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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2008 Bosnia Herzegovina Team Selection Test
2008 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(3)
2
2
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Divisibility
Find all pairs of positive integers
m
m
m
and
n
n
n
that satisfy (both) following conditions: (i) m^{2}\minus{}n divides m\plus{}n^{2} (ii) n^{2}\minus{}m divides n\plus{}m^{2}
Circumcenter lies on EF
Let
A
D
AD
A
D
be height of triangle
△
A
B
C
\triangle ABC
△
A
BC
and
R
R
R
circumradius. Denote by
E
E
E
and
F
F
F
feet of perpendiculars from point
D
D
D
to sides
A
B
AB
A
B
and
A
C
AC
A
C
. If AD\equal{}R\sqrt{2}, prove that circumcenter of triangle
△
A
B
C
\triangle ABC
△
A
BC
lies on line
E
F
EF
EF
.
1
2
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simple inequality
Prove that in an isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
with AC\equal{}BC\equal{}b following inequality holds
b
>
π
r
b> \pi r
b
>
π
r
, where
r
r
r
is inradius.
very simple
8
8
8
students took part in exam that contains
8
8
8
questions. If it is known that each question was solved by at least
5
5
5
students, prove that we can always find
2
2
2
students such that each of questions was solved by at least one of them.
3
2
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30 persons (round table)
30
30
30
persons are sitting at round table. 30 \minus{} N of them always speak true ("true speakers") while the other
N
N
N
of them sometimes speak true sometimes not ("lie speakers"). Question: "Who is your right neighbour - "true speaker" or "lie speaker" ?" is asked to all 30 persons and 30 answers are collected. What is maximal number
N
N
N
for which (with knowledge of these answers) we can always be sure (decide) about at least one person who is "true speaker".
Nice functional equation
Find all functions
f
:
R
→
R
f: \mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
satisfying f(f(x) \plus{} y) \equal{} f(x^2 \minus{} y) \plus{} 4f(x)y for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.