AMO 2027 Grades 2-3 SET 23 - Self-study

A, B, C, D and E were standing in a line. A was standing between B and D. E was not standing in the last place. B was standing first in line. Who was standing in the last place?

For this self-study question, use the source solution’s intended assumption: B, A and D occupy three consecutive places, with A immediately between B and D.



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A number gives a remainder of 1 when divided by 2, 3 or 6, and a remainder of 4 when divided by 5. If this number is divisible by 7, find the smallest such whole number.



Enter one whole number, without units.

Grace and Freddy each tossed a coin three times. Grace got a head on her first toss, and Freddy got a tail on his last toss. In how many different ways can they get the same outcome? “The same outcome” means that their three results match toss by toss, in the same order.



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What is the sum of the digits of the result of $5654+6545+5465+4556$?



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The lamp is more costly than the watch and the candle stand. The candle stand is less expensive than the watch but more expensive than the basket. Which is the second most expensive item? Use the booklet options below.



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What should be added to the number formed by 5 hundreds, 3 tens and 8 ones so that the new number has the same digits in reverse order? For example, reversing 123 gives 321.



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In the following addition, different letters represent different digits. What is the sum of A, B and D? The addends are 7A, 3B5, 1C70 and D209; their sum is 8026.



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Some children were playing cricket. Each child brought 4 balls. During the game, 17 balls were hit outside the ground and lost. In the end, 55 balls remained altogether. How many children were playing?



Enter one whole number, without units.

Freddy has 1-dollar, 2-dollar, 5-dollar and 10-dollar coins. How many different ways can he pay exactly 10 dollars using one or more coins? Assume he has enough coins of each value. A way is determined by how many coins of each value are used; changing their order does not create a new way.



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Paul can jog 1 km in 4 minutes and 30 seconds. How many minutes will he take to jog 10 km? Assume he keeps the same pace for the whole distance.



Enter one whole number, without units.

Patrick started jumping every day. He jumped 10 times on Day 1, then 8 more times on Day 2 than on Day 1, and 8 more times on Day 3 than on Day 2. Each day he continued to jump 8 more times than the previous day. Which day is the first on which he jumps at least 100 times in that one day? Give the day number, not the total jumps over several days.



Enter one whole number, without units.

The numbers $20,21,22,\ldots,63,64$ are written together as the 90-digit number $2021222324\ldots626364$. All the odd digits are removed. How many digits remain? Remove individual digits 1, 3, 5, 7 and 9, not whole numbers that contain them.



Enter one whole number, without units.

Bret has fewer marbles than Ruby but more than Teresa. Patrick has fewer marbles than Suzy and fewer than Bret. Teresa has fewer marbles than Suzy, but she does not have the least number of marbles. Who has the least number of marbles?



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How many times altogether do the hour hand and minute hand cross an even-numbered mark on a clock from 1 a.m. to 1 p.m.? Count each hand separately, even if both cross at the same instant. Count crossings after the starting time and up to and including the ending time. The source clock is an illustration of the face, not a picture of the stated starting time.



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The sum of three consecutive whole numbers is 81. Find the smallest number.



Enter one whole number, without units.