MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile Classification NMO Juniors
2017 Chile Classification NMO Juniors
2017 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
(1)
1
Hide problems
2017 Chile Classification / Qualifying NMO Juniors XXIX
p1. Inside a rectangle a point
P
P
P
is marked. Segments are drawn that join the vertices with
P
P
P
. In an alternate way, the sectors that are formed are colored. Show that the sum of the areas of the painted sectors is equal to the sum of the unpainted sectors. p2. Determine the smallest integer
n
n
n
such that the decimal representation of
15
⋅
n
15\cdot n
15
⋅
n
has only
0
0
0
and
2
2
2
. p3. Find all functions
f
:
Z
→
Z
f:Z \to Z
f
:
Z
→
Z
with the property that for every pair of integers
x
,
y
x,y
x
,
y
the equation holds
f
(
x
+
f
(
f
(
y
)
)
)
=
y
+
f
(
f
(
x
)
)
f(x + f(f(y))) = y+ f(f(x))
f
(
x
+
f
(
f
(
y
)))
=
y
+
f
(
f
(
x
))
p4. Calculate the maximum number of different paths that can be built on a pool table to match two balls with a
3
3
3
-cushion shot. No ball is touching a band. A band is defined when a ball bounces off one side of the table, with the angle of incidence equal to the angle of departure. A path in two bands is shown in the following figure: https://cdn.artofproblemsolving.com/attachments/9/b/e3e4be3b7f127f61f07be48bfc5ff4ac810e20.jpgPS. Juniors p1 was also [url=https://artofproblemsolving.com/community/c4h2691326p23361117]Seniors p1.