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PrintJapan Junior Mathematical Olympiad
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Problem
A positive integer whose 1's digit is not 0 is called a palindromic number if the number remains the same when its digits are read in reverse order. For example, the number is a palindromic number, since the number obtained by reading its digits in reverse order is the same , while is not since the number is obtained if its digits are read in reverse order. How many palindromic numbers are there which are less than or equal to ?
Solution
Let us call a palindromic number a p.d.number. We count p.d.numbers less than by classifying them according to the number of digits.
A 4-digit number is a p.d.number if and only if its thousand's digit and one's digit coincide and also its hundred's digit and ten's digit coincide. Therefore, is the only p.d.number with its thousand's digit equal to (and less than ). For a 4-digit number with its thousand's digit to be a p.d.number, its hundred's digit can be any of the numbers between and ( and inclusive), and therefore there are such numbers. Thus there are p.d.numbers with digits less than .
A 3-digit number is a p.d.number if and only if its hundred's digit and one's digit coincide. Hundred's digit of a 3-digit number can be any number between and ( and inclusive), and ten's digit of a 3-digit p.d.number can be any of the numbers lying in between and . Therefore, there are p.d.numbers with digits.
A 2-digit number is a p.d.number if and only if its ten's digit and one's digit coincide. Since the ten's digit of a 2-digit number can take any number lying in between and , there are p.d.numbers with digits.
Any 1-digit non-zero number is a p.d.number, so there are such numbers.
Thus the answer we seek is .
A 4-digit number is a p.d.number if and only if its thousand's digit and one's digit coincide and also its hundred's digit and ten's digit coincide. Therefore, is the only p.d.number with its thousand's digit equal to (and less than ). For a 4-digit number with its thousand's digit to be a p.d.number, its hundred's digit can be any of the numbers between and ( and inclusive), and therefore there are such numbers. Thus there are p.d.numbers with digits less than .
A 3-digit number is a p.d.number if and only if its hundred's digit and one's digit coincide. Hundred's digit of a 3-digit number can be any number between and ( and inclusive), and ten's digit of a 3-digit p.d.number can be any of the numbers lying in between and . Therefore, there are p.d.numbers with digits.
A 2-digit number is a p.d.number if and only if its ten's digit and one's digit coincide. Since the ten's digit of a 2-digit number can take any number lying in between and , there are p.d.numbers with digits.
Any 1-digit non-zero number is a p.d.number, so there are such numbers.
Thus the answer we seek is .
Final answer
119
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