Some one arrives in a city with very interesting news and within 10 minutes tells it to two others. Each of these tells the news within 10 minutes to two others(who have not heard it yet), and so on. How long will it take before everyone in the city has heard the news if the city has three million inhabitants?

Problem 2:

A cyclist and a horseman have a race in a stadium. The course is five laps long. They spend the same time on the first lap. The cyclist travels each succeeding lap 1.1 times more slowly than he does the preceding one. On each lap the horseman spends d minutes more than he spent on the preceding lap. They each arrive at the finish line at the same time. Which of them spends the greater amount of time on the fifth lap and how much greater is this amount of time?

I hope you enjoy “mathematizing” every where you see…

Six boxes are numbered 1 through 6. How many ways are there to put 20 identical balls into these boxes so that none of them is empty?

Problem 2:

How many ways are there to distribute n identical balls in m numbered boxes so that none of the boxes is empty?

Problem 3:

Six boxes are numbered 1 through 6. How many ways are there to distribute 20 identical balls between the boxes (this time some of the boxes can be empty)?

Pre-RMO days are back again. Here is a list of some of my random thoughts:

Problem 1:

There are five different teacups, three saucers, and four teaspoons in the “Tea Party” store. How many ways are there to buy two items with different names?

Problem 2:

We call a natural number “odd-looking” if all of its digits are odd. How many four-digit odd-looking numbers are there?

Problem 3:

We toss a coin three times. How many different sequences of heads and tails can we obtain?

Problem 4:

Each box in a 2 x 2 table can be coloured black or white. How many different colourings of the table are there?

Problem 5:

How many ways are there to fill in a Special Sport Lotto card? In this lotto, you must predict the results of 13 hockey games, indicating either a victory for one of two teams, or a draw.

Problem 6:

The Hermetian alphabet consists of only three letters: A, B and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain?

Problem 7:

A captain and a deputy captain must be elected in a soccer team with 11 players. How many ways are there to do this?

Problem 8:

How many ways are there to sew one three-coloured flag with three horizontal strips of equal height if we have pieces of fabric of six colours? We can distinguish the top of the flag from the bottom.

Problem 9:

How many ways are there to put one white and one black rook on a chessboard so that they do not attack each other?

Problem 10:

How many ways are there to put one white and one black king on a chessboard so that they do not attack each other?

Here, in finding the LCM of the denominators, it must be observed that there are not six different compound factors to be considered; for, three of them differ from the other three only in sign.

Thus,

Hence, replacing the second factor in each denominator by its equivalent, we may write the expression in the form

call this expression 1

Now, the LCM is

and the expression is .,

Some Remarks:

There is a peculiarity in the arrangement of this example, which is desirable to notice. In the expression 1, the letters occur in what is known as cyclic order; that is, b follows a, a follows c, c follows b. Thus, if a, b, c are arranged round the circumference of a circle, if we may start from any letter and move round in the direction of the arrows, the other letters follow in cyclic order; namely, abc, bca, cab.

The observance of this principle is especially important in a large class of examples in which the differences of three letters are involved. Thus, we are observing cyclic order when we write , , , whereas we are violating order by the use of arrangements such as , , , etc. It will always be found that the work is rendered shorter and easier by following cyclic order from the beginning, and adhering to it throughout the question.

Prove that the median of a triangle which lies between two of its unequal sides forms a greater angle with the smaller of those sides.

Point A is given inside a triangle. Draw a line segment with end-points on the perimeter of the triangle so that the point divides the segment in half.

If the sides of a triangle are longer than 1000 inches, can its area be less than one inch?