MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2016 Latvia Baltic Way TST
2
2
Part of
2016 Latvia Baltic Way TST
Problems
(1)
0 <= mu + nv <= 2 \sqrt{X}
Source: 2016 Latvia BW TST P2
12/17/2022
Given natural numbers
m
,
n
m, n
m
,
n
and
X
X
X
such that
X
≥
m
X \ge m
X
≥
m
and
X
≥
n
X \ge n
X
≥
n
. Prove that one can find two integers
u
u
u
and
v
v
v
such that
∣
u
∣
+
∣
v
∣
>
0
|u| + |v| > 0
∣
u
∣
+
∣
v
∣
>
0
,
∣
u
∣
≤
X
|u| \le \sqrt{X}
∣
u
∣
≤
X
,
∣
v
∣
≤
X
|v| \le \sqrt{X}
∣
v
∣
≤
X
and
0
≤
m
u
+
n
v
≤
2
X
.
0 \le mu + nv \le 2 \sqrt{X}.
0
≤
m
u
+
n
v
≤
2
X
.
number theory
inequalities
algebra