MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2019 Chile National Olympiad
2019 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(4)
2
1
Hide problems
k moves on right or lest in 2019 cells
Javiera and Claudio play on a board consisting of a row with
2019
2019
2019
cells. Claudio starts by placing a token anywhere on the board. Next Javiera says a natural number
k
k
k
,
1
≤
k
≤
n
1 \le k \le n
1
≤
k
≤
n
and Claudio must move the token to the right or to the left at your choice
k
k
k
squares and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest
n
n
n
such that Javiera always wins after a finite number of moves.
3
1
Hide problems
3^x -5^y = z^2 diophanine
Find all solutions
x
,
y
,
z
x,y,z
x
,
y
,
z
in the positive integers of the equation
3
x
−
5
y
=
z
2
3^x -5^y = z^2
3
x
−
5
y
=
z
2
1
1
Hide problems
1-9 into 3x3 board, difference of digits in cells that share a side is <=3
A square of
3
×
3
3 \times 3
3
×
3
is subdivided into 9 small squares of
1
×
1
1 \times 1
1
×
1
. It is desired to distribute the nine digits
1
,
2
,
.
.
.
,
9
1, 2, . . . , 9
1
,
2
,
...
,
9
in each small square of
1
×
1
1 \times 1
1
×
1
, a number in each small square. Find the number of different distributions that can be formed in such a way that the difference of the digits in cells that share a side in common is less than or equal to three. Two distributions are distinct even if they differ by rotation and/or reflection.
4
1
Hide problems
EF//CD wanted, <ADC = <BCD > 90^o, // (Chile 2019 L2 P4)
In the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
∠
A
D
C
=
∠
B
C
D
>
9
0
o
\angle ADC = \angle BCD > 90^o
∠
A
D
C
=
∠
BC
D
>
9
0
o
. Let
E
E
E
be the intersection of the line
A
C
AC
A
C
with the line parallel to
A
D
AD
A
D
that passes through
B
B
B
. Let
F
F
F
be the intersection of line
B
D
BD
B
D
with the line parallel to
B
C
BC
BC
passing through
A
A
A
. Prove that
E
F
EF
EF
is parallel to
C
D
CD
C
D
.