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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1939 Eotvos Mathematical Competition
1939 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
3 semicircles on sides of acute triangle
A
B
C
ABC
A
BC
is an acute triangle. Three semicircles are constructed outwardly on the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively. Construct points
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
on these semicìrcles respectively so that
A
B
′
=
A
C
′
AB' = AC'
A
B
′
=
A
C
′
,
B
C
′
=
B
A
′
BC' = BA'
B
C
′
=
B
A
′
and
C
A
′
=
C
B
′
CA'= CB'
C
A
′
=
C
B
′
.
1
1
Hide problems
(a_1 + a_2)(c_1 + c_2) >= (b_1 + b_2)^2
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,
c
1
c_1
c
1
and
c
2
c_2
c
2
be real numbers for which
a
1
a
2
>
0
a_1a_2 > 0
a
1
a
2
>
0
,
a
1
c
1
≥
b
1
2
a_1c_1 \ge b^2_1
a
1
c
1
≥
b
1
2
and
a
2
c
2
>
b
2
2
a_2c_2 > b^2_2
a
2
c
2
>
b
2
2
. Prove that
(
a
1
+
a
2
)
(
c
1
+
c
2
)
≥
(
b
1
+
b
2
)
2
(a_1 + a_2)(c_1 + c_2) \ge (b_1 + b_2)^2
(
a
1
+
a
2
)
(
c
1
+
c
2
)
≥
(
b
1
+
b
2
)
2
2
1
Hide problems
max power of 2 that divides 2^n!.
Determine the highest power of
2
2
2
that divides
2
n
!
2^n!
2
n
!
.