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.
Examples:
- 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,
u0 := number of seconds since the birth of our universe.
- If 31025 is divided by 5, what is the remainder?
- If a, b > 0, show that (ab)1/2 ≤ (a + b)/2.
- Is the square root of 7 a rational number?
- 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
2 x?
- 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
> 1.
- (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:
- 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.
- 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.]
- 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.
- 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.