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OlympiadBench / 1747 / Determine the smallest number M such that the inequality

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

Determine the smallest number MM such that the inequality |ab(a2−b2)+bc(b2−c2)+ca(c2−a2)|≤M(a2+b2+c2)2 \left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} holds for all real numbers a,b,ca, b, c.
Plain-text mathematical notation (without MathML)
Determine the smallest number M such that the inequality

|ab(a²−b²)+bc(b²−c²)+ca(c²−a²)|≤M(a²+b²+c²)²

holds for all real numbers a,b,c.
Original LaTeX notation
Determine the smallest number $M$ such that the inequality

$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$

holds for all real numbers $a, b, c$.

question type

Open-ended

subject

Math

Discussion

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Source and history

upstream_id / Original problem ID
1747
task_id
83e6d245-58e4-5beb-b883-f8c1d3c7c8f6
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1747
task_revision_id
2
config / split
OE_TO_maths_en_COMP / train

Official source

initial import