AMO 2027 Grades 4-5 SET 29 - Self-study

Gina has 1350 dollars and receives 11 dollars every week. James has 1650 dollars and spends 19 dollars per week. After how many weeks will they have the same amount of money?

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The source asks how many ways we can go from the dark square at the top to the dark square at the bottom, moving only right or down, with no backtracking and no repeated section. It also says “only along the grid lines”.

**Movement clarification for this edition:** start in the top-left dark cell and finish in the bottom-right dark cell. Each move goes to the neighboring cell immediately right or down, across a shared side. Count cell-to-cell routes, rather than routes between grid vertices. The original wording mixes these two descriptions.



Enter one whole number, without units.

What is the largest three-digit number divisible by 32?

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Last holiday, Jonathan received 134 dollars more than three times the money Sally received. They received 710 dollars altogether. How much did Jonathan receive?

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Charlie's age is half the combined age of Suzy and Brad. Suzy is 4 years younger than Brad, who is 10 years old. What is the sum of all three ages?

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The cube shown is constructed from identical rectangular slabs. How many slabs does the cube contain? The divisions in the original drawing indicate the same unit scale on all three directions; there are no gaps or overlaps.



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How many digits does $N$ have if $N=2\times3\times4\times5\times12\times25$?

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Find $117+104+91+78+65+52+39+26+13$.

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Jean has a collection of chips. Arranging them in piles of 12 leaves 11 chips over. Arranging them in piles of 21 leaves 20 chips over. What is the least number of chips Jean can own?

Enter one whole number, without units.

Find $A+B+C$ if different letters stand for different digits in the multiplication below. A and B are leading digits and cannot be zero. The diagram reads: the two-digit number A8 multiplied by 8 equals the three-digit number BCC.



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Each letter represents a different digit in the addition below. What is the least possible value of the three-digit number SUM? A cannot be zero. The four rows are ADD + ADD + ADD + ADD = SUM; each word denotes a three-digit number.



Enter one whole number, without units.

The five-digit number 653A2 is a multiple of 3. How many possible digits A are there?

Enter one whole number, without units.

Each position in a four-digit PIN code can be any digit from 0 to 9. How many possible codes are there? Leading zeros and repeated digits are allowed. The standalone source phrases this as the number of possible outcomes when guessing a four-digit ATM PIN; the mathematical conditions are the same.

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Ann read a book with 1000 pages. How many digits were used to print its page numbers? Assume pages are numbered 1 through 1000, each number printed once without leading zeros.

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When 500 chairs were distributed equally among 18 rooms, 14 chairs were left over. How many chairs were placed in each room?

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