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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2002 Poland - Second Round
1
1
Part of
2002 Poland - Second Round
Problems
(2)
Periodic
Source: Poland
6/11/2005
Prove that all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfying, for all real
x
x
x
,
f
(
x
)
=
f
(
2
x
)
=
f
(
1
−
x
)
f(x)=f(2x)=f(1-x)
f
(
x
)
=
f
(
2
x
)
=
f
(
1
−
x
)
are periodic.
function
algebra
All expressions of p,q,r are primes
Source:
2/2/2011
Find all numbers
p
≤
q
≤
r
p\le q\le r
p
≤
q
≤
r
such that all the numbers
p
q
+
r
,
p
q
+
r
2
,
q
r
+
p
,
q
r
+
p
2
,
r
p
+
q
,
r
p
+
q
2
pq+r,pq+r^2,qr+p,qr+p^2,rp+q,rp+q^2
pq
+
r
,
pq
+
r
2
,
q
r
+
p
,
q
r
+
p
2
,
r
p
+
q
,
r
p
+
q
2
are prime.
modular arithmetic
number theory proposed
number theory