MathDB
Problems
Contests
International Contests
APMO
2019 APMO
2019 APMO
Part of
APMO
Subcontests
(5)
3
1
Hide problems
Variable point on the median
Let
A
B
C
ABC
A
BC
be a scalene triangle with circumcircle
Γ
\Gamma
Γ
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. A variable point
P
P
P
is selected in the line segment
A
M
AM
A
M
. The circumcircles of triangles
B
P
M
BPM
BPM
and
C
P
M
CPM
CPM
intersect
Γ
\Gamma
Γ
again at points
D
D
D
and
E
E
E
, respectively. The lines
D
P
DP
D
P
and
E
P
EP
EP
intersect (a second time) the circumcircles to triangles
C
P
M
CPM
CPM
and
B
P
M
BPM
BPM
at
X
X
X
and
Y
Y
Y
, respectively. Prove that as
P
P
P
varies, the circumcircle of
△
A
X
Y
\triangle AXY
△
A
X
Y
passes through a fixed point
T
T
T
distinct from
A
A
A
.
1
1
Hide problems
Easy integer functional equation
Let
Z
+
\mathbb{Z}^+
Z
+
be the set of positive integers. Determine all functions
f
:
Z
+
→
Z
+
f : \mathbb{Z}^+\to\mathbb{Z}^+
f
:
Z
+
→
Z
+
such that
a
2
+
f
(
a
)
f
(
b
)
a^2+f(a)f(b)
a
2
+
f
(
a
)
f
(
b
)
is divisible by
f
(
a
)
+
b
f(a)+b
f
(
a
)
+
b
for all positive integers
a
,
b
a,b
a
,
b
.
2
1
Hide problems
Strange Conditional Sequence
Let
m
m
m
be a fixed positive integer. The infinite sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
is defined in the following way:
a
1
a_1
a
1
is a positive integer, and for every integer
n
≥
1
n\geq 1
n
≥
1
we have
a
n
+
1
=
{
a
n
2
+
2
m
if
a
n
<
2
m
a
n
/
2
if
a
n
≥
2
m
a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}
a
n
+
1
=
{
a
n
2
+
2
m
a
n
/2
if
a
n
<
2
m
if
a
n
≥
2
m
For each
m
m
m
, determine all possible values of
a
1
a_1
a
1
such that every term in the sequence is an integer.
4
1
Hide problems
Terrifying "2018 \times 2019" board
Consider a
2018
×
2019
2018 \times 2019
2018
×
2019
board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?
5
1
Hide problems
Entirely wrapped by f
Determine all the functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
2
+
f
(
y
)
)
=
f
(
f
(
x
)
)
+
f
(
y
2
)
+
2
f
(
x
y
)
f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy)
f
(
x
2
+
f
(
y
))
=
f
(
f
(
x
))
+
f
(
y
2
)
+
2
f
(
x
y
)
for all real numbers
x
x
x
and
y
y
y
.