MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2020 Philippine MO
2020 Philippine MO
Part of
Philippine MO
Subcontests
(4)
4
1
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Right-angle revelation
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle with circumcircle
Γ
\Gamma
Γ
and
D
D
D
the foot of the altitude from
A
A
A
. Suppose that
A
D
=
B
C
AD=BC
A
D
=
BC
. Point
M
M
M
is the midpoint of
D
C
DC
D
C
, and the bisector of
∠
A
D
C
\angle ADC
∠
A
D
C
meets
A
C
AC
A
C
at
N
N
N
. Point
P
P
P
lies on
Γ
\Gamma
Γ
such that lines
B
P
BP
BP
and
A
C
AC
A
C
are parallel. Lines
D
N
DN
D
N
and
A
M
AM
A
M
meet at
F
F
F
, and line
P
F
PF
PF
meets
Γ
\Gamma
Γ
again at
Q
Q
Q
. Line
A
C
AC
A
C
meets the circumcircle of
△
P
N
Q
\triangle PNQ
△
PNQ
again at
E
E
E
. Prove that
∠
D
Q
E
=
9
0
∘
\angle DQE = 90^{\circ}
∠
D
QE
=
9
0
∘
.
3
1
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Raving recurrence relation
Define the sequence
{
a
i
}
\{a_i\}
{
a
i
}
by
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
4
a_1=4
a
1
=
4
, and
a
n
+
1
=
5
a
n
−
a
n
−
1
a_{n+1}=5a_n-a_{n-1}
a
n
+
1
=
5
a
n
−
a
n
−
1
for all
n
≥
1
n\geq 1
n
≥
1
. Show that all terms of the sequence are of the form
c
2
+
3
d
2
c^2+3d^2
c
2
+
3
d
2
for some integers
c
c
c
and
d
d
d
.
2
1
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Krazy equality
Determine all positive integers
k
k
k
for which there exist positive integers
r
r
r
and
s
s
s
that satisfy the equation
(
k
2
−
6
k
+
11
)
r
−
1
=
(
2
k
−
7
)
s
.
(k^2-6k+11)^{r-1}=(2k-7)^{s}.
(
k
2
−
6
k
+
11
)
r
−
1
=
(
2
k
−
7
)
s
.
1
1
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Tantalizing T-tetromino tiling
A T-tetromino is formed by adjoining three unit squares to form a
1
×
3
1 \times 3
1
×
3
rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a
202
×
202
202 \times 202
202
×
202
grid so that it can be tiled using T-tetrominoes.