20 problems.
- Sequences showing each Arzelà–Ascoli hypothesis is needed
- Three unit complex numbers with zero sum form an equilateral triangle
- Integral of \(x^3-3xy^2\) over the region between two circles
- Roots of \(z^5-6z+3\): distinct, exactly three real — Fall 1986 Problem 4; repeated exactly as Fall 2013 Problem 4A
- Jordan form of \(X\mapsto XA-AX\) for a nilpotent \(2\times2\) matrix \(A\)
- Decay of solutions of \(f'+qf=0\) versus \(f'+pf=0\) when \(|q|\le|p|\)
- Quadratic extensions of \(\mathbb Q\) are \(\mathbb Q(\sqrt m)\) for a unique squarefree \(m\)
- A continuous function with nonnegative upper right Dini derivative is nondecreasing
- The integral \(\int_0^\infty \frac{\log x}{(x^2+1)(x^2+4)}\,dx\)
- Solutions of \(\Delta^n f=0\) are polynomials with period-one coefficients
- Row rank equals column rank
- Shrinking an open cover of \(\mathbb R^n\) to one meeting each compact set finitely often
- Points of differentiability of \(x^{4/3}\sin(y/x)\)
- Orthogonal unit vectors in \(\mathbb R^3\) with prescribed third coordinates
- Irreducibility of \((x^p-1)/(x-1)\) over \(\QQ\) — Fall 1986 Problem 15; irreducibility of the $p$th cyclotomic polynomial over $\mathbb{Q}$
- Existence and uniqueness for \((e^x\sin y)(y')^3+(e^x\cos y)y'+e^y\tan x=0\), \(y(0)=0\)
- A subgroup of \(S_5\) containing a \(5\)-cycle and a transposition is \(S_5\)
- The contour integral \(\frac{1}{2\pi i}\oint_{|z|=1}\frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}\,dz\)
- Six people include three mutual friends or three mutual strangers
- If \(f(x)+\int_0^x f\) converges then \(f(x)\to0\)