Some Problems for August

Some of the following are immediate; others may take more time — and require ingenuity and a deeper understanding. "Elementary" is not the same as "Simple." I hope you will even find it fun to think about some of these. You may find these interesting to discuss with friends.

  1. Prove that the product of two odd integers is also odd.
  2. If k > 0 is an integer, is   k(k+1)(k+2)   always divisible by 6?
  3. List these numbers from smallest to largest:   2121,   955,   788,   u0 := number of seconds since the birth of our universe.
  4. If   31025   is divided by 5, what is the remainder?
  5. If a, b > 0, show that   (ab)1/2 ≤ (a + b)/2.
  6. Is the square root of 7 a rational number?
  7. Can cos nx be written in the form   a0 + a1cos x + a2cos2x + ...+ ancosnx  ?
  8. Can the function sin x be written as a polynomial in x? How about  2 x?
  9. Find a formula for   Sn := 12 + 22 +... + n2.
  10. Does   0.99999... = 1.0000...?
  11. If c > 0 is a real number, prove there is an integer N so that Nc > 1.
  12. (a) Among all triangles inscribed in a fixed circle, show that equilateral triangles have the largest area.
    (b) Among all triangles in the plane with fixed area A0, show that equilateral triangles have the smallest perimeter. [Equivalently, if you fix the perimeter, then equilateral triangles have the largest area.]

    Rust Remover:
  1. Since Math 103 is a prerequisite, you should feel comfortable with all the material from it. Here are two old Final Exams: Fall 2007,   Fall 2008.
  2. Describe the real numbers x that satisfy   |x − 2| < 3.
  3. Sketch the points (x,y) in the plane where   |x − y| > 1.
  4. a). How many real roots does   x4 + x2 − 2x + 2 = 0   have?
    b). Find all points (x,y) in the plane that satisfy x2 - 2xy + 5y2 = 0.
  5. Show that   √(7 + 2√6) − √(7 − 2√6) = 2.
  6. Solve   log9(5 − 3x) = -1/2   for  x.
  7. Let A = (-6,3), B = (2,7), and C be the vertices of a triangle. Say the altitudes through the vertices A and B intersect at Q = (2,-1). Find the coordinates of C.
    [ The altitude through a vertex of a triangle is a straight line through the vertex that is perpendicular to the opposite side — or an extension of the opposite side. Although not needed here, the three altitudes always intersect in a single point, sometimes called the orthocenter of the triangle.]
  8. Let y = f(x) describe a smooth curve in the plane (-∞ < x <∞) that does not pass through the origin. Say the point P = (a,b) on the curve is closest to the origin. Show that the straight line from the origin to P is perpendicular to the curve.
  9. Let f(x) be a continuous function that satisfies 0x f(t) dt = c - cos(x2). Find the function f(t) and the constant c.