Index
One line per note. Scan this first; open only what is relevant.
Format: - [Title](path) — one-line hook. `tag`, `tag`
algebra
- An odd solution count across two quadratics pins the parameter at a degenerate case — Roots come in pairs, so an odd total means a double root or a discarded one — an equation in the parameter, not an inequality.
quadratic,discriminant,counting - A piecewise bijection fills each gap in the range exactly — in either direction — The middle piece must be monotone with its endpoint values at the gap's ends; the reversed order counts too, so a bijection need not be monotone.
function,bijection,piecewise - A quadratic through two given points is the chord plus $a(x-p)(x-q)$ — Two points leave one free coefficient, so a third condition finishes it — at the midpoint $g=\ell-ah^2$, $h$ the half-width.
quadratic,interpolation,remainder-theorem - Two functions are equal when they agree at every point of their domain — A function is its input–output pairs: on a finite domain equality is one equation per element, and different formulas can agree.
function,equality,domain - An absolute-value equation solved by cases keeps only the roots inside each case — Each case solves against a whole line; a root on the wrong half hits the dashed extension, not the graph.
absolute-value,equation,case-analysis - An increasing function meets its inverse only on $y=x$ — So $f(x)=f^{-1}(x)$ reduces to $f(x)=x$; a decreasing $f$ can also meet its inverse in mirror pairs off that line.
function,inverse-function,monotone
analysis
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calculus
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discrete
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geometry
- A circle tangent to both coordinate axes has center $(\pm r, \pm r)$ — Tangency turns into $|a|=|b|=r$, so the quadrant alone fixes the center's signs.
circle,tangency,coordinate-geometry - A curve symmetric about a line meets every perpendicular line in mirror pairs — Intersections and tangencies come two at a time, and an odd count needs a point on the axis; $\sqrt{x-a}+b$ and $b-(x-a)^2$ mirror across $x+y=a+b$.
symmetry,reflection,counting
linear-algebra
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number-theory
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probability
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statistics
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