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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2020 Chile National Olympiad
2020 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(4)
4
1
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x + y -z = 6 , x^3 + y^3 -z^3 = 414, diophantine
Determine all three integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that are solutions of the system
x
+
y
−
z
=
6
x + y -z = 6
x
+
y
−
z
=
6
x
3
+
y
3
−
z
3
=
414
x^3 + y^3 -z^3 = 414
x
3
+
y
3
−
z
3
=
414
2
1
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no of different squares in a lattice of 100x100 points
The points of this lattice
4
×
4
=
16
4\times 4 = 16
4
×
4
=
16
points can be vertices of squares. [asy] unitsize(1 cm);int i, j;for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); }draw((1,1)--(2,2)--(1,3)--(0,2)--cycle);for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy]Calculate the number of different squares that can be formed in a lattice of
100
×
100
100\times 100
100
×
100
points.
1
1
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6^n + 1 has all digits same
Determine all positive integers
n
n
n
such that the decimal representation of the number
6
n
+
1
6^n + 1
6
n
+
1
has all its digits the same.
3
1
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ratio of areas (PQA): (ABC) wanted, AB=BC,AQ=QP=PC (Chile 2020 L2 P3)
Given the isosceles triangle
A
B
C
ABC
A
BC
with
∣
A
B
∣
=
∣
A
C
∣
=
10
| AB | = | AC | = 10
∣
A
B
∣
=
∣
A
C
∣
=
10
and
∣
B
C
∣
=
15
| BC | = 15
∣
BC
∣
=
15
. Let points
P
P
P
in
B
C
BC
BC
and
Q
Q
Q
in
A
C
AC
A
C
chosen such that
∣
A
Q
∣
=
∣
Q
P
∣
=
∣
P
C
∣
| AQ | = | QP | = | P C |
∣
A
Q
∣
=
∣
QP
∣
=
∣
PC
∣
. Calculate the ratio of areas of the triangles
(
P
Q
A
)
:
(
A
B
C
)
(PQA): (ABC)
(
PQ
A
)
:
(
A
BC
)
.