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OlympiadBench / 2261 / Let ⌊x⌋ denote the greatest integer less than or equal to x.…

Problem

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

answer type

Numerical

is multiple answer

false

language

English

question

Let

\lfloor x\rfloor

denote the greatest integer less than or equal to

x

. For example,

\lfloor 3.1\rfloor=3

and

\lfloor-1.4\rfloor=-2

. Suppose that

f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor

and

g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor

for each positive integer

n

. Determine a value of

n

for which

f(n)=100

Plain-text mathematical notation (without MathML)

Let ⌊x⌋ denote the greatest integer less than or equal to x. For example, ⌊3.1⌋=3 and ⌊−1.4⌋=−2.

Suppose that f(n)=2n−⌊(1+√(8n−7))/(2)⌋ and g(n)=2n+⌊(1+√(8n−7))/(2)⌋ for each positive integer n.
Determine a value of n for which f(n)=100.

Original LaTeX notation

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.

Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determine a value of $n$ for which $f(n)=100$.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2261
task_id: 148fe168-6722-5706-98aa-0a4af55f00de
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2261
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

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