Back to VTRMC list
Problem source: HTML (archive) and PDF (archive)

Problem 1

Three pasture fields have areas of 10/310/3, 1010 and 2424 acres, respectively. The fields initially are covered with grass of the same thickness and new grass grows on each at the same rate per acre. If 1212 cows eat the first field bare in 44 weeks and 2121 cows eat the second field bare in 99 weeks, how many cows will eat the third field bare in 1818 weeks? Assume that all cows eat at the same rate. (From Math Can be Fun by Ya Perelman.)

Problem 2

A person is engaged in working a jigsaw puzzle that contains 10001000 pieces. It is found that it takes 33 minutes to put the first two pieces together and that when xx pieces have been connected it takes 3(1000−x)1000+x\dfrac{3(1000 - x)}{1000 + x} minutes to connect the next piece. Determine an accurate estimate of the time it takes to complete the puzzle. Give both a formula and an approximate numerical value in hours. (You may find useful the approximate value ln⁡2=.69\ln 2 = .69.)

Problem 3

Let ff be defined on the natural numbers as follows: f(1)=1f(1) = 1 and for n>1n > 1, f(n)=f(f(n−1))+f(n−f(n−1))f(n) = f(f(n - 1)) + f(n - f(n - 1))

Find, with proof, a simple explicit expression for f(n)f(n) which is valid for all n=1,2,…n = 1, 2, \dots.

Problem 4

Suppose that P(x)P(x) is a polynomial of degree 33 with integer coefficients and that P(1)=0P(1) = 0, P(2)=0P(2) = 0. Prove that at least one of its four coefficients is equal to or less than −2-2.

Problem 5

Determine all real values of pp for which the following series converge.

  1. ∑n=1∞(sin⁡1n)p\displaystyle{\sum_{n=1}^{\infty} \left( \sin \frac{1}{n} \right)^p}
  2. ∑n=1∞∣sin⁡n∣p\displaystyle{\sum_{n=1}^{\infty} \left| \sin n \right|^p}

Problem 6

The number of individuals in a certain population (in arbitrary real units) obeys, at discrete time intervals, the equation yn+1=yn(2−yn)for n=0,1,2,…, y_{n+1} = y_n (2 - y_n) \quad\text{for } n = 0, 1, 2, \dots, where y0y_0 is the initial population.

  1. Find all “steady-state” solutions y∗y^* such that, if y0=y∗y_0 = y^*, then yn=y∗y_n = y^* for n=1,2,…n = 1, 2, \dots.
  2. Prove that if y0y_0 is any number in (0,1)(0, 1), then the sequence {yn}\lbrace y_n \rbrace converges monotonically to one of the steady-state solutions found in (a).

Problem 7

Let the following conditions be satisfied:

  1. f=f(x)f = f(x) and g=g(x)g = g(x) are continuous functions on [0,1][0, 1],
  2. there exists a number aa such that 0<f(x)≤a<10 < f(x) \le a < 1 on [0,1][0, 1],
  3. there exists a number uu such that max⁡0≤x≤1(g(x)+uf(x))=u\displaystyle{\max_{0 \le x \le 1} (g(x) + u f(x)) = u}.

Find constants AA and BB such that F(x)=Ag(x)f(x)+BF(x) = \dfrac{Ag(x)}{f(x) + B} is a continuous function on [0,1][0, 1] satisfying max⁡0≤x≤1F(x)=u\displaystyle{\max_{0 \le x \le 1} F(x) = u}, and prove that your function has the required properties.

Problem 8

Ten points in space, no three of which are collinear, are connected, each one to all the others, by a total of 4545 line segments. The resulting framework FF will be “disconnected” into two disjoint nonempty parts by the removal of one point from the interior of each of the 99 segments emanating from any one vertex of FF. Prove that FF cannot be similarly disconnected by the removal of only 88 points from the interiors of the 4545 segments.