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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2022 El Salvador Correspondence
2022 El Salvador Correspondence
Part of
El Salvador Correspondence
Subcontests
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2022 El Salvador Correspondence / Qualifying NMO XXII
p1. Ana is thinking of two numbers
a
a
a
,
b
b
b
and Beto tries to guess them. Beto asks for a clue and Ana tells him: "Just knowing that they comply
a
2
+
a
=
b
2
+
3
b
+
2
a^2 + a = b^2 + 3b + 2
a
2
+
a
=
b
2
+
3
b
+
2
you will be able to determine them." After listening to the clue, Beto is able to deduce the values of
a
a
a
and
b
b
b
. Determine these values. p2. In the following figure the circles have a radius of
1
1
1
cm and
G
H
=
5
GH = 5
G
H
=
5
cm. Determine the length of the sides of the rectangle. https://cdn.artofproblemsolving.com/attachments/a/4/6c79ac93ba8e2f7c94045c54cd5f3d25162051.pngp3. There are three gears in the position shown in the figure. It is known that they have an entire perimeter
P
1
P_1
P
1
,
P
2
P_2
P
2
and
P
3
P_3
P
3
(from left to right) and that if the first is given a complete turn, then one turn complete turn to the second and finally a complete turn to the third (all clockwise), the three gears remain in their original position. Determine the possible values of
(
P
1
,
P
2
,
P
3
)
(P_1, P_2, P_3)
(
P
1
,
P
2
,
P
3
)
. https://cdn.artofproblemsolving.com/attachments/2/e/fa7cf45b5419acec4a64fcba0d08087ddbaa23.pngp4. In a regular pyramid-shaped container with a square base,
120
120
120
cm
3
^3
3
of water is placed. If the pyramid is placed with the base on the ground, the water covers up to
5
5
5
cm in height and if placed upside down, The water covers
10
10
10
cm in height. Determine the side of the base and the height of the container. p5. A frog is at the origin of the plane. In the second
k
k
k
, the frog decides which direction walk (up, down, left or right) and walk
2
2
2
units. Prove that (a) The frog can reach
(
4
,
0
)
(4, 0)
(
4
,
0
)
and
(
4
,
4
)
(4, 4)
(
4
,
4
)
. (b) There is a path that the frog can take to pass through all the points of the shape
(
4
a
,
4
b
)
(4a, 4b)
(
4
a
,
4
b
)
with
a
a
a
,
b
b
b
integers. (c) The frog can reach any integer coordinate point on the plane.PS. You should use hide for answers.