| A1. A standard parabola has an equation of the form y = x2 + ax + b. Three standard parabolas have vertices V1, V2, V3 and intersect pairwise at the points A1, A2, A3. P → r(P) is reflection in the x-axis. Show that the standard parabolas with vertices r(A1), r(A2), r(A3) intersect pairwise at r(V1), r(V2), r(V3). | |
| A2. Is there a function f: R → R with continuous derivative such that f(x) > 0 and f '(x) = f(f(x)) for all x? | |
| A3. Put an = 1/nCk, bn = 1/2n-k for k = 1, 2, ... , n (where nCk is the binomial coefficient). Show that ∑ (ai - bi)/i = 0. | |
| A4. Let f: [a,b] → [a,b] be a continuous function. For p ∈ [a,b] define p0 = p, pn+1 = f(pn). The set Tp = {p0, p1, p2, ... } is closed. Show that it has only finitely many elements. | |
| A5. Does there exist a monotonic function f: [0,1] → [0,1] such that f(x) = k has uncountably many solutions for each k ∈ [0,1]? Does there exist such a function which also has a continuous derivative? | |
| A6. For a real n x n matrix M define |M| = sup x≠0 |Mx|/|x| (where |x| is the standard Euclidean norm for x ∈ Rn). If the matrix A satisfies |Ak - A-k| ≤ 1/(2002k) for all positive integers k, show that |Ak| ≤ 2002 for all k. | |
| B1. The matrix A = (aij) is defined by aij = 2 if i=j, (-1)|i-j| if i≠j. Find det A. | |
| B2. 200 students did an exam with 6 questions. Every question was correctly answered by at least 120 students. Show that there must be two students such that every question was correctly answered by at least one of them. | |
| B3. Show that (∑k=0∞ kn/k!)(∑k=0∞ (-1)k kn/k!) is an integer. | |
| B4. OABC is a tetrahedron. ∠BOC = α, ∠COA = β, ∠AOB = γ. The angle between the faces OAB and OAC is σ, and the angle between faces OAB and OBC is τ. Show that γ > β cos σ + α cos τ. | |
| B5. A is a complex n x n matrix for n > 1. A' is the complex conjugate of A (each element is the complex conjugate of the corresponding element of A). Show that AA' = 1 iff A = S(S')-1 for some S. | |
| B6. f: Rn → R is convex. ∇f exists at every point and for some L > 0 we have |∇f(x1) - ∇f(x2)| ≤ L|x1 - x2| for all x1, x2. Show that |∇f(x1) - ∇f(x2)|2 ≤ L (∇f(x1) - ∇f(x2)).(x1 - x2) (the dot product). |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
1 Dec 2003