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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1953 Kurschak Competition
1953 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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<A=<D, <B=<E, <C=<F if ABCDEF is convex 6-gon, <A+<C+<E=<B+<D+<F, equal sides
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a convex hexagon with all its sides equal. Also
∠
A
+
∠
C
+
∠
E
=
∠
B
+
∠
D
+
∠
F
\angle A + \angle C + \angle E = \angle B + \angle D + \angle F
∠
A
+
∠
C
+
∠
E
=
∠
B
+
∠
D
+
∠
F
. Show that
∠
A
=
∠
D
\angle A = \angle D
∠
A
=
∠
D
,
∠
B
=
∠
E
\angle B = \angle E
∠
B
=
∠
E
and
∠
C
=
∠
F
\angle C = \angle F
∠
C
=
∠
F
.
2
1
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n^2 + d not a square if d divides 2n^2
n
n
n
and
d
d
d
are positive integers such that
d
d
d
divides
2
n
2
2n^2
2
n
2
. Prove that
n
2
+
d
n^2 + d
n
2
+
d
cannot be a square.
1
1
Hide problems
A,B subsets of 1-n |A|+|b|> n-
A
A
A
and
B
B
B
are any two subsets of
{
1
,
2
,
.
.
.
,
n
−
1
}
\{1, 2,...,n - 1\}
{
1
,
2
,
...
,
n
−
1
}
such that
∣
A
∣
+
∣
B
∣
>
n
−
1
|A| +|B|> n - 1
∣
A
∣
+
∣
B
∣
>
n
−
1
. Prove that one can find
a
a
a
in
A
A
A
and
b
b
b
in
B
B
B
such that
a
+
b
=
n
a + b = n
a
+
b
=
n
.