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International Contests
Austrian-Polish
1997 Austrian-Polish Competition
7
7
Part of
1997 Austrian-Polish Competition
Problems
(1)
largest real constant c, such p^2 + q^2 + 1 >= bp(q + 1(
Source: Austrian - Polish 1997 APMC
5/3/2020
(a) Prove that
p
2
+
q
2
+
1
>
p
(
q
+
1
)
p^2 + q^2 + 1 > p(q + 1)
p
2
+
q
2
+
1
>
p
(
q
+
1
)
for any real numbers
p
,
q
p, q
p
,
q
, . (b) Determine the largest real constant
b
b
b
such that the inequality
p
2
+
q
2
+
1
≥
b
p
(
q
+
1
)
p^2 + q^2 + 1 \ge bp(q + 1)
p
2
+
q
2
+
1
≥
b
p
(
q
+
1
)
holds for all real numbers
p
,
q
p, q
p
,
q
(c) Determine the largest real constant c such that the inequality
p
2
+
q
2
+
1
≥
c
p
(
q
+
1
)
p^2 + q^2 + 1 \ge cp(q + 1)
p
2
+
q
2
+
1
≥
c
p
(
q
+
1
)
holds for all integers
p
,
q
p, q
p
,
q
.
inequalities
max
algebra