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
be points on the segments
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
, and
D
A
DA
D
A
, respectively. It is given that the segments
P
R
PR
PR
and
Q
S
QS
QS
dissect
A
B
C
D
ABCD
A
BC
D
into four quadrilaterals, each of which has perpendicular diagonals. Show that the points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
are concyclic.
5
1
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Minimal Numbers of Sticks
Fix an integer
n
≥
2
n \geq 2
n
≥
2
. An
n
×
n
n\times n
n
×
n
sieve is an
n
×
n
n\times n
n
×
n
array with
n
n
n
cells removed so that exactly one cell is removed from every row and every column. A stick is a
1
×
k
1\times k
1
×
k
or
k
×
1
k\times 1
k
×
1
array for any positive integer
k
k
k
. For any sieve
A
A
A
, let
m
(
A
)
m(A)
m
(
A
)
be the minimal number of sticks required to partition
A
A
A
. Find all possible values of
m
(
A
)
m(A)
m
(
A
)
, as
A
A
A
varies over all possible
n
×
n
n\times n
n
×
n
sieves. Palmer Mebane
3
1
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Proper Subset
Let
n
n
n
be an integer greater than
1
1
1
and let
X
X
X
be an
n
n
n
-element set. A non-empty collection of subsets
A
1
,
.
.
.
,
A
k
A_1, ..., A_k
A
1
,
...
,
A
k
of
X
X
X
is tight if the union
A
1
∪
⋯
∪
A
k
A_1 \cup \cdots \cup A_k
A
1
∪
⋯
∪
A
k
is a proper subset of
X
X
X
and no element of
X
X
X
lies in exactly one of the
A
i
A_i
A
i
s. Find the largest cardinality of a collection of proper non-empty subsets of
X
X
X
, no non-empty subcollection of which is tight.Note. A subset
A
A
A
of
X
X
X
is proper if
A
≠
X
A\neq X
A
=
X
. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.
2
1
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Monic Polynomial
Determine all positive integers
n
n
n
satisfying the following condition: for every monic polynomial
P
P
P
of degree at most
n
n
n
with integer coefficients, there exists a positive integer
k
≤
n
k\le n
k
≤
n
and
k
+
1
k+1
k
+
1
distinct integers
x
1
,
x
2
,
⋯
,
x
k
+
1
x_1,x_2,\cdots ,x_{k+1}
x
1
,
x
2
,
⋯
,
x
k
+
1
such that
P
(
x
1
)
+
P
(
x
2
)
+
⋯
+
P
(
x
k
)
=
P
(
x
k
+
1
)
P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})
P
(
x
1
)
+
P
(
x
2
)
+
⋯
+
P
(
x
k
)
=
P
(
x
k
+
1
)
.Note. A polynomial is monic if the coefficient of the highest power is one.
1
1
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Unique Representation
(a) Prove that every positive integer
n
n
n
can be written uniquely in the form
n
=
∑
j
=
1
2
k
+
1
(
−
1
)
j
−
1
2
m
j
,
n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},
n
=
j
=
1
∑
2
k
+
1
(
−
1
)
j
−
1
2
m
j
,
where
k
≥
0
k\geq 0
k
≥
0
and
0
≤
m
1
<
m
2
⋯
<
m
2
k
+
1
0\le m_1<m_2\cdots <m_{2k+1}
0
≤
m
1
<
m
2
⋯
<
m
2
k
+
1
are integers. This number
k
k
k
is called weight of
n
n
n
.(b) Find (in closed form) the difference between the number of positive integers at most
2
2017
2^{2017}
2
2017
with even weight and the number of positive integers at most
2
2017
2^{2017}
2
2017
with odd weight.