Math · 02
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Solve \(\log_2 x+\log_x 2=5/2\).
Timer not started. Twenty minutes of real work before a solution.
Let \(t=\log_2 x\), so \(t\ne 0\) and \(x>0\), \(x\ne 1\). Then \(t+1/t=5/2\). Multiply by \(2t\): \(2t^2-5t+2=0\), \((2t-1)(t-2)=0\). So \(t=1/2\) or \(t=2\), hence \(x=\sqrt{2}\) or \(x=4\). Both check.
Attack 2
How many real roots does \(x^5-5x+1=0\) have? Do not approximate them. Count them.
Timer not started. Twenty minutes of real work before a solution.
\(f'(x)=5x^4-5=5(x^4-1)\), so the only real critical points are \(x=\pm 1\). \(f(-1)=5>0\) is a local max and \(f(1)=-3<0\) is a local min, because \(f''\) or the first-derivative sign chart says so: \(f'\) is negative on \((-1,1)\) and positive outside. \(f(-2)<0\), \(f(2)>0\). So there is one root in \((-\infty,-1)\), one in \((-1,1)\), and one in \((1,\infty)\). Three real roots, and therefore two nonreal complex roots.
Attack 3
Prove that for positive \(a,b\), \((a+b)/2\ge\sqrt{ab}\), and characterize equality.
Timer not started. Twenty minutes of real work before a solution.
\((\sqrt{a}-\sqrt{b})^2\ge 0\) expands to \(a+b\ge 2\sqrt{ab}\). Equality holds exactly when \(\sqrt{a}=\sqrt{b}\), that is \(a=b\).
This page is the same functions, trigonometry, and analytic geometry, pushed until the standard method is not enough. Competition precalculus adds complex numbers as a working tool, polynomials as objects with structure rather than expressions to factor, functional equations, inequalities with equality cases, and trigonometry used as algebra on the circle. The College Board will not ask most of this. Later analysis and olympiad calculus will assume you already paid for it.
The failure mode is collecting tricks. A trick that you cannot derive is a fact you will misapply the first time the problem is rotated. Every identity on this page has to be something you can rebuild: De Moivre from multiplication of complex numbers, AM-GM from a square, the factor theorem from division. Gelfand’s short books were written exactly in this spirit. The free scans of them are access-restricted on the Internet Archive, so they are not linked here as downloads. Use the sources below, which are open.
When a problem splits into cases, write the cases down before you compute. Most lost points at this level are a case that was never named.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Complex arithmetic. \(i^2=-1\). In polar form, \(re^{i\theta}\cdot \rho e^{i\phi}=r\rho e^{i(\theta+\phi)}\). De Moivre: \((\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)\). Nonreal roots of real polynomials come in conjugate pairs.
Multiplication adds angles because that is how \(e^{i\theta}=\cos\theta+i\sin\theta\) multiplies, or by expanding the sine and cosine addition formulas inductively. The conjugate-root theorem is the observation that conjugating \(p(z)=0\) with real coefficients yields \(p(\bar z)=0\).
Polynomial structure. Over \(\mathbb{C}\), a degree-\(n\) polynomial has exactly \(n\) roots, counted with multiplicity. If the coefficients are real and \(a+bi\) is a root, so is \(a-bi\). A symmetric polynomial in the roots can be written in the elementary symmetric sums, which are the coefficients up to sign.
The existence half is the fundamental theorem of algebra; you are not asked to prove it here, only to use the counting consequence after unique factorization in \(\mathbb{C}[x]\), which you may take as known once the theorem is stated. Vieta is equating coefficients in \(a_n\prod(x-r_i)\).
Inequalities. AM-GM for \(n\) positive numbers, Cauchy–Schwarz in the form \((a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)\ge(a_1b_1+\cdots+a_nb_n)^2\), and the trivial bound from squares: a sum of squares is nonnegative.
The \(n=2\) AM-GM is the attack above. Cauchy–Schwarz is the discriminant of \(\sum(a_i t-b_i)^2\ge 0\) being nonpositive, or the expansion of \(\sum_{i
Trigonometric form. The linear combination \(a\cos\theta+b\sin\theta\) equals \(R\cos(\theta-\alpha)\) with \(R=\sqrt{a^2+b^2}\) and \(\cos\alpha=a/R\), \(\sin\alpha=b/R\). Product-to-sum identities are the addition formulas solved for products.
Divide by \(R\) and match the addition formula for cosine. Product-to-sum is the sum and difference of \(\cos(\alpha+\beta)\) and \(\cos(\alpha-\beta)\), rearranged.
Conics. A conic is a curve \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\). After rotation to kill the \(xy\) term and translation to complete the square, the type is ellipse, parabola, or hyperbola according to the discriminant \(B^2-4AC\).
The rotation by an angle satisfying \(\cot 2\theta=(A-C)/B\) removes \(xy\). Completing the square is algebra. The discriminant is invariant under rotation, which is why it classifies the conic before you compute the angle.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Paul’s Online Math Notes, Algebra
Read once, do not study from this. Skim the section list so you can see which algebra moves you still hesitate on. Then close it and go to the book.
The thing you work through.
Stitz and Zeager, the complex, trig, and conic chapters you rushed
Same free book as the previous page. The hardening is in which exercises you choose: every proof-style exercise, every complex-number section, every inequality.
The problem book.
AoPS Precalculus, plus Alcumus on hard PAID
Paid textbook aimed at contests. Paired with Alcumus: https://artofproblemsolving.com/alcumus . Better than another college precalculus book because the problems are the point.
The hardest questions that exist at this level.
AIME problems and Sharygin’s plane geometry
AIME is the ceiling for precalculus technique. Sharygin, for the coordinate slice only: https://archive.org/details/SharyginProblemsInPlaneGeometryScienceForEveryone . Gelfand’s problem books are the right spirit; legal free downloads were not verified, so they are not linked.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: eight AIME problems, numbered 8 or higher, from at least four different years, covering at least three of complex numbers, trigonometry, polynomials, and inequalities. Full writeups. You may use the contest’s answer format as a checksum only after the solution is written. Fail if you needed a solution video to find the idea.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Explain polar form, prove AM-GM for two numbers, and count the real roots of a quintic using only the derivative sign chart. No notes.