AMO 2017 Secondary 3 (Grade 9)

Negative integers are Cold Cubes and positive integers are Hot Cubes. Adding five Hot Cubes raises a liquid’s temperature by five degrees; adding five Cold Cubes lowers it by five degrees. How many Cold Cubes must be removed to raise the temperature from $-97$ degrees to $-42$ degrees?

Find $k$ if $k=5.6(5.6+0.4)+0.4(5.6+0.4)$.

A palindrome reads the same forwards and backwards. Find the sum of all digits $x$ for which the sum of the four-digit numbers $\overline{43xx}+\overline{xx34}$ is a palindrome.

Evaluate $\sqrt{3^6}-\sqrt[3]{-3^6}$.

The positive integers are arranged five per row in alternating directions, as shown. Find the sum of the row number and column number containing $2019$.

A wire forms a square of area $324\text{ cm}^2$. It is straightened, cut into equal lengths, and fastened together to make the twelve edges of a cube, with no waste. Find the cube’s volume in $\text{cm}^3$.

The six numbered squares in the net are folded into a cube. Find the greatest product of the numbers on three faces meeting at one vertex.

How many perfect squares lie strictly between $7^6$ and $4^8$?

Some edges of a cube are coloured red so that each face has exactly one red edge. How many edges are red?

$1001$ is the product of three distinct primes. How many positive factors of $1001$ are not prime?

How many ways can digits fill the two blanks in $9\_5\_$ to make a multiple of $11$? Digits may repeat.

If $2016=P^A Q^B R^C$, where $P,Q,R$ are primes and $A,B,C$ are positive integers, find $A+B+C$.

For all integer solution pairs $(x,y)$ of $x^y=256$, find the positive difference between the sum of the $x$-coordinates and the sum of the $y$-coordinates, counting each coordinate once per solution pair.

Clarification: this is the counting convention used by the source solution; repeated $y$-values are counted again.

Write $\sqrt{\frac1{64}-\frac1{289}}=\frac ab$ in lowest terms, with $a,b$ positive integers. Find $b-a$.

Six boys of different heights stand in two rows of three. Each boy is taller than every boy to his left in his row and taller than the boy directly in front of him, if any. How many arrangements are possible?

Place the integers $1$ through $10$ in the ten circles, each once. Each number in the upper three rows is the positive difference of the two numbers directly below it. Find $A+B$.

A spinner has five equal sectors labelled $1,2,3,4,5$. Spin once. If it lands on a perfect square, toss a fair coin once; if it lands on a prime, toss it twice. All tosses are independent. The probability of at least one head is $p/q$ in lowest terms. Find $p+q$.

Jethro tiles a $10\text{ cm}\times15\text{ cm}$ floor with the pictured J-shaped tiles. Each tile consists of six $1\text{ cm}\times1\text{ cm}$ squares. Tiles may be turned, but may not be cut or overlap. What is the smallest uncovered area, in $\text{cm}^2$?

Interpretation: tiles follow the unit-square grid, as in the source construction.

How many four-digit positive integers have distinct nonzero digits whose sum is $15$?

Two squares with positive integer side lengths overlap as shown, with two sides of the smaller lying along two sides of the larger. The shaded L-shaped region has area $45$ square units. Find the sum of the three possible side lengths of the smaller square.