Pauley Morph

My wife and I drive a Kia Spectra. Often she needs to take a wheel chair with her. She can walk, but she has siezures occassionally and, when she does, walking is difficult if not impossible for a long while. But it's hard to put both groceries and a whel chair in the back of a Kia Spectira. Is there any way to put a whell chair mount on the back of the car?

I noticed that the four digit number 6834 is both

68x100 + 68/2

and

67*100 + 2*67

What other four digit numbers have this property?

Suppose f(f(t)) = t for all real t.

Consider the graph of (x, y) = ( t - f(t), t + f(t)).

Show that the graph is even, that is,

for every point (x, y) on the graph, there is a point (-x, y) on the graph.

I'm getting notified that I have best answers but when I click on notifications I don't see them. Yet I can click on the reference in my email and go to it.

Consider the numbers

11, 111, 1111, 11111, ...

Prove that every prime number but 2 and 5 is a divisor of at least one of them.

Say that N is dense if all of its prime factors are less than √N. Say that N is 2-dense if N and N+1 are dense. What is the smallest 2-dense greater than 200?

The triangle with sides 13, 14, 15 has one integer altitude. Which one is it?

Suppose

x,y, and z are positive integers

GCD(x, y, z) = 1

1/x + 1/y = 1/z

Prove x + y is a perfect square.

I have a big sack full of socks. It contains at least 900 but no more than 1000 individual socks. Each one of the socks is either black or blue, and there are more blue socks than black ones. They all mixed up in the sack, but if I reach into it and grab two socks at random, then I expect half of the time to pull out a matching pair.

I have always been very careful when doing my laundry, and have only ever lost a single sock in my entire life.

What color was that lost sock?

Categorize all positive integers x, y, and z such that 1/x + 1/y = 1/z.

Prove that the lengths of the sides of a right triangle are rational, if and only if there exists a rational numbers α > 1 such that the sides of the triangle are in the ratio 2 : α - 1/α : α + 1/α.

Let ABC be a right triangle with BC = 1, AC = √5 and AB = 2. Bisect AB at D and drop a perpendicular segment from D to segment AC at E. Prove that the measure of angle CEB is 45°.

I used to alway tell my students that there are as many r's in interger as there are in sherbert. I was wondering if there were any other bad examples like this out there.

Let a, b, h, and k be positive real numbers. In the coordinate plane, let A = (-a, 0), B = (h, k), and C = (c, 0). Find the locus of all points D such that quadrilateral ABCD contains an inscribed circle.