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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1960 Kurschak Competition
1960 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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FG touches the inscribed circle of the square.
E
E
E
is the midpoint of the side
A
B
AB
A
B
of the square
A
B
C
D
ABCD
A
BC
D
, and
F
,
G
F, G
F
,
G
are any points on the sides
B
C
BC
BC
,
C
D
CD
C
D
such that
E
F
EF
EF
is parallel to
A
G
AG
A
G
. Show that
F
G
FG
FG
touches the inscribed circle of the square.
2
1
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every n = sum a_ if a_k < 1 + a_1 + a2_ +... + a_{k-1
Let
a
1
=
1
,
a
2
,
a
3
,
.
.
.
a_1 = 1, a_2, a_3,...
a
1
=
1
,
a
2
,
a
3
,
...
: be a sequence of positive integers such that
a
k
<
1
+
a
1
+
a
2
+
.
.
.
+
a
k
−
1
a_k < 1 + a_1 + a_2 +... + a_{k-1}
a
k
<
1
+
a
1
+
a
2
+
...
+
a
k
−
1
for all
k
>
1
k > 1
k
>
1
. Prove that every positive integer can be expressed as a sum of
a
i
a_i
a
i
s.
1
1
Hide problems
Among any 4 people at a party
Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party