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OlympiadBench / 2091 / For any integer n≥2, let N(n) be the maximal number of triples…

Problem

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

question

For any integer

n \geq 2

, let

N(n)

be the maximal number of triples

\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)

, consisting of nonnegative integers

a_{i}, b_{i}

and

c_{i}

such that the following two conditions are satisfied: (1)

a_{i}+b_{i}+c_{i}=n

for all

i=1, \ldots, N(n)

, (2) If

i \neq j

, then

a_{i} \neq a_{j}, b_{i} \neq b_{j}

and

c_{i} \neq c_{j}

. Determine

N(n)

for all

n \geq 2

Plain-text mathematical notation (without MathML)

For any integer n≥2, let N(n) be the maximal number of triples (a_(i),b_(i),c_(i)),i=1,…,N(n), consisting of nonnegative integers a_(i),b_(i) and c_(i) such that the following two conditions are satisfied:

(1) a_(i)+b_(i)+c_(i)=n for all i=1,…,N(n),

(2) If i≠j, then a_(i)≠a_(j),b_(i)≠b_(j) and c_(i)≠c_(j).

Determine N(n) for all n≥2.

Original LaTeX notation

For any integer $n \geq 2$, let $N(n)$ be the maximal number of triples $\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i}$ and $c_{i}$ such that the following two conditions are satisfied:

(1) $a_{i}+b_{i}+c_{i}=n$ for all $i=1, \ldots, N(n)$,

(2) If $i \neq j$, then $a_{i} \neq a_{j}, b_{i} \neq b_{j}$ and $c_{i} \neq c_{j}$.

Determine $N(n)$ for all $n \geq 2$.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2091
task_id: a7a6c56b-477a-55b0-af91-8eacd5b0859d
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2091
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

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