Question bank AMO 2019

Find the sum: $2+4+6+8+10+20+40+60+80+100$.

How many three-digit numbers have a digit sum of 4? The hundreds digit cannot be 0.

Suppose today is Wednesday. How many Saturdays are there in the next 40 days?

A ladder has 3 rungs. An ant climbs along the ladder without retracing any part of its path. It starts at A, the middle of the bottom rung, and wants to reach B, the middle of the top rung. How many different paths can it take?

Jason has more money than Tom and Edison. Brandon has more money than Edison but less than Jason. Tom has less money than Edison but more than Alice. Who has the most money?
Enter 001 for Jason, 002 for Tom, 003 for Edison, 004 for Brandon, or 005 for Alice.

The solid shown is made of cubes. If all its exposed surfaces are painted black, how many square faces are painted?

Hannah is thinking of a three-digit number that reads the same forward and backward. Multiplying it by 101 gives 24442. What is her number?

Emma fills the 3 by 3 grid with the numbers 1 through 9, each used once. Every row, column, and diagonal must have the same sum. The grid shows the numbers placed so far. What number must go in the box labeled X?

Here is a method for multiplying a number ending in 5 by itself.
$25\times25=625$: $2\times(2+1)=6$, followed by $25$.
$35\times35=1225$: $3\times(3+1)=12$, followed by $25$.
$45\times45=2025$: $4\times(4+1)=20$, followed by $25$.
The result always ends in 25, since $5\times5=25$. Find the sum of the digits of $255\times255$.

Isabella has stacked unit cubes in a corner of a room as shown. If she continues the pattern to make the stack five cubes high instead of four, how many more cubes does she need?

Stacy read 60 pages of a book in 5 days. Each day she read 2 more pages than the day before. How many pages did she read on the first day?

Two players each start a game with 10 points. In every round, the winner gains 3 points and the loser loses 3 points. William wins exactly four rounds, and Abigail finishes with 16 points. How many rounds did they play altogether?

The place cards shown are folded along the dotted line so that only a number or a letter is visible. Chrissy sees all five cards, some showing numbers and some showing letters. The numbers she sees add up to 10. How many different sets of numbers are possible?

The four-digit number $\overline{A4A6}$ is divisible by 11. What is the digit A?

In $\overline{AB}\times\overline{BA}=\overline{57B}$, A and B are different digits, $\overline{AB}$ and $\overline{BA}$ are two-digit numbers, and $\overline{57B}$ is a three-digit number. If $\overline{AB}<\overline{BA}$, what is $\overline{AB}$?

Find the value of $100-80+60-40+20-10+8-6+4-2$.

How many three-digit numbers have a digit sum of 5? The hundreds digit cannot be 0.

I am thinking of a number. Subtract 17 from it and then multiply by 7 to get 91. What is half of my number?

Here is a method for squaring a number ending in 5. For $25\times25=625$, calculate $2\times(2+1)=6$ and append 25. Similarly, $35\times35=1225$ uses $3\times4=12$, and $45\times45=2025$ uses $4\times5=20$. These squares always end in 25. Find the sum of the digits of $455\times455$.

Isabella has stacked unit cubes in a corner as shown. If she continues the pattern until the stack is six cubes high instead of four, how many more cubes does she need?