Some Problems for August - A few solutions

Some of the following are immediate; most others may take more time — and require ingenuity and a deeper understanding. They are intended to be challenging. A complete solution to some may require ideas we will cover later in the course.
"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 others. Remember, the first person you need to convince is yourself.

    General Problems

  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 [but before computing, make your own intuitive estimate].
    2121,   955,   788,  
    K := an estimate for when   1 + 1/2 + 1/3 + 1/4 + ··· + 1/K > 100.
    N := number of seconds since the birth of our universe (about 14 billion years ago),
    Solution:
    For the first three numbers, one approach uses that their exponents are all multiples of 11. Thus
    2121 = 211×11= 204811 = [2.048×103]11 = 2.04811 × 1033
    955 = 95×11= 5904911 = [5.9049×104]11 = 5.904911 × 1044
    788=78×11 = 576480111 = [5.764801×106]11 = 5.76480111× 1066

    K:   Use the integral test estimate to get an upper and lower bound for the sum. Compare SK with the area under the curve y = 1/x gives   ln(K+1) < SK < 1+ ln(K)
    Since we want ln(K+1) < SK ≥ 100 so we can use any K with ln(K+1) ≥ 100, that is, K +1 ≥ e100 ≈ 2.7×1043
    N:  1 year × 365 days/year × 24 hours/day × 60 min/hour × 60 sec/min = 365×24×60×60 =31,536,000 sec =3.1536 × 107 sec
    N ≈ 14 billion years = 14×109 years = 1.4×1010 years
    so
    N ≈ (1.4×1010)×(3.15360×107) ≈ 4.41504×1017 seconds. This is by far the smallest. Were you surprised?

  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? Generalize?

  7. In Illinois, there are two broods of cicadas. Brood 1 emerges from the ground every 17 years, Brood 2 every 13 years. They both appeared in 2024. When is the next year in which they will again both emerge?

  8. A Giant Supermarket poster, July 2013   Solve the Problem of the Week:.
    Suppose f is a function from positive integers to positive integers satisfying
        f(1)= 1, f(2n) = f(n), and f(2n+1) = f(2n) + 1 for all positive integers n.
    Find the maximum of f(n) when n is greater than or equal to 1 and less than or equal to 1984.

  9. Can cos nx be written in the form   a0 + a1cos x + a2cos2x + ...+ ancosnx  ?

  10. Can the function sin x be written as a polynomial (not power series) in x? How about  2 x?

  11. Find a formula for   Sn := 12 + 22 +... + n2.

  12. Does   0.99999... = 1.0000...?

  13. If c > 0 is a real number, prove there is an integer N so that Nc > 1.

  14. A function f:R -->R is called even if f(-x)=f(x) for all x and odd if f(-x)=f(x) for all x.
    a). Show that for any f there is an even function, feven and an odd fnnction, fodd so that
    f(x)= feven(x) + fodd(x)
    b). Compute this decomposition for f(x):= 1/(1+x), and for f(x):=p(x)/q(x), where p and q are any polynomials.
    c). Show that a polynomial p(x) is even if and only if it contains only even powers of x.
    d). Show that the derivative of a (differentiable) even function is odd and that the derivative of a (differentiable) odd function is even.
    e). Let f(x) be defined by a convergent power series f(x) := Σ akxk. Show that f is even if and only if the power series contains only even powers of x.

  15. (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.]
    (c) Generalize?

    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.]
    Solution: Let V = Q − A,   W = Q − B. Pick V⊥ and W⊥ perpendicular to V and W, respectively. Then the sides of the triangle are the lines
    X(t) = A + tW⊥,   Y(s) = B + sV⊥.
    C is the intersection of these lines. At this point <X(t),V> = <Y(s),V> = <B,V>,   so   t = ≤B−A,V>/<W⊥,V>.
    Hence C = A + [<B−A,V>/<W⊥,V>]W⊥.
    In this problem B−A=(8,4), V=(8,-4), W=(0,-8) so we can let V⊥=(1,2) and W⊥=(1,0).
    Then t = 48/8 = 6 so C = (-6,3) + 6(1,0) = (0,3).