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Contests
National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1928 Eotvos Mathematical Competition
1928 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
min of longer of lengths AP,BP
Let
ℓ
\ell
ℓ
be a given line,
A
A
A
and
B
B
B
given points of the plane. Choose a point
P
P
P
on
ℓ
\ell
ℓ
so that the longer of the segments
A
P
AP
A
P
,
B
P
BP
BP
is as short as possible. (If
A
P
=
B
P
,
AP = BP,
A
P
=
BP
,
either segment may be taken as the longer one.)
2
1
Hide problems
numbers 1-n on the circumference of a circle
Put the numbers
1
,
2
,
3
,
.
.
.
,
n
1,2,3,...,n
1
,
2
,
3
,
...
,
n
on the circumference of a circle in such a way that the difference between neighboring numbers is at most
2
2
2
. Prove that there is just one solution (if regard is paid only to the order in which the numbers are arranged).
1
1
Hide problems
there is one that differs from an integer by at most 1/n
Prove that, among the positive numbers
a
,
2
a
,
.
.
.
,
(
n
−
1
)
a
.
a,2a, ...,(n - 1)a.
a
,
2
a
,
...
,
(
n
−
1
)
a
.
there is one that differs from an integer by at most
1
/
n
1/n
1/
n
.