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National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2021 El Salvador Correspondence
2021 El Salvador Correspondence
Part of
El Salvador Correspondence
Subcontests
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2021 El Salvador Correspondence / Qualifying NMO XXI
p1. A group of mechanics have inspected a lot of old vehicles, and together they determined that
45
45
45
of these have engine problems,
30
30
30
have electrical faults and
35
35
35
have oil leaks. There are exactly
10
10
10
vehicles that have engine problems and electrical faults,
10
10
10
with engine problems and oil leakage, and
10
10
10
with electrical faults and engine leaks. But nevertheless, of all vehicles, only two have all three defects at the same time, while the last
18
18
18
vehicles they are in perfect condition. Determine how many total cars are on that lot. p2. A number is said fourfriend when he and all the numbers that are obtained by rearranging their digits in any order they are multiples of
4
4
4
. Determine how many
7
7
7
-digit numbers are fourfriends. Note: A friend number four cannot have
0
0
0
digits in its decimal representation. p3. A sequence of numbers is formed, each of which is equal to the product of
3
3
3
consecutive numbers, as follows:
1
×
2
×
3
1 \times 2 \times 3
1
×
2
×
3
,
2
×
3
×
4
2 \times 3 \times 4
2
×
3
×
4
,
3
×
4
×
5
3 \times 4 \times 5
3
×
4
×
5
, ... ,
98
×
×
99
×
10
98 \times× 99 \times 10
98
×
×
99
×
10
. Juan adds the inverses of all these numbers and obtains as a result a rational number, whose simplified expression is
p
/
q
p / q
p
/
q
(that is,
p
p
p
and
q
q
q
have no divisors in common), with natural
p
p
p
and
q
q
q
. Find the value of
p
+
q
p + q
p
+
q
. p4. There is a trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
D
AD
A
D
and
B
C
BC
BC
. A point
E
E
E
moves along side
A
B
AB
A
B
and let
O
1
O_1
O
1
,
O
2
O_2
O
2
be the centers of the circles that pass through the vertices of the triangles
A
E
D
AED
A
E
D
and
B
E
C
BEC
BEC
, respectively. Show that the distance from
O
1
O_1
O
1
to
O
2
O_2
O
2
is constant, regardless of the position of point
E
E
E
. p5. Determine the value of the following expression:
5
+
5
2
−
9
−
11
+
1
1
2
−
9
+
17
+
1
7
2
−
9
−
23
+
2
3
2
−
9
+
29
+
2
9
2
−
9
−
.
.
.
+
797
+
79
7
2
−
9
.
\sqrt{5 + \sqrt{5^2 - 9}} -\sqrt{11 + \sqrt{11^2 - 9}} + \sqrt{17 + \sqrt{17^2 - 9}} - \sqrt{23 + \sqrt{23^2 - 9}} + \sqrt{29 + \sqrt{29^2 - 9}} - ... +\sqrt{797 + \sqrt{797^2-9}}.
5
+
5
2
−
9
−
11
+
1
1
2
−
9
+
17
+
1
7
2
−
9
−
23
+
2
3
2
−
9
+
29
+
2
9
2
−
9
−
...
+
797
+
79
7
2
−
9
.