MathDB
Problems
Contests
International Contests
APMO
2018 APMO
2018 APMO
Part of
APMO
Subcontests
(5)
4
1
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Reflecting a ray wrt the sides of an equilateral triangle
Let
A
B
C
ABC
A
BC
be an equilateral triangle. From the vertex
A
A
A
we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle
α
\alpha
α
, it leaves with a directed angle
18
0
∘
−
α
180^{\circ}-\alpha
18
0
∘
−
α
. After
n
n
n
bounces, the ray returns to
A
A
A
without ever landing on any of the other two vertices. Find all possible values of
n
n
n
.
3
1
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Tri-connected squares
A collection of
n
n
n
squares on the plane is called tri-connected if the following criteria are satisfied:(i) All the squares are congruent. (ii) If two squares have a point
P
P
P
in common, then
P
P
P
is a vertex of each of the squares. (iii) Each square touches exactly three other squares.How many positive integers
n
n
n
are there with
2018
≤
n
≤
3018
2018\leq n \leq 3018
2018
≤
n
≤
3018
, such that there exists a collection of
n
n
n
squares that is tri-connected?
2
1
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Bounding difference of sum of fractions in an interval
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be given by
f
(
x
)
=
1
x
+
1
x
−
2
+
1
x
−
4
+
⋯
+
1
x
−
2018
f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}
f
(
x
)
=
x
1
+
x
−
2
1
+
x
−
4
1
+
⋯
+
x
−
2018
1
g
(
x
)
=
1
x
−
1
+
1
x
−
3
+
1
x
−
5
+
⋯
+
1
x
−
2017
g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}
g
(
x
)
=
x
−
1
1
+
x
−
3
1
+
x
−
5
1
+
⋯
+
x
−
2017
1
.Prove that
∣
f
(
x
)
−
g
(
x
)
∣
>
2
|f(x)-g(x)| >2
∣
f
(
x
)
−
g
(
x
)
∣
>
2
for any non-integer real number
x
x
x
satisfying
0
<
x
<
2018
0 < x < 2018
0
<
x
<
2018
.
5
1
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P(s) and P(t) integers implies P(st) integer
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that for all real numbers
s
s
s
and
t
t
t
, if
P
(
s
)
P(s)
P
(
s
)
and
P
(
t
)
P(t)
P
(
t
)
are both integers, then
P
(
s
t
)
P(st)
P
(
s
t
)
is also an integer.
1
1
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Circles tangent at orthocenter
Let
H
H
H
be the orthocenter of the triangle
A
B
C
ABC
A
BC
. Let
M
M
M
and
N
N
N
be the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Assume that
H
H
H
lies inside the quadrilateral
B
M
N
C
BMNC
BMNC
and that the circumcircles of triangles
B
M
H
BMH
BM
H
and
C
N
H
CNH
CN
H
are tangent to each other. The line through
H
H
H
parallel to
B
C
BC
BC
intersects the circumcircles of the triangles
B
M
H
BMH
BM
H
and
C
N
H
CNH
CN
H
in the points
K
K
K
and
L
L
L
, respectively. Let
F
F
F
be the intersection point of
M
K
MK
M
K
and
N
L
NL
N
L
and let
J
J
J
be the incenter of triangle
M
H
N
MHN
M
H
N
. Prove that
F
J
=
F
A
F J = F A
F
J
=
F
A
.