Packet 10: Bonus 8
A famous unsolved problem in geometry asks whether one of these polygons, whose side lengths and area are rational, can have irrational diagonals. For 10 points each:
[10e] Name these five-sided polygons whose angles sum to 540 degrees.
ANSWER: pentagons
[10m] This mathematician’s formula can find a solution for triangles. This mathematician’s namesake formula is used to find the area of a triangle given the semiperimeter and the three side lengths.
ANSWER: Hero of Alexandria [or Heron of Alexandria; or Heron's formula; or Heron's method]
[10h] Pentagon, whose diagonals are all rational, must have this property. Ptolemy's theorem relates the four sides and two diagonals of a quadrilateral with this property.
ANSWER: cyclic quadrilaterals
<Science - Math, Vikram Narasimhan> | VAULT II Packet 10
| Heard | PPB | E % | M % | H % |
|---|---|---|---|---|
| 37 | 20.27 | 100% | 73% | 30% |
Conversion
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total | Parts |
|---|---|---|---|---|---|---|
| Bath | Birmingham B | 10 | 0 | 0 | 10 | E |
| Birmingham A | Warperial | 10 | 0 | 0 | 10 | E |
| Cambridge | Southampton | 10 | 10 | 10 | 30 | EMH |
| Oxford | Southperial | 10 | 10 | 0 | 20 | EM |
Summary
| Tournament | Edition | Match | Heard | PPB | E % | M % | H % |
|---|---|---|---|---|---|---|---|
| VAULT II at Georgia Tech | 2026-01-24 | ✓ | 6 | 28.33 | 100% | 100% | 83% |
| VAULT II at Eden Prairie HS | 2026-03-13 | ✓ | 3 | 23.33 | 100% | 100% | 33% |
| VAULT II at McGill | 2026-03-13 | ✓ | 4 | 20.00 | 100% | 75% | 25% |
| VAULT II at UCSB | 2026-03-13 | ✓ | 5 | 16.00 | 100% | 40% | 20% |
| VAULT II Northern UK | 2026-04-24 | ✓ | 3 | 13.33 | 100% | 33% | 0% |
| VAULT II Southern UK | 2026-04-24 | ✓ | 4 | 17.50 | 100% | 50% | 25% |
| VAULT II at Harvard | 2026-04-30 | ✓ | 4 | 22.50 | 100% | 75% | 50% |
| VAULT II at MINT | 2026-06-19 | ✓ | 1 | 10.00 | 100% | 0% | 0% |
| VAULT II Stanford Online | 2026-07-17 | ✓ | 7 | 20.00 | 100% | 100% | 0% |