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Problems
Contests
International Contests
APMO
2020 APMO
2020 APMO
Part of
APMO
Subcontests
(5)
5
1
Hide problems
Possible values of t/s for stones in buckets
Let
n
≥
3
n \geq 3
n
≥
3
be a fixed integer. The number
1
1
1
is written
n
n
n
times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers
a
a
a
and
b
b
b
, replacing them with the numbers
1
1
1
and
a
+
b
a+b
a
+
b
, then adding one stone to the first bucket and
gcd
(
a
,
b
)
\gcd(a, b)
g
cd
(
a
,
b
)
stones to the second bucket. After some finite number of moves, there are
s
s
s
stones in the first bucket and
t
t
t
stones in the second bucket, where
s
s
s
and
t
t
t
are positive integers. Find all possible values of the ratio
t
s
\frac{t}{s}
s
t
.
4
1
Hide problems
Polynomial values contain some contiguous subsequence sum
Let
Z
\mathbb{Z}
Z
denote the set of all integers. Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients that satisfy the following property:For any infinite sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dotsc
…
of integers in which each integer in
Z
\mathbb{Z}
Z
appears exactly once, there exist indices
i
<
j
i < j
i
<
j
and an integer
k
k
k
such that
a
i
+
a
i
+
1
+
⋯
+
a
j
=
P
(
k
)
a_i +a_{i+1} +\dotsb +a_j = P(k)
a
i
+
a
i
+
1
+
⋯
+
a
j
=
P
(
k
)
.
3
1
Hide problems
All sufficiently large integers have k representations as subset sum
Determine all positive integers
k
k
k
for which there exist a positive integer
m
m
m
and a set
S
S
S
of positive integers such that any integer
n
>
m
n > m
n
>
m
can be written as a sum of distinct elements of
S
S
S
in exactly
k
k
k
ways.
2
1
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Sequence satisfying sqrt inequality eventually alternates
Show that
r
=
2
r = 2
r
=
2
is the largest real number
r
r
r
which satisfies the following condition:If a sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
of positive integers fulfills the inequalities
a
n
≤
a
n
+
2
≤
a
n
2
+
r
a
n
+
1
a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}
a
n
≤
a
n
+
2
≤
a
n
2
+
r
a
n
+
1
for every positive integer
n
n
n
, then there exists a positive integer
M
M
M
such that
a
n
+
2
=
a
n
a_{n+2} = a_n
a
n
+
2
=
a
n
for every
n
≥
M
n \geq M
n
≥
M
.
1
1
Hide problems
AC, BF, DE concurrent
Let
Γ
\Gamma
Γ
be the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
. Let
D
D
D
be a point on the side
B
C
BC
BC
. The tangent to
Γ
\Gamma
Γ
at
A
A
A
intersects the parallel line to
B
A
BA
B
A
through
D
D
D
at point
E
E
E
. The segment
C
E
CE
CE
intersects
Γ
\Gamma
Γ
again at
F
F
F
. Suppose
B
B
B
,
D
D
D
,
F
F
F
,
E
E
E
are concyclic. Prove that
A
C
AC
A
C
,
B
F
BF
BF
,
D
E
DE
D
E
are concurrent.