Packet 6: Bonus 4
Brouwer and Katukani 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 Katukani 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
| Heard | PPB | E % | M % | H % |
|---|---|---|---|---|
| 20 | 16.50 | 95% | 40% | 30% |
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| Albanian Rift in the Sky | Prat Pack | 10 | 10 | 10 | 30 | MHE |
| Big Brown Ballers | Basileus II: Electric Boogaloo | 10 | 0 | 10 | 20 | ME |
| Heauton Timorumenos | With Less Than 2 Years of Combined Quizbowl Experience | 0 | 0 | 10 | 10 | E |
| Midnight Bus to Vista | Leif Eriksen's Striking Appearance | 0 | 10 | 10 | 20 | HE |
| Taman's Nepo Babies | whatever | 0 | 10 | 10 | 20 | HE |
Summary
| Tournament | Edition | Match | Heard | PPB | E % | M % | H % |
|---|---|---|---|---|---|---|---|
| VAULT II at Georgia Tech | 2026-01-24 | ✓ | 5 | 18.00 | 100% | 60% | 20% |
| Triangle Cup | 2026-03-13 | ✓ | 3 | 10.00 | 100% | 0% | 0% |
| VAULT II at Eden Prairie HS | 2026-03-13 | ✓ | 3 | 23.33 | 100% | 67% | 67% |
| VAULT II at McGill | 2026-03-13 | ✓ | 4 | 10.00 | 75% | 25% | 0% |
| VAULT II at UCSB | 2026-03-13 | ✓ | 5 | 20.00 | 100% | 40% | 60% |
| VAULT II Northern UK | 2026-04-24 | ✕ | 3 | 16.67 | 100% | 33% | 33% |
| VAULT II Southern UK | 2026-04-24 | ✕ | 4 | 22.50 | 100% | 100% | 25% |
| VAULT II at Harvard | 2026-04-30 | ✕ | 3 | 16.67 | 100% | 33% | 33% |
| VAULT II at MINT | 2026-06-19 | ✕ | 5 | 8.00 | 80% | 0% | 0% |
| VAULT II Stanford Online | 2026-07-17 | ✕ | 8 | 16.25 | 88% | 38% | 38% |