MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
USAMO
2023 USAMO
2023 USAMO
Part of
USAMO
Subcontests
(4)
5
1
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Double dose of cyanide on day 2
Let
n
≥
3
n\geq3
n
≥
3
be an integer. We say that an arrangement of the numbers
1
1
1
,
2
2
2
,
…
\dots
…
,
n
2
n^2
n
2
in a
n
×
n
n \times n
n
×
n
table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of
n
n
n
is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
3
1
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I rushed on this problem for an hour
Consider an
n
n
n
-by-
n
n
n
board of unit squares for some odd positive integer
n
n
n
. We say that a collection
C
C
C
of identical dominoes is a maximal grid-aligned configuration on the board if
C
C
C
consists of
(
n
2
−
1
)
/
2
(n^2-1)/2
(
n
2
−
1
)
/2
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap:
C
C
C
then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let
k
(
C
)
k(C)
k
(
C
)
be the number of distinct maximal grid-aligned configurations obtainable from
C
C
C
by repeatedly sliding dominoes. Find all possible values of
k
(
C
)
k(C)
k
(
C
)
as a function of
n
n
n
.Proposed by Holden Mui
6
1
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force overlay inversion vibes
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
and excenters
I
a
I_a
I
a
,
I
b
I_b
I
b
, and
I
c
I_c
I
c
opposite
A
A
A
,
B
B
B
, and
C
C
C
, respectively. Let
D
D
D
be an arbitrary point on the circumcircle of
△
A
B
C
\triangle{ABC}
△
A
BC
that does not lie on any of the lines
I
I
a
II_a
I
I
a
,
I
b
I
c
I_bI_c
I
b
I
c
, or
B
C
BC
BC
. Suppose the circumcircles of
△
D
I
I
a
\triangle{DII_a}
△
D
I
I
a
and
△
D
I
b
I
c
\triangle{DI_bI_c}
△
D
I
b
I
c
intersect at two distinct points
D
D
D
and
F
F
F
. If
E
E
E
is the intersection of lines
D
F
DF
D
F
and
B
C
BC
BC
, prove that
∠
B
A
D
=
∠
E
A
C
\angle{BAD} = \angle{EAC}
∠
B
A
D
=
∠
E
A
C
.Proposed by Zach Chroman
2
1
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How many approaches you got? (A lot)
Let
R
+
\mathbb{R}^+
R
+
be the set of positive real numbers. Find all functions
f
:
R
+
→
R
+
f \colon \mathbb{R}^+ \to \mathbb{R}^+
f
:
R
+
→
R
+
such that, for all
x
,
y
∈
R
+
x,y \in \mathbb{R}^+
x
,
y
∈
R
+
,
f
(
x
y
+
f
(
x
)
)
=
x
f
(
y
)
+
2.
f(xy+f(x))=xf(y)+2.
f
(
x
y
+
f
(
x
))
=
x
f
(
y
)
+
2.