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Omni-MATH / Let a=2001. Consider the set A of all pairs of integers (m,n) with n≠0 such that (i) m<2a; (ii) 2n|(2am−m²+n²); (iii) n²−m²+2mn≤2a(n−m). For (m,n)∈A, let f(m,n)=(2am−m²−mn)/(n). De…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let

a=2001

. Consider the set

A

of all pairs of integers

(m,n)

with

n\neq0

such that (i)

m<2a

; (ii)

2n|(2am-m^2+n^2)

; (iii)

n^2-m^2+2mn\leq2a(n-m)

. For

(m, n)\in A

, let

f(m,n)=\frac{2am-m^2-mn}{n}.

Determine the maximum and minimum values of

f

Plain-text mathematical notation (without MathML)

Let a=2001. Consider the set A of all pairs of integers (m,n) with n≠0 such that
(i) m<2a;
(ii) 2n|(2am−m²+n²);
(iii) n²−m²+2mn≤2a(n−m).
For (m,n)∈A, let f(m,n)=(2am−m²−mn)/(n).
Determine the maximum and minimum values of f.

Original LaTeX notation

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that
(i) $m<2a$;
(ii) $2n|(2am-m^2+n^2)$;
(iii) $n^2-m^2+2mn\leq2a(n-m)$.
For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\]
Determine the maximum and minimum values of $f$.

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task_id: f94ef29f-53a0-59c1-80ad-7ee6d9431820
task_key: test--f94ef29f-53a0-59c1-80ad-7ee6d9431820
task_revision_id: 4
config / split: / test

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