MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2023 Mexico National Olympiad
2023 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
4
1
Hide problems
Magic trick with black and white cards with numbers
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer. For every number from
1
1
1
to
n
n
n
, there is a card with this number and which is either black or white. A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card. For instance, when
n
=
5
n=5
n
=
5
and the initial configuration is
(
1
B
,
2
B
,
3
W
,
4
B
,
5
B
)
(1B, 2B, 3W, 4B,5B)
(
1
B
,
2
B
,
3
W
,
4
B
,
5
B
)
, the magician can choose
1
B
,
3
W
1B, 3W
1
B
,
3
W
on the first move to obtain
(
3
W
,
2
B
,
3
W
,
4
B
,
5
B
)
(3W, 2B, 3W, 4B, 5B)
(
3
W
,
2
B
,
3
W
,
4
B
,
5
B
)
and then
3
W
,
4
B
3W, 4B
3
W
,
4
B
on the second move to obtain
(
4
B
,
2
B
,
3
W
,
4
B
,
5
B
)
(4B, 2B, 3W, 4B, 5B)
(
4
B
,
2
B
,
3
W
,
4
B
,
5
B
)
. Determine in terms of
n
n
n
all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.
6
1
Hide problems
NT FE with two conditions
Find all functions
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb {N}
f
:
N
→
N
such that for all positive integers
m
,
n
m, n
m
,
n
,
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
f(m+n)\mid f(m)+f(n)
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
and
f
(
m
)
f
(
n
)
∣
f
(
m
n
)
f(m)f(n) \mid f(mn)
f
(
m
)
f
(
n
)
∣
f
(
mn
)
.
5
1
Hide problems
Two circles meet on AB
Let
A
B
C
ABC
A
BC
be an acute triangle,
Γ
\Gamma
Γ
is its circumcircle and
O
O
O
is its circumcenter. Let
F
F
F
be the point on
A
C
AC
A
C
such that the
∠
C
O
F
=
∠
A
C
B
\angle COF=\angle ACB
∠
COF
=
∠
A
CB
, such that
F
F
F
and
B
B
B
lie in opposite sides with respect to
C
O
CO
CO
. The line
F
O
FO
FO
cuts
B
C
BC
BC
at
G
G
G
. The line parallel to
B
C
BC
BC
through
A
A
A
intersects
Γ
\Gamma
Γ
again at
M
M
M
. The lines
C
O
CO
CO
and
M
G
MG
MG
meet at
K
K
K
. Show that the circumcircles of the triangles
B
G
K
BGK
BG
K
and
A
O
K
AOK
A
O
K
meet on
A
B
AB
A
B
.
3
1
Hide problems
Quadrilateral geo
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. If
M
,
N
,
K
M, N, K
M
,
N
,
K
are the midpoints of the segments
A
B
,
B
C
AB, BC
A
B
,
BC
, and
C
D
CD
C
D
, respectively, and there is also a point
P
P
P
inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
such that,
∠
B
P
N
=
∠
P
A
D
\angle BPN= \angle PAD
∠
BPN
=
∠
P
A
D
and
∠
C
P
N
=
∠
P
D
A
\angle CPN=\angle PDA
∠
CPN
=
∠
P
D
A
. Show that
A
B
⋅
C
D
=
4
P
M
⋅
P
K
AB \cdot CD=4PM\cdot PK
A
B
⋅
C
D
=
4
PM
⋅
P
K
.
2
1
Hide problems
Numbers on a polygon
The numbers from
1
1
1
to
2000
2000
2000
are placed on the vertices of a regular polygon with
2000
2000
2000
sides, one on each vertex, so that the following is true: If four integers
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
satisfy that
1
≤
A
<
B
<
C
<
D
≤
2000
1 \leq A<B<C<D \leq 2000
1
≤
A
<
B
<
C
<
D
≤
2000
, then the segment that joins the vertices of the numbers
A
A
A
and
B
B
B
and the segment that joins the vertices of
C
C
C
and
D
D
D
do not intersect inside the polygon. Prove that there exists a perfect square such that the number diametrically opposite to it is not a perfect square.
1
1
Hide problems
Sum of squares is twice sum of digits
Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits.