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:
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,   u0 := number of seconds since the birth of our universe.
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?
7. Can cos nx be written in the form   a0 + a1cos x + a2cos2x + ...+ ancosnx  ?
8. Can the function sin x be written as a polynomial in x? How about  2 x?
9. Find a formula for   Sn := 12 + 22 +... + n2.
10. Does   0.99999... = 1.0000...?
11. If c > 0 is a real number, prove there is an integer N so that Nc > 1.
12. (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:
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.]
8. 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.
9. 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.