Pre-RMO (PRMO) Practice Problems

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?

I will post the answers in a couple of days.

Nalin Pithwa.

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