MathDB
Problems
Contests
International Contests
IMO Longlists
1966 IMO Longlists
1966 IMO Longlists
Part of
IMO Longlists
Subcontests
(3)
50
1
Hide problems
Soviet Union 7
For any quadrilateral with the side lengths
a
,
a,
a
,
b
,
b,
b
,
c
,
c,
c
,
d
d
d
and the area
S
,
S,
S
,
prove the inequality
S
≤
a
+
c
2
⋅
b
+
d
2
.
S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.
S
≤
2
a
+
c
⋅
2
b
+
d
.
18
1
Hide problems
Hungary 1
Solve the equation
1
sin
x
+
1
cos
x
=
1
p
,
\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p},
s
i
n
x
1
+
c
o
s
x
1
=
p
1
,
where
p
p
p
is a real parameter. Investigate for which values of
p
p
p
solutions exist and how many solutions exist.(Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo
2
π
2\pi
2
π
''.)
2
1
Hide problems
The Democratic Republic Of Germany 1
Given
n
n
n
positive numbers
a
1
,
a_{1},
a
1
,
a
2
,
a_{2},
a
2
,
.
.
.
,
...,
...
,
a
n
a_{n}
a
n
such that
a
1
⋅
a
2
⋅
.
.
.
⋅
a
n
=
1.
a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.
a
1
⋅
a
2
⋅
...
⋅
a
n
=
1.
Prove
(
1
+
a
1
)
(
1
+
a
2
)
.
.
.
(
1
+
a
n
)
≥
2
n
.
\left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.
(
1
+
a
1
)
(
1
+
a
2
)
...
(
1
+
a
n
)
≥
2
n
.