MathDB
Problems
Contests
National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2007 El Salvador Correspondence
2007 El Salvador Correspondence
Part of
El Salvador Correspondence
Subcontests
(1)
1
Hide problems
2007 El Salvador Correspondence / Qualifying NMO VII
p1. Let
A
B
C
ABC
A
BC
be a right triangle at
B
B
B
, side BC is
72
72
72
cm and side AC is
78
78
78
cm. Consider
D
D
D
on side
A
B
AB
A
B
such that
2
A
D
=
B
D
2AD = BD
2
A
D
=
B
D
. If
O
O
O
is the center of the circle that is tangent to side
B
C
BC
BC
and passes through
A
A
A
and
D
D
D
. Find the measure of
O
B
OB
OB
. p2.The increasing sequence
3
,
15
,
24
,
48
,
.
.
.
3, 15, 24, 48, ...
3
,
15
,
24
,
48
,
...
consists of all positive multiples of
3
3
3
which are one less than a perfect square number. Determine the remainder of the division per
1000
1000
1000
of the term that occupies position
2007
2007
2007
in the sequence. p3. Given
x
2
+
y
2
x
2
−
y
2
+
x
2
−
y
2
x
2
+
y
2
=
k
\frac{x^2+y^2}{x^2-y^2}+\frac{x^2-y^2}{x^2+y^2}=k
x
2
−
y
2
x
2
+
y
2
+
x
2
+
y
2
x
2
−
y
2
=
k
, express
x
8
+
y
8
x
8
−
y
8
+
x
8
−
y
8
x
8
+
y
2
\frac{x^8+y^8}{x^8-y^8}+\frac{x^8-y^8}{x^8+y^2}
x
8
−
y
8
x
8
+
y
8
+
x
8
+
y
2
x
8
−
y
8
in terms of k. p4. Two of the squares on a
7
×
7
7\times 7
7
×
7
board are painted white, and the rest are painted blue. We will say that two colorings are equal if one can be obtained from another by applying a rotation to the board. Determine the number of different colorings that exist. p5. In each square of a giant board there is written a natural number, according to the following rules: the numbers in the first column form an arithmetic sequence of first term
6
6
6
and difference
3
3
3
, that is,
6
,
9
,
12
,
15
,
.
.
.
6, 9, 12, 15, ...
6
,
9
,
12
,
15
,
...
The numbers of the first row form an arithmetic sequence of difference
3
3
3
, the numbers in the second row form an arithmetic progression of difference
5
5
5
, and in general, the numbers in row number
k
k
k
they form an arithmetic progression of difference
2
k
+
1
2k + 1
2
k
+
1
. Find all the cells that they have the number
2007
2007
2007
written on them.