Question
Muirhead’s inequality generalizes this inequality that gives that the cube in Hoffman’s packing puzzle has enough volume to fit all of the pieces. For 10 points each:
[10h] Name this inequality that generalizes the statement that [read slowly] twice the square root of a times b [pause] is bounded above by a plus b.
ANSWER: AM–GM inequality [or inequality of arithmetic and geometric means or arithmetic mean–geometric mean inequality]
[10e] The AM–GM inequality can be proved via Jensen’s (“YEN-sen’s”) inequality, which applies to functions with this property, the opposite of concave.
ANSWER: convexity [prompt on concave up]
[10m] This mean, the reciprocal of the arithmetic mean of the reciprocals, is bounded above by the aforementioned means. This mean shares its name with a series that sums the reciprocals of the positive integers.
ANSWER: harmonic mean [or harmonic series]
<Edward Liu, Science - Math> ~22180~ <Editor: David Bass>
Data
| Team | Opponent | Part 1 | Part 2 | Part 3 | Total |
|---|---|---|---|---|---|
| Heights | Innovation Academy A | 0 | 10 | 10 | 20 |
| Johns Creek A | Hunter A | 10 | 10 | 10 | 30 |
| Kinkaid | Belmont | 10 | 10 | 10 | 30 |
| Winston Churchill A | Hoover A | 0 | 10 | 10 | 20 |
Summary
| Tournament | Edition | Exact Match? | Heard | PPB | Easy % | Medium % | Hard % |
|---|---|---|---|---|---|---|---|
| 2024 PACE NSC | 06/08/2024 | Y | 4 | 25.00 | 100% | 100% | 50% |