1
Sum all page numbers · 6 points
Not answeredA copy of Olympiad Course has 120 pages, numbered consecutively from 1 through 120. Xiaoming adds every page number. What is the sum?
total
Chapter 13 Exercise — Page Number Problems
A complete English worksheet on page-number sums, digit blocks, digit occurrences, missing physical leaves, calendar blocks, and working backward from a digit budget. All information needed to solve the problems is included here.
Exercise directions
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Tips for this exercise
Keep the units separateA page count, a page-number value, and the number of written digit symbols are three different quantities.
Use the book structureThroughout this worksheet, numbering starts at 1 with no gaps or leading zeros, and each page label is counted once unless a problem describes a counting error. One physical leaf has an odd-numbered front and the next even-numbered back. Removing leaves does not renumber the pages left behind.
Extended responseQuestions 11–14 are worth 15 points each. Give a final response and show enough reasoning to explain your method.
Section I — Fill in the blanks
Questions 1–10 · 6 points each · 60 points
2
Digit symbols through page 180 · 6 points
Not answeredA copy of Manga World has 180 pages. Page numbers are printed from 1 through 180 without leading zeros. How many individual digit symbols are needed?
digits
Optional scratch work
3
One page number was repeated · 6 points
Not answeredA book is numbered from page 1 through page 60. Xiaojun adds all the page numbers, but accidentally includes one page number twice. His reported sum is 1,845. Which page number was repeated?
page
Optional scratch work
4
One page number was omitted · 6 points
Not answeredA book is numbered from page 1 through page 55. Xiaohong adds all the page numbers, but accidentally leaves out one page number. Her reported sum is 1,500. Which page number was omitted?
page
Optional scratch work
5
Count every written 4 · 6 points
Not answeredBlue Cat Legends has 96 pages. When the page numbers from 1 through 96 are printed, how many times does the digit 4 appear?
Count occurrences, not labels.A page number such as 44 contributes two occurrences of the digit 4.
times
Optional scratch work
6
Count every written 1 · 6 points
Not answeredConsider all natural numbers from 1 through 750, inclusive. Across all of those written numbers, how many times is the digit 1 used?
times
Optional scratch work
7
Length of a concatenated even-number string · 6 points
Not answeredThe even natural numbers from 2 through 2,010 are written consecutively with no spaces:
2468101214···20082010How many digit symbols are in the complete string?
digits
Optional scratch work
8
Front page of a missing physical leaf · 6 points
Not answeredA storybook is missing one physical leaf from somewhere inside the book. The two page numbers printed on that leaf have sum 95. What is the page number on the front side of the missing leaf?
odd pagefront
next even pageback
Front and back of the same leaf, shown side by side; these are not facing pages.
page
Optional scratch work
9
The central leaf of a book · 6 points
Not answeredExcept for the front and back covers, both sides of every physical leaf in a book are numbered consecutively from page 1 onward. The central leaf has the same number of physical leaves before and after it. The product of its two page numbers is 2,450.
What is the sum of all page numbers in the book?
total
Optional scratch work
10
Maximum digit sum in a February calendar block · 6 points
Not answeredUse the displayed 28-day February, with February 1 on Saturday. A valid block uses two neighboring weekday columns and two neighboring week rows, with a date in all four cells; it cannot include blanks or wrap from Sunday to Monday. A frame may cover any four dates in a valid adjacent 2 × 2 block. In the illustrated example, the framed dates are 4, 5, 11, and 12, and the sum of all their written digits is:
4 + 5 + 1 + 1 + 1 + 2 = 14
Among all possible valid 2 × 2 date blocks, what is the greatest possible digit sum?
| Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|
| 1 | 2 | |||||
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| 17 | 18 | 19 | 20 | 21 | 22 | 23 |
| 24 | 25 | 26 | 27 | 28 |
digit sum
Optional scratch work
Section II — Extended response
Questions 11–14 · 15 points each · 60 points
11
Can 15 torn leaves have total 2,014? · 15 points
Not answeredA 300-page book has 15 physical leaves torn out. Could the sum of all page numbers printed on those 15 leaves equal 2,014? Explain why or why not.
Physical-leaf conventionEach torn leaf removes its odd-numbered front and the next even-numbered back.
12
Recover one torn leaf from the remaining sum · 15 points
Not answeredOne physical leaf is torn from a 90-page book. The sum of all page numbers still remaining in the book is 4,012. Which leaf was torn out? Give both page numbers printed on it.
13
Work backward from 2,049 digit symbols · 15 points
Not answeredA science-fiction novel uses exactly 2,049 digit symbols to print all its page numbers, beginning with page 1 and using no leading zeros. How many pages does the novel have?
pages
14
Locate the 2,016th digit in a counting decimal · 15 points
Not answeredThe decimal
A = 0.1234567891011121314···is formed by writing the positive integers consecutively after the decimal point. What is the 2,016th digit after the decimal point?
digit