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Brazil Team Selection Test
2015 Brazil Team Selection Test
2015 Brazil Team Selection Test
Part of
Brazil Team Selection Test
Subcontests
(2)
1
2
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f(x + a) is even, and f(x + b) is odd => f(x) is periodic
Let's call a function
f
:
R
→
R
f : R \to R
f
:
R
→
R
cool if there are real numbers
a
a
a
and
b
b
b
such that
f
(
x
+
a
)
f(x + a)
f
(
x
+
a
)
is an even function and
f
(
x
+
b
)
f(x + b)
f
(
x
+
b
)
is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.
walking over the edges of a connected graph, stuck at same vertexat the end
Starting at a vertex
x
0
x_0
x
0
, we walk over the edges of a connected graph
G
G
G
according to the following rules:1. We never walk the same edge twice in the same direction.2. Once we reach a vertex
x
≠
x
0
x \ne x_0
x
=
x
0
, never visited before, we mark the edge by which we come to
x
x
x
. We can use this marked edge to leave vertex
x
x
x
only if we already have traversed, in both directions, all other edges incident to
x
x
x
.Show that, regardless of the path followed, we will always be stuck at
x
0
x_0
x
0
and that all other edges will have been traveled in both directions.
3
1
Hide problems
y^2-P(x)| <= 2|x| iff |x^2-P(y)|<=2|y|. P(0) = 1
Determine all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients and which satisfy the following properties: i)
P
(
0
)
=
1
P(0) = 1
P
(
0
)
=
1
ii) for any real numbers
x
x
x
and
y
,
y,
y
,
|y^2-P(x)|\le 2|x| \text{if and only if} |x^2-P(y)|\le 2|y|.