AMO 2027 Grade 4 SET 4 - Self-study

The keys to nine rooms are all mixed up in a box. What is the least number of key trials needed to guarantee that every key can be matched to its room lock?

There are exactly nine distinct keys, each opening exactly one different lock. Count physical key-in-lock trials; a match known by deduction needs no trial. Find the worst-case guarantee, not a lucky outcome. Enter one whole number.

In the pattern $15,17,21,A,35,45,B,\ldots$, what is $A+B$? Use the intended rule that the successive increases are consecutive positive even numbers, starting with 2.

Choose one option.

When the product of 9 and a number is divided by 4 and then multiplied by 25, the result is 19800. Find the number.

Enter one whole number, without units.

A small cube of side 6 units is placed inside a large cubical container of side 8 units. Liquid is then poured into the large container. What is the volume of the liquid, in cubic units?



Assume 8 units is the inside edge length, the smaller cube is solid and fully submerged, and liquid fills all remaining space to the brim. Ignore wall thickness. Enter one whole number without units.

Robin thought of a five-digit number that begins with 8. Her number is the smallest such number divisible by 9, and all its digits are different. Find the last four digits of the number.

Enter exactly four digits, retaining any leading zero. Do not insert spaces.

Brian takes a day off from work every 4th day, and David every 5th day. They both have today off, and their next shared day off is a Monday. What day is today?

Choose one option.

Yasmine sets an alarm to ring every 4 hours. When the 12th alarm rings, the hour hand points exactly to 7. What clock number did it point to when the first alarm rang?

Enter one whole number, without units.

The following $4\times4\times4$ cube is painted red on all its faces. Find the number of smaller cubes with at most two faces painted.





Choose one option.

Find the sum of the numbers in the 12th group of the sequence $(1,4,6),(2,6,10),(3,8,14),\ldots$.

Continue the displayed pattern: corresponding entries increase by 1, 2 and 4 in successive groups. Enter one whole number.

Find the ones digit of $100\times102\times104\times106\times108-99\times101\times103\times105\times107$.

Enter one whole number, without units.

What is the largest three-digit number divisible by 24?

Choose one option.

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?

Choose one option.

How many times does the digit 5 appear in the page numbers of a 350-page book? Count page numbers 1 through 350 once each; count both 5s in a number such as 55.

Choose one option.

On Monday, only $\frac56$ of the students were present in a class. All the absent students were girls. If $\frac23$ of the students present were boys and 15 girls were present, how many students are in the class?

Enter one whole number, without units.

Patrick scored 82 and 85 marks respectively in English and Science. If the total marks of these papers are 100, how many marks should he score in Mathematics so that his average marks in these three subjects becomes 87?

Each paper is marked out of 100; all three subjects have equal weight.

Enter only the whole-number answer, without units.