MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2017 Mexico National Olympiad
2017 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
Vote manipulation
Let
n
≥
2
n \geq 2
n
≥
2
and
m
m
m
be positive integers.
m
m
m
ballot boxes are placed in a line. Two players
A
A
A
and
B
B
B
play by turns, beginning with
A
A
A
, in the following manner. Each turn,
A
A
A
chooses two boxes and places a ballot in each of them. Afterwards,
B
B
B
chooses one of the boxes, and removes every ballot from it.
A
A
A
wins if after some turn of
B
B
B
, there exists a box containing
n
n
n
ballots. For each
n
n
n
, find the minimum value of
m
m
m
such that
A
A
A
can guarantee a win independently of how
B
B
B
plays.
5
1
Hide problems
Midpoint on diagonal of cyclic quadrilateral
On a circle
Γ
\Gamma
Γ
, points
A
,
B
,
N
,
C
,
D
,
M
A, B, N, C, D, M
A
,
B
,
N
,
C
,
D
,
M
are chosen in a clockwise order in such a way that
N
N
N
and
M
M
M
are the midpoints of clockwise arcs
B
C
BC
BC
and
A
D
AD
A
D
respectively. Let
P
P
P
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
, and let
Q
Q
Q
be a point on line
M
B
MB
MB
such that
P
Q
PQ
PQ
is perpendicular to
M
N
MN
MN
. Point
R
R
R
is chosen on segment
M
C
MC
MC
such that
Q
B
=
R
C
QB = RC
QB
=
RC
, prove that the midpoint of
Q
R
QR
QR
lies on
A
C
AC
A
C
.
4
1
Hide problems
Triangle side set
A subset
B
B
B
of
{
1
,
2
,
…
,
2017
}
\{1, 2, \dots, 2017\}
{
1
,
2
,
…
,
2017
}
is said to have property
T
T
T
if any three elements of
B
B
B
are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property
T
T
T
may contain.
3
1
Hide problems
Parallel induced by intersections with circles
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocenter
H
H
H
. The circle through
B
,
H
B, H
B
,
H
, and
C
C
C
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively, and segment
D
E
DE
D
E
intersects
H
B
HB
H
B
and
H
C
HC
H
C
at
P
P
P
and
Q
Q
Q
respectively. Two points
X
X
X
and
Y
Y
Y
, both different from
A
A
A
, are located on lines
A
P
AP
A
P
and
A
Q
AQ
A
Q
respectively such that
X
,
H
,
A
,
B
X, H, A, B
X
,
H
,
A
,
B
are concyclic and
Y
,
H
,
A
,
C
Y, H, A, C
Y
,
H
,
A
,
C
are concyclic. Show that lines
X
Y
XY
X
Y
and
B
C
BC
BC
are parallel.
2
1
Hide problems
Integer average set
A set of
n
n
n
positive integers is said to be balanced if for each integer
k
k
k
with
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
, the average of any
k
k
k
numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to
2017
2017
2017
.
1
1
Hide problems
Moving pairs of knights
A knight is placed on each square of the first column of a
2017
×
2017
2017 \times 2017
2017
×
2017
board. A move consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers
k
k
k
with
1
≤
k
≤
2017
1 \leq k \leq 2017
1
≤
k
≤
2017
such that it is possible for each square in the
k
k
k
-th column to contain one knight after a finite number of moves.Note: Two squares are a knight-step away if they are opposite corners of a
2
×
3
2 \times 3
2
×
3
or
3
×
2
3 \times 2
3
×
2
board.