AMC 10 · 2014 · #15
Grade 7 number-theorycountingPick an answer.
Tool #15 (Reorganize) is the whole idea: adding 900 five-digit numbers one at a time is hopeless, but adding them slot by slot is three short computations. Tool #4 (Variable) gives the handle — write a palindrome as 10001a + 1010b + 100c so the slots become separate. Tool #7 (Subproblems) splits the job into "total of all a's", "total of all b's", "total of all c's". Tool #16 (Change Focus) supplies an independent second route: pair each palindrome with its digit-complement instead of summing slots, and every pair adds to the same amount.
Count the palindromes
There are 900 such palindromes.
The last two digits are echoes of the first two, so counting palindromes is really just counting the first three digits.
7.SP.C.8Introduce A VariableExpand by place value
Place value turns one into a weighted sum.
Because the two copies of a always travel together, they can be merged into a single weight 10001, and likewise b carries weight 1010.
4.NBT.A.2Introduce A VariableAdd by column, not by number
So the total adds up column by column.
It is the same move as adding a long addition problem column by column instead of row by row.
6.EE.A.3Organize Information In More WaysTotal each digit slot
Each digit slot totals easily.
Every allowed digit value shows up equally often in its slot, so a slot total is just (sum of allowed values) times (how many times each appears).
7.SP.C.8Identify SubproblemsMultiply out and add
Multiplying out gives a round total.
Three multiplications and one addition replace nine hundred additions.
5.NBT.B.5Identify SubproblemsCross-check by pairing complements
Pairing complements confirms it.
Marry every palindrome to its digit-complement; each couple adds to the same 110,000, so the grand total is just the number of couples times 110,000.
Marry every palindrome to its digit-complement and each couple adds to the same total.
▸ Why?
Replacing each digit by nine minus itself sends every palindrome to exactly one partner and back again.
▸ Why?
Every such pair totals the same amount, so the grand total is that amount times the number of couples.
Add the digits of S
Its digits add to 18, choice (B).
Trailing zeros contribute nothing, so only the three leading digits matter.
4.NBT.A.2Organize Information In More WaysAdding 900 numbers one at a time is hopeless, so add them column by column: each digit slot has a total you can get in one line. That gives S = 49,500,000, and its digits add to 18 — choice (B).
- Count the palindromes
- Expand by place value
- Add by column, not by number
- Total each digit slot
- Multiply out and add
- Cross-check by pairing complements
- Add the digits of S