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Problems
Contests
International Contests
APMO
2022 APMO
2022 APMO
Part of
APMO
Subcontests
(5)
4
1
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Game with marbles
Let
n
n
n
and
k
k
k
be positive integers. Cathy is playing the following game. There are
n
n
n
marbles and
k
k
k
boxes, with the marbles labelled
1
1
1
to
n
n
n
. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say
i
i
i
, to either any empty box or the box containing marble
i
+
1
i+1
i
+
1
. Cathy wins if at any point there is a box containing only marble
n
n
n
. Determine all pairs of integers
(
n
,
k
)
(n,k)
(
n
,
k
)
such that Cathy can win this game.
3
1
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APMO 2022 P3
Find all positive integers
k
<
202
k<202
k
<
202
for which there exist a positive integers
n
n
n
such that \bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}
5
1
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Inequality on APMO P5
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real numbers such that
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2+b^2+c^2+d^2=1
a
2
+
b
2
+
c
2
+
d
2
=
1
. Determine the minimum value of
(
a
−
b
)
(
b
−
c
)
(
c
−
d
)
(
d
−
a
)
(a-b)(b-c)(c-d)(d-a)
(
a
−
b
)
(
b
−
c
)
(
c
−
d
)
(
d
−
a
)
and determine all values of
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
such that the minimum value is achived.
2
1
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Line passes through fixed point, as point varies
Let
A
B
C
ABC
A
BC
be a right triangle with
∠
B
=
9
0
∘
\angle B=90^{\circ}
∠
B
=
9
0
∘
. Point
D
D
D
lies on the line
C
B
CB
CB
such that
B
B
B
is between
D
D
D
and
C
C
C
. Let
E
E
E
be the midpoint of
A
D
AD
A
D
and let
F
F
F
be the seconf intersection point of the circumcircle of
△
A
C
D
\triangle ACD
△
A
C
D
and the circumcircle of
△
B
D
E
\triangle BDE
△
B
D
E
. Prove that as
D
D
D
varies, the line
E
F
EF
EF
passes through a fixed point.
1
1
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Pair of multiples
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers such that
a
3
a^3
a
3
is multiple of
b
2
b^2
b
2
and
b
−
1
b-1
b
−
1
is multiple of
a
−
1
a-1
a
−
1
.