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OlympiadBench / 2349 / Without using a calculator, determine positive integers m and n for which

Problem

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

question

Without using a calculator, determine positive integers

m

and

n

for which

\sin ^{6} 1^{\circ}+\sin ^{6} 2^{\circ}+\sin ^{6} 3^{\circ}+\cdots+\sin ^{6} 87^{\circ}+\sin ^{6} 88^{\circ}+\sin ^{6} 89^{\circ}=\frac{m}{n}

(The sum on the left side of the equation consists of 89 terms of the form

\sin ^{6} x^{\circ}

, where

x

takes each positive integer value from 1 to 89.)

Plain-text mathematical notation (without MathML)

Without using a calculator, determine positive integers m and n for which

sin⁶1^(∘)+sin⁶2^(∘)+sin⁶3^(∘)+⋯+sin⁶87^(∘)+sin⁶88^(∘)+sin⁶89^(∘)=(m)/(n)

(The sum on the left side of the equation consists of 89 terms of the form sin⁶x^(∘), where x takes each positive integer value from 1 to 89.)

Original LaTeX notation

Without using a calculator, determine positive integers $m$ and $n$ for which

$$
\sin ^{6} 1^{\circ}+\sin ^{6} 2^{\circ}+\sin ^{6} 3^{\circ}+\cdots+\sin ^{6} 87^{\circ}+\sin ^{6} 88^{\circ}+\sin ^{6} 89^{\circ}=\frac{m}{n}
$$

(The sum on the left side of the equation consists of 89 terms of the form $\sin ^{6} x^{\circ}$, where $x$ takes each positive integer value from 1 to 89.)

answer type

Numerical

is multiple answer

true

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2349
task_id: cf87ced3-eb6d-5181-aace-3000fa755c9b
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2349
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import