Some Problems for August

Some of the following are immediate; most others may take (much) more time — and require ingenuity and a deeper understanding. For the most part, only High School mathematics is needed. Some will 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.

An important aspect of these problems is what you do after you have "solved" the problem. The goal is to understand more deeply.

• Can you simplify your solution? Make it shorter or clearer?
• Can you generalize problem? Part of this might be finding a related problem that is simpler. Then seek a pattern.
• Explain your solution to a friend. This will force you to better understand your own intuition.
• Can you give another different solution, perhaps more geometic or more algebraic?

Examples:
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. a). Roughly how many digits are neeeded to write   3¹⁰²⁵   (base 10, as usual)?
   b). 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? This assumes you already know how to prove that √ 2  is irrational. If not, first try it yourself -- or look it up.
   Generalize?
7. a).  Let a > 0 be a rational number with the property that a² < 2. Find a rational number b > a with the property a² < b² < 2, so b is closer to √ 2 .
   b).  Similarly, let c > 0 be a rational number with the property that 2 < c². Find a rational number 0 < d < c with the property 2 < d² < c²,
8. If you repeat the process in problem 7a), do your rational numbers converge to √ 2 ?
9. 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.
10. Can cos nx be written in the form   a₀ + a₁cos x + a₂cos²x + ...+ a_ncosⁿx  ?
11. Can the function sin x be written as a polynomial in x? How about  2 ^x?
12. Find a formula for   S_n := 1² + 2² +... + n².
13. Does   0.99999... = 1.0000...?
14. If c > 0 is a real number, prove there is an integer N so that Nc > 1.
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. 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 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.]