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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2018 Philippine MO
2018 Philippine MO
Part of
Philippine MO
Subcontests
(4)
4
1
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An exponential diophantine equation
Determine all ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of nonnegative integers that satisfy the equation
3
x
2
+
2
⋅
9
y
=
x
(
4
y
+
1
−
1
)
.
3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).
3
x
2
+
2
⋅
9
y
=
x
(
4
y
+
1
−
1
)
.
3
1
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Diagonalized Latin squares
Let
n
n
n
be a positive integer. An
n
×
n
n \times n
n
×
n
matrix (a rectangular array of numbers with
n
n
n
rows and
n
n
n
columns) is said to be a platinum matrix if:[*] the
n
2
n^2
n
2
entries are integers from
1
1
1
to
n
n
n
; [*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from
1
1
1
to
n
n
n
exactly once; and [*] there exists a collection of
n
n
n
entries containing each of the numbers from
1
1
1
to
n
n
n
, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix.Determine all values of
n
n
n
for which there exists an
n
×
n
n \times n
n
×
n
platinum matrix.
2
1
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Prime linear recurrences are constant
Suppose
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
is a sequence of integers, and
d
d
d
is some integer. For all natural numbers
n
n
n
, \begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*} Show that the sequence is constant.
1
1
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5AB = 4BC, <B = 60
In triangle
A
B
C
ABC
A
BC
with
∠
A
B
C
=
6
0
∘
\angle ABC = 60^{\circ}
∠
A
BC
=
6
0
∘
and
5
A
B
=
4
B
C
5AB = 4BC
5
A
B
=
4
BC
, points
D
D
D
and
E
E
E
are the feet of the altitudes from
B
B
B
and
C
C
C
, respectively.
M
M
M
is the midpoint of
B
D
BD
B
D
and the circumcircle of triangle
B
M
C
BMC
BMC
meets line
A
C
AC
A
C
again at
N
N
N
. Lines
B
N
BN
BN
and
C
M
CM
CM
meet at
P
P
P
. Prove that
∠
E
D
P
=
9
0
∘
\angle EDP = 90^{\circ}
∠
E
D
P
=
9
0
∘
.