7th IMC 2000

7th IMC 2000 problems


A1. Does every monotone increasing function f:[0,1] → [0,1] have a fixed point? What about every monotone decreasing function?
A2. p(x) ≡ x⁵ + x, q(x) ≡ x⁵ + x². Find all pairs (w,z) of complex numbers with w ≠ z such that p(w) = p(z) and q(w) = q(z).
A3. A, B are square complex matrices and rank(AB - BA) = 1. Show that (AB - BA)² = 0.
A4. xi is a decreasing sequence of positive reals. Show that √(∑₁ⁿ x_i²) ≤ ∑₁ⁿ x_i/√i.
A5. R is a ring of characteristic zero. e, f, g are elements of R such that e + f + g = 0, e² = e, f² = f, g² = g. Show that e = f = g = 0.
A6. f: R → (0,∞) is an increasing differentiable function such that f(x) → ∞ as x → ∞, and f' is bounded. Let F(x) = &inf;₀^x f(t) dt. Define a₀ = 1, a_n+1 = a_n + 1/f(a_n), and b_n = F^-1(n). Prove that lim_n→∞ (a_n - b_n) = 0.
B1. Show that a square may be partitioned into n smaller squares for sufficiently large n. Show that for some constant N(d), a d-dimensional cube can be partitioned into n smaller cubes if n ≥ N(d).
B2. f is continuous on [0,1]. There is no open subinterval of [0,1] on which f is monotone. Show that the set of points on which f attains a local minimum is dense in [0,1].
B3. p(z) is a polynomial of degree n>0 with complex coefficients. Show that p(z) is 0 or 1 for at least n+1 complex numbers z.
B4. The graph of a polynomial of degree 6 is tangent to a straight line at A, B, and C where B lies between A and C. If B is the midpoint of AC show that the area bounded by AB and the graph equals the area bounded by BC and the graph. If BC/AC = k, show that the ratio K of these areas satisfies 2k⁵/7 < K < 7k⁵/2.
B5. R⁺ is the set of positive real numbers. Find all functions f: R⁺ → R⁺ such that f(x)f(yf(x)) = f(x+y) for all x,y.
B6. For ayn m x m real matrix A, define e^A = ∑₀^∞ Aⁿ/n! . Prove or disprove that for any real polynomial p(x), p(e^AB) is nilpotent iff p(e^BA) is nilpotent.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.