MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1969 Poland - Second Round
1
a^2 + b^2 = 1, c^2 + d^2 = 1, ac + bd = - 1/2
a^2 + b^2 = 1, c^2 + d^2 = 1, ac + bd = - 1/2
Source: Polish MO second round 1969 p1
August 28, 2024
algebra
system of equations
Problem Statement
Prove that if the real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy the equations
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
,
a
c
+
b
d
=
−
1
2
,
\; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},
a
2
+
b
2
=
1
,
c
2
+
d
2
=
1
,
a
c
+
b
d
=
−
2
1
,
then
a
2
+
a
c
+
c
2
=
b
2
+
b
d
+
d
2
.
a^2 + ac + c^2 = b^2 + bd + d^2.
a
2
+
a
c
+
c
2
=
b
2
+
b
d
+
d
2
.
Back to Problems
View on AoPS