Sharing your mathematics in MathYouLike

THE BICYCLES AND THE FLY

Two boys on bicycles, 20 miles apart, began racing directly toward each other. The instant they started, a fly on the handle bar of one bicycle started flying straight toward the other cyclist. As soon as it reached the other handle bar it turned and started back. The fly flew back and forth in this way, from handle bar to handle bar, until the two bicycles met.

If each bicycle had a constant speed of 10 miles an hour, and the fly flew at a constant speed of 15 miles an hour, how far did the fly fly?

PICK YOUR PAY

Suppose you take a new job and the boss offers you a choice between:

(A) $4,000 for your first year of work, and a raise of $800 for each year thereafter;

(B) $2,000 for your first six months of work, and a raise of $200 every six months thereafter.

Which offer would you take and why?

NO CHANGE

- «Give me change for a dollar, please," said the customer.

- "I'm sorry," said Miss Jones, the cashier, after searching through the cash register, "but I can't do it with the coins I have here."

- "Can you change a half dollar then?"

- Miss Jones shook her head. In fact, she said, she couldn't even make change for a quarter, dime, or nickel!

- "Do you have any coins at all?" asked the customer.

- "Oh yes," said Miss Jones. "I have $1.15 in coins."

Exactly what coins were in the cash register?

ERROR LIES

"I seem to have overdrawn my account," said Mr. Green to the bank president, «though I can't for the life of me understand how it could have happened. You see, I originally had $100 in the bank. Then I made six withdrawals. These withdrawals add up to $100, but according to my records, there was only $99 in the bank to draw from. Let me show you the figures." Mr. Green handed the bank president a sheet of paper on which he had written:

«As you see," said Mr. Green, "I seem to owe the bank a dollar."

The bank president looked over the figures and smiled. "I appreciate your honesty, Mr. Green. But you owe us nothing."

"Then there is a mistake in the figures"

"No, your figures are correct."

Can you explain where the error lies?

If three cats catch three rats in three minutes, how many cats will catch 100 rats in 100 minutes?

. A coordinate system is devised in which the positive y-axis makes a 60 degree angle with the positive x-axis. Using this coordinate system, A(6,1), B(2,-3) and C(-2,5) are the coordinates of vertices of a triangle. Find:

a) the perimeter of the triangle, and

b) the area of the triangle.

Suppose that the absolute value of w < 1. Find a simple expression for the sum:

$1+\left(w+w^2\right)+\left(w^2+w^3+w^4\right)+\left(w^3+w^4+w^5+w^6\right)+\left(w^4+w^5+w^6+w^7+w^8\right)+...$

Tim and Nancy have children Alex and Morgan with Morgan 4 years older than Alex. Tim is 2 years older than Nancy and the sum of ages of the 4 family members is now 96 years. Seven years ago the sum of the ages of all of the family members was 69 years. What is Nancy's present age?

For f(x) = | a + 1 - ax | , the sum of the roots of f(x) = x is 5/2 = 2.5. Find any such a.

Find all three digit numbers n for which n = 100a + 10b + c = a! + b! + c!

Noting that: n! = 1 . 2 ... (n-1) . n

Let S be a subset of {1, 2, 3, 4, ... , 1998} for which no two elements of S differ by 4 or by 7. What is the largest number of elements that S can have?

Triangle ABC is an isosceles triangle with AC = BC. Point D lies on AC and point E lies on BC such that DE $\perp$ AC. The extensions of DE and AB meet at F so that CD = BF. Find the ratio of the area of $\Delta$ BEF to the area of $\Delta$CDE.

In Mr. Chaffee's math class, he noted that when the attendance was exactly 84%, then there were 13 empty seats. What could be the number of empty seats when all students attend?

In the pictured isosceles triangle ABC $\angle A=20^o$, side BE = AE and $\angle ACD=50^o$. Without a calculator or computer find the measure of $\angle BED$.

Let a, b, and c be non-zero real numbers for which $\dfrac{a+b-c}{c}=\dfrac{a+c-b}{b}=\dfrac{c+b-a}{a}$ . For $x=\dfrac{\left(a+b\right)\left(a+c\right)\left(b+c\right)}{abc}$ with x < 0, find x.

If m, n, and q are distinct roots of $x^3-19x^2+99x-1999=0$ , then find the numerical value of $m^3+n^3+q^3$ .

Determine the unique pair of real numbers ( x y, ) that satisfy the equation

(4x² + 6x + 4)(4y² – 12y + 25) = 28 .

Determine the leftmost three digits of the number:

1¹ + 2² + 3³ + … + 999⁹⁹⁹ + 1000¹⁰⁰⁰ .

Let f be a polynomial of degree 98, such that $f\left(k\right)=\dfrac{1}{k}$ for k = 1, 2, 3, ..., 99. Determine f(100).

Prove that if n is an odd positive integer, then N = 2269ⁿ + 1779ⁿ + 1730ⁿ – 1776ⁿ is an integer multiple of 2001.