MathDB
Problems
Contests
National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2017 El Salvador Correspondence
2017 El Salvador Correspondence
Part of
El Salvador Correspondence
Subcontests
(1)
1
Hide problems
2017 El Salvador Correspondence / Qualifying NMO XVII
p1. Determine the number of positive integers from
1
1
1
to
1
0
2017
10^{2017}
1
0
2017
that satisfy that the sum of their digits is exactly
2
2
2
. p2. A positive integer is such that both adding
2000
2000
2000
and subtracting
17
17
17
results in a perfect square. Determine that number. p3. Each integer number is painted red or blue according to the following rules:
∙
\bullet
∙
The number
1
1
1
is red.
∙
\bullet
∙
If
a
a
a
and
b
b
b
are two red numbers, not necessarily different, then the numbers
a
−
b
a-b
a
−
b
and
a
+
b
a + b
a
+
b
have different colors. Determine the color of the number
2017
2017
2017
. p4. Determine all positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
that satisfy the system of equations
1
x
+
y
+
z
=
x
+
1
y
+
z
=
x
+
y
+
1
x
=
3
\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y +\frac{1}{x} = 3
x
1
+
y
+
z
=
x
+
y
1
+
z
=
x
+
y
+
x
1
=
3
p5. Let
A
B
C
D
ABCD
A
BC
D
be a rectangle and
M
M
M
a point on segment
B
C
BC
BC
. The bisector of angle
∠
D
A
M
\angle DAM
∠
D
A
M
intersect segment
C
D
CD
C
D
at
N
N
N
. If
A
M
=
B
M
+
D
N
AM = BM +DN
A
M
=
BM
+
D
N
, prove that
A
B
C
D
ABCD
A
BC
D
is a square.