AMO 2027 Grades 4-5 SET 27 - Self-study

The product of $k$ and 1980 is a square number. What is the least possible positive whole number $k$? Editorial clarification: take $k>0$.

Enter one whole number, without units.

Find the sum of the numbers in the 101st group: $(1,4,6),(2,6,10),(3,8,14),\ldots$. Continue the pattern in which the first, second and third entries increase by 1, 2 and 4 respectively.

Enter one whole number, without units.

In a town of 10 905 residents, there are 2500 more male voters than female voters. If 911 residents are not allowed to vote, how many female voters are there? Here voters means residents eligible to vote.

Choose one option.

Sixteen trees stand in a straight row on one side of a road. Two cars are parked in every gap between consecutive trees. How many cars are there? Count only these gaps.

Enter one whole number, without units.

Find the ones digit of $99\times101\times103\times105\times107-100\times102\times104\times106\times108$. For a negative result, use the ones digit of its absolute value (ignore the minus sign).

Enter one whole number, without units.

Each different letter represents a different digit in the six-digit numbers below, and B is not zero. Find the last three digits of the smallest possible sum. For example, if a sum is 12345, enter 345.

**Solution-derived reconstruction:** the question's cryptarithm is missing in the source. BETTER + BITTER + BATTER + BUTTER is reconstructed from its worked solution. The original row order is unknown; row order does not affect the sum.



Enter exactly three digits, including a leading zero if needed.

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

Enter one whole number, without units.

Three pentagons are joined as shown. Every side is 3 cm. What is the difference between the sum of the three individual perimeters and the perimeter of the joined figure?



Choose one option.

How many digits are in the least positive multiple of 41 whose digits are all 1?

Enter one whole number, without units.

Ten equal short ropes are knotted end to end in one chain to make a rope 81 cm long. Each knot reduces the combined length by 1 cm. How long was each short rope? This uses the overlap interpretation of the source solution.

Enter one whole number, without units.

How many triangles are in the figure? Count triangles of every size whose complete straight sides follow drawn line segments.



Choose one option.

The picture shows a $4\times4$ grid. How many grid-aligned squares of any size contain the small square marked X? Count only squares whose sides follow grid lines.



Choose one option.

Fill the $3\times3$ grid using each number from 1 to 9 exactly once. Every row and every column must sum to 15. Find the number in the cell marked *. Diagonals are not required to sum to 15.



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.

Find the 10th term of $550,490,520,460,490,430,\ldots$. Continue the alternating rule: subtract 60, then add 30, and repeat.

Choose one option.