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Packet 6: Bonus 4

Brouwer and Katutani name theorems for finding these values. For 10 points each:
[10m] Name these values, which do not change under transformation. For functions, these values map to themselves for a given mapping.
ANSWER: fixed points [accept invariant points; accept Brouwer fixed point theorem or Katutani fixed point theorem]
[10h] For a function to satisfy the Brouwer fixed-point theorem, it must be continuous and map a convex set with this property to itself. The Heine–Borel theorem states that this property is equivalent to being closed and bounded.
ANSWER: compactness
[10e] Because fixed points of functions map to themselves, functions with fixed points must intersect this line. Inverse functions are reflections of functions over this line, which has a slope of 1.
ANSWER: y=x (“y equals x”) [accept identity function; prompt on x]
<Science - Math, Aveek Sarkar> | VAULT II Packet 06

HeardPPBE %M %H %
2315.6591%39%26%

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Conversion

TeamOpponentPart 1Part 2Part 3TotalParts
Georgia AWashington001010E
Georgia BNew York001010E
Georgia CNorth Carolina001010E
Texas ACalifornia0000
Texas BKentucky001010E

Summary

TournamentEditionMatchHeardPPBE %M %H %
VAULT II at Georgia Tech2026-01-24✕518.00100%60%20%
Triangle Cup2026-03-13✕310.00100%0%0%
VAULT II at Eden Prairie HS2026-03-13✕323.33100%67%67%
VAULT II at McGill2026-03-13✕410.0075%25%0%
VAULT II at UCSB2026-03-13✕520.00100%40%60%
VAULT II Northern UK2026-04-24✓316.67100%33%33%
VAULT II Southern UK2026-04-24✓422.50100%100%25%
VAULT II at Harvard2026-04-30✓316.67100%33%33%
VAULT II at MINT2026-06-19✓58.0080%0%0%
VAULT II Stanford Online2026-07-17✓816.2588%38%38%