Competition math — in-game stage & study lens
Competition math — in-game stage & study lens
First Principles includes a dedicated Competition math stage (near the end of level select) that uses y = a\ln(k(x-d)) + c on a shifted domain — the same natural logarithm family you use in AP Calculus, TMUA/MAT, and contest prep, but framed for inequalities, bounding, and “is the function bending the right way?”
This page is unofficial study context only — not affiliated with the MAA (AMC/AIME), UKMT, Oxford MAT, or any olympiad organiser. For real contests, use official syllabi, past papers, and coaches.
What “competition math” usually stresses
| Theme | Calculus link | Game hook |
|---|---|---|
| Bounding & estimates | Compare to tangents/secants; mean value mood | (\ln) is concave: chord/tangent tricks are standard toolkit |
| AM–GM (arithmetic–geometric mean) | Often proved via (\ln) or convexity | Same “turn products into sums” spirit as (\sum \ln) |
| Smoothness & case splits | Domain of (\ln), removable vs essential mood | Stage stays on a safe branch of (\ln) so the graph is clean |
| Clever substitution | Chain rule / change of variables | Parallel to “replace (u) to simplify” in integrals |
Why (\ln) for this stage
- Concavity: (f’‘(x)<0) on ((0,\infty)) for (f(x)=\ln x) — Jensen’s inequality for concave functions (and the tangent line upper bound) shows up constantly in olympiad-style inequalities.
- Derivative: (\frac{d}{dx}\ln x = \frac{1}{x}) — reciprocals and harmonic-flavored estimates appear in many discrete bounds.
- Domain hygiene: (\ln) forces you to think where an expression is defined — a contest habit of checking positivity before taking logs.
The derivative-driven platform rule in the game is unchanged: where the sampled derivative is large enough, you tend to get platforms; where it is not, you may see gaps / hazards — you are still “walking on slopes.”
How this fits next to other docs
| Resource | Role |
|---|---|
| Math concepts | Broad in-game curriculum index |
| AP Calculus BC — prep | Syllabus-shaped BC map (includes (\ln), concavity, series) |
| AMC 10 & 12 — prep | MAA AMC multiple-choice map (unofficial); algebra/geo/NT + graph bridges |
| TMUA — calculus / MAT — calculus | UK admissions multiple-choice / reasoning pace |
| This page | Contest style: bounds, concavity, (\ln) as a hub |
In-game title
The sample lineup no longer dedicates a level to this ln run — use any natural-log stage in your own graphs or study (\ln) concavity here; Mandelbrot remains the final boss after the economics pair in Advanced & boss.
Unofficial · First Principles (proprietary) — for learning mood only.