Math 320: Some Problems for August

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:   2¹²¹,   9⁵⁵,   7⁸⁸,   N := number of seconds since the birth of our universe.

4. If   3¹⁰²⁵   is divided by 5, what is the remainder?

5. Let c=271.4 in base 10. Write c in base 2.

6. If a, b > 0, show that   (ab)^1/2 ≤ (a + b)/2.

7. Is the square root of 7 a rational number? Generalize?

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 + ...+ a_ncosⁿx  ?

10. Can the function sin x be written as a polynomial in x? How about  2 ^x?

11. Find a formula for   S_n := 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) 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 (Why?), if you fix the perimeter, then equilateral triangles have the largest area.]
    (c) Generalize?

Rust Remover Problems
1. Describe the real numbers x that satisfy   |x − 2| < 3.

2. Sketch the points (x,y) in the plane where   |x − y| > 1.

3. 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.

4. Show that   √(7 + 2√6) − √(7 − 2√6) = 2.

5. Solve   log₉(5 − 3x) = -1/2   for  x.

6. 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.

7. Let f(x) be a continuous function that satisfies ∫₀^x f(t) dt = c - cos(x²). Find the function f(t) and the constant c.

8. Let S_N = 1 + 1/2 + 1/3 + ... + 1/N
       a). Write a computer program (in any language) that computes this for N = 10,000. Have your program also print the amount of time it took.
       b). Find an N so that S_N > 100. Justify your assertion. How does this number compare with the number of seconds since the birth of the universe?