MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2016 Mexico National Olmypiad
2016 Mexico National Olmypiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
Problem 6
Let
A
B
C
D
ABCD
A
BC
D
a quadrilateral inscribed in a circumference,
l
1
l_1
l
1
the parallel to
B
C
BC
BC
through
A
A
A
, and
l
2
l_2
l
2
the parallel to
A
D
AD
A
D
through
B
B
B
. The line
D
C
DC
D
C
intersects
l
1
l_1
l
1
and
l
2
l_2
l
2
at
E
E
E
and
F
F
F
, respectively. The perpendicular to
l
1
l_1
l
1
through
A
A
A
intersects
B
C
BC
BC
at
P
P
P
, and the perpendicular to
l
2
l_2
l
2
through
B
B
B
cuts
A
D
AD
A
D
at
Q
Q
Q
. Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be the circumferences that pass through the vertex of triangles
A
D
E
ADE
A
D
E
and
B
F
C
BFC
BFC
, respectively. Prove that
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
are tangent to each other if and only if
D
P
DP
D
P
is perpendicular to
C
Q
CQ
CQ
.
5
1
Hide problems
Problem 5
The numbers from
1
1
1
to
n
2
n^2
n
2
are written in order in a grid of
n
×
n
n \times n
n
×
n
, one number in each square, in such a way that the first row contains the numbers from
1
1
1
to
n
n
n
from left to right; the second row contains the numbers
n
+
1
n + 1
n
+
1
to
2
n
2n
2
n
from left to right, and so on and so forth. An allowed move on the grid consists in choosing any two adjacent squares (i.e. two squares that share a side), and add (or subtract) the same integer to both of the numbers that appear on those squares. Find all values of
n
n
n
for which it is possible to make every squares to display
0
0
0
after making any number of moves as necessary and, for those cases in which it is possible, find the minimum number of moves that are necessary to do this.
4
1
Hide problems
Problem 4
We say a non-negative integer
n
n
n
"contains" another non-negative integer
m
m
m
, if the digits of its decimal expansion appear consecutively in the decimal expansion of
n
n
n
. For example,
2016
2016
2016
contains
2
2
2
,
0
0
0
,
1
1
1
,
6
6
6
,
20
20
20
,
16
16
16
,
201
201
201
, and
2016
2016
2016
. Find the largest integer
n
n
n
that does not contain a multiple of
7
7
7
.
2
1
Hide problems
Painting positive integers
A pair of positive integers
m
,
n
m, n
m
,
n
is called guerrera, if there exists positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that
m
=
a
b
m=ab
m
=
ab
,
n
=
c
d
n=cd
n
=
c
d
and
a
+
b
=
c
+
d
a+b=c+d
a
+
b
=
c
+
d
. For example the pair
8
,
9
8, 9
8
,
9
is guerrera cause
8
=
4
⋅
2
8= 4 \cdot 2
8
=
4
⋅
2
,
9
=
3
⋅
3
9= 3 \cdot 3
9
=
3
⋅
3
and
4
+
2
=
3
+
3
4+2=3+3
4
+
2
=
3
+
3
. We paint the positive integers if the following order:We start painting the numbers
3
3
3
and
5
5
5
. If a positive integer
x
x
x
is not painted and a positive
y
y
y
is painted such that the pair
x
,
y
x, y
x
,
y
is guerrera, we paint
x
x
x
. Find all positive integers
x
x
x
that can be painted.
1
1
Hide problems
Prove QR=RT
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two circumferences externally tangents at
S
S
S
such that the radius of
C
2
C_2
C
2
is the triple of the radius of
C
1
C_1
C
1
. Let a line be tangent to
C
1
C_1
C
1
at
P
≠
S
P \neq S
P
=
S
and to
C
2
C_2
C
2
at
Q
≠
S
Q \neq S
Q
=
S
. Let
T
T
T
be a point on
C
2
C_2
C
2
such that
Q
T
QT
QT
is diameter of
C
2
C_2
C
2
. Let the angle bisector of
∠
S
Q
T
\angle SQT
∠
SQT
meet
S
T
ST
ST
at
R
R
R
. Prove that
Q
R
=
R
T
QR=RT
QR
=
RT
3
1
Hide problems
Floor function inequality
Find the minimum real
x
x
x
that satisfies
⌊
x
⌋
<
⌊
x
2
⌋
<
⌊
x
3
⌋
<
⋯
<
⌊
x
n
⌋
<
⌊
x
n
+
1
⌋
<
⋯
\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots
⌊
x
⌋
<
⌊
x
2
⌋
<
⌊
x
3
⌋
<
⋯
<
⌊
x
n
⌋
<
⌊
x
n
+
1
⌋
<
⋯