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Problems
Contests
International Contests
APMO
2021 APMO
2021 APMO
Part of
APMO
Subcontests
(5)
5
1
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Perfect Square Function
Determine all Functions
f
:
Z
→
Z
f:\mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that
f
(
f
(
a
)
−
b
)
+
b
f
(
2
a
)
f(f(a)-b)+bf(2a)
f
(
f
(
a
)
−
b
)
+
b
f
(
2
a
)
is a perfect square for all integers
a
a
a
and
b
b
b
.
2
1
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Bounding Divisibility
For a polynomial
P
P
P
and a positive integer
n
n
n
, define
P
n
P_n
P
n
as the number of positive integer pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
<
b
≤
n
a<b \leq n
a
<
b
≤
n
and
∣
P
(
a
)
∣
−
∣
P
(
b
)
∣
|P(a)|-|P(b)|
∣
P
(
a
)
∣
−
∣
P
(
b
)
∣
is divisible by
n
n
n
. Determine all polynomial
P
P
P
with integer coefficients such that
P
n
≤
2021
P_n \leq 2021
P
n
≤
2021
for all positive integers
n
n
n
.
3
1
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Collinear Centers and Midarcs
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic convex quadrilateral and
Γ
\Gamma
Γ
be its circumcircle. Let
E
E
E
be the intersection of the diagonals of
A
C
AC
A
C
and
B
D
BD
B
D
. Let
L
L
L
be the center of the circle tangent to sides
A
B
AB
A
B
,
B
C
BC
BC
, and
C
D
CD
C
D
, and let
M
M
M
be the midpoint of the arc
B
C
BC
BC
of
Γ
\Gamma
Γ
not containing
A
A
A
and
D
D
D
. Prove that the excenter of triangle
B
C
E
BCE
BCE
opposite
E
E
E
lies on the line
L
M
LM
L
M
.
1
1
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Number of Solutions is 2
Prove that for each real number
r
>
2
r>2
r
>
2
, there are exactly two or three positive real numbers
x
x
x
satisfying the equation
x
2
=
r
⌊
x
⌋
x^2=r\lfloor x \rfloor
x
2
=
r
⌊
x
⌋
.
4
1
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Mouse eating cheese in a square grid
Given a
32
×
32
32 \times 32
32
×
32
table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a subset of cells containing cheese is good if, during this process, the mouse tastes each piece of cheese exactly once and then falls off the table. Show that:(a) No good subset consists of 888 cells. (b) There exists a good subset consisting of at least 666 cells.