AMO 2019 – Primary 5 (Grade 5)

What is twice the value of $9+7\times7+5\times5+3\times3+2$?

Find the sum of the digits of $3517+5173+1735+7351$.

The sum of three consecutive whole numbers is $443$ more than the greatest of them. What is the least number?

How many four-digit numbers have a digit sum of $4$?

What is the $100$th number in the sequence $17,24,31,38,45,52,\ldots$?

Hannah is thinking of a five-digit number that reads the same forward and backward. Its product with $101$ is $2488842$. What is the sum of its digits?

Tom’s PIN is a five-digit number with all digits different. The thousands digit is twice the ten-thousands digit; the hundreds digit is $50\%$ more than the thousands digit; the tens digit is $1$ more than the ten-thousands digit; and the ones digit is $2$ more than the tens digit. Find the PIN divided by $65$.

The odd numbers from $1$ through $17$ are placed in the magic square shown. Every row, column and diagonal has the same sum. What number belongs in the cell marked $A$?

Adding $52$ to two-fifths of a number gives three times the number. What is the number?

To square a number ending in $5$, multiply the preceding number by its successor and append $25$. For example, $25^2=625$ because $2\times3=6$; $35^2=1225$ because $3\times4=12$; and $45^2=2025$ because $4\times5=20$. If $625\times625=1000N+625$, find $N$.

Jessica folds a paper rectangle in half to form another rectangle, then folds that rectangle in half to form a $3$ cm by $3$ cm square. What is the greatest possible perimeter, in centimetres, of the original rectangle?

The diagram uses $17$ toothpicks to form a $2$-square by $3$-square rectangle. How many toothpicks form a $10$-square by $12$-square rectangle?

Find the whole number $X<5000$ whose digits strictly increase from left to right, for which $X-3$ is a perfect cube and $X$ is not divisible by $2$.

Clarification: interpret $X-3$ as a positive perfect cube, as intended by the supplied solution.

All $300$ people going to an event are in couples. No couple may be split between buses, each bus holds at most $65$ people, and every bus must carry the same number of couples. What is the least number of buses needed?

In $\overline{AB}\times\overline{BA}=\overline{A45B}$, $A$ and $B$ are different digits, the factors are two-digit numbers, and the product is four-digit. If $\overline{AB}<\overline{BA}$, find $\overline{AB}$.

The six place cards shown are folded along the dotted lines so that only a number or a letter is visible on each card. Chrissy sees all six cards. The visible numbers sum to $10$. How many different sets of visible numbers are possible?

The shaded region in the square-grid diagram has area $18\text{ cm}^2$. Find the area of triangle $CAB$, in square centimetres.

A MOEMS-tile is shaped like an M, as shown: a $5$ by $5$ square with two $4$ by $1$ rectangles removed. What is the maximum number of these tiles Jimmy can place on a $6$ by $30$ board without overlap?

A car has four tyres and one spare. Each tyre can travel at most $200$ km before wearing out. What is the greatest distance, in kilometres, the car can travel? Tyres may be exchanged along the journey.

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