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National and Regional Contests
Russia Contests
All-Russian Olympiad
1971 All Soviet Union Mathematical Olympiad
157
157
Part of
1971 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 157 All Soviet Union MO 1971 f_a(x,y) = x^2 + axy + y^2
Source:
7/3/2019
a) Consider the function
f
(
x
,
y
)
=
x
2
+
x
y
+
y
2
f(x,y) = x^2 + xy + y^2
f
(
x
,
y
)
=
x
2
+
x
y
+
y
2
Prove that for the every point
(
x
,
y
)
(x,y)
(
x
,
y
)
there exist such integers
(
m
,
n
)
(m,n)
(
m
,
n
)
, that
f
(
(
x
−
m
)
,
(
y
−
n
)
)
≤
1
/
2
f((x-m),(y-n)) \le 1/2
f
((
x
−
m
)
,
(
y
−
n
))
≤
1/2
b) Let us denote with
g
(
x
,
y
)
g(x,y)
g
(
x
,
y
)
the least possible value of the
f
(
(
x
−
m
)
,
(
y
−
n
)
)
f((x-m),(y-n))
f
((
x
−
m
)
,
(
y
−
n
))
for all the integers
m
,
n
m,n
m
,
n
. The statement a) was equal to the fact
g
(
x
,
y
)
≤
1
/
2
g(x,y) \le 1/2
g
(
x
,
y
)
≤
1/2
. Prove that in fact,
g
(
x
,
y
)
≤
1
/
3
g(x,y) \le 1/3
g
(
x
,
y
)
≤
1/3
Find all the points
(
x
,
y
)
(x,y)
(
x
,
y
)
, where
g
(
x
,
y
)
=
1
/
3
g(x,y)=1/3
g
(
x
,
y
)
=
1/3
. c) Consider function
f
a
(
x
,
y
)
=
x
2
+
a
x
y
+
y
2
(
0
≤
a
≤
2
)
f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)
f
a
(
x
,
y
)
=
x
2
+
a
x
y
+
y
2
(
0
≤
a
≤
2
)
Find any
c
c
c
such that
g
a
(
x
,
y
)
≤
c
g_a(x,y) \le c
g
a
(
x
,
y
)
≤
c
. Try to obtain the closest estimation.
algebra
polynomial