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
- Prove that the product of two odd integers is also odd.
- If k > 0 is an integer, is k(k+1)(k+2) always
divisible by 6?
- List these numbers from smallest to largest:
2121, 955, 788,
N := number of seconds since the birth of our universe.
- a). Roughly how many digits are neeeded to write
31025 (base 10, as usual)?
b). If 31025 is divided by 5, what is
- If a, b > 0, show that (ab)1/2 ≤ (a + b)/2.
- Is the square root of 7 a rational number? This assumes you
already know how to prove that
is irrational. If not, first try it yourself -- or look it up.
- 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,
- If you repeat the process in problem 7a), do your rational numbers
√ 2 ?
- A Giant Supermarket poster, July 2013
Solve the Problem of the Week:.
Suppose f is a function from positive integers to positive
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.
- Can cos nx be written in the form
a0 + a1cos x + a2cos2x +
...+ ancosnx ?
- Can the function sin x be written as a polynomial in x? How about
- Find a formula for
Sn := 12 + 22 +... + n2.
- Does 0.99999... = 1.0000...?
- If c > 0 is a real number, prove there is an integer N so that Nc
- (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.]
- Describe the real numbers x that satisfy |x − 2| < 3.
- Sketch the points (x,y) in the plane where |x − y| > 1.
- 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.
- Show that
√(7 + 2√6) − √(7 − 2√6) = 2.
- Solve log9(5 − 3x) = -1/2 for x.
- 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.]