AMO 2017 Secondary 2 (Grade 8)

Find the sum of the digits of the product $2017\times99999$.

The first five instances of a pattern involving the sum of three squares are shown. Find $B+D-A-C$.

Write $\sqrt{3^6}-\sqrt[3]{3^6}$ as an integer.

Find $(9.87)^2+19.74\times0.13+0.0169$.

The whole numbers are arranged five at a time in each row as shown. What is the sum of the row number and the column number of 2017?

If $2016=2^A\times3^B\times7^C$ for positive integers $A,B,C$, find $A^B+C$.

How many different sums can be obtained by adding two different integers chosen from $\{-102,-101,-100,\ldots,86,87,88\}$?

The spinner is divided into five equal parts. Spin once. If the needle lands on a perfect square, flip a fair coin once; if it lands on a prime number, do not flip the coin. What is the probability that you did not flip a coin? A fair coin lands on heads half the time. If the probability is $p/q$ in simplest form, write $p+q$.

Find the positive difference between the sum of all possible integers $x$ and the sum of all possible integers $y$ that satisfy $x^y=81$. Clarification for this edition: count each coordinate once for every integer solution pair $(x,y)$.

If the simplest form of $\frac{98}{100}\times\frac{97}{99}\times\frac{96}{98}\times\cdots\times\frac35\times\frac24\times\frac13$ is $a/b$, find $a+b-4000$.

The fraction $a/b$ is in simplest form and $\sqrt{\frac1{49}-\frac1{625}}=\frac ab$. Find $a+b$.

A survey of 45 students at the MOEMS Prep Academy found that 32 take Spanish and 40 take Algebra. What is the least possible number taking both subjects?



Diagram key: S is Spanish and A is Algebra.

Some of the 12 edges of a cube are coloured red so that each face has exactly three red edges. How many edges are red?

There are exactly $n$ whole-number cubes strictly between $7^6$ and $5^9$. Find $n$; do not include either endpoint.

The cardboard pattern shown consists of six numbered squares and is folded into a cube. What is the greatest product of the numbers on three faces meeting at one vertex?

If 450 times the positive integer $N$ is a perfect cube, what is the least possible $N$?

In rectangle $ABCD$, $X$ is the midpoint of $AB$ and $Y$ is the midpoint of $AD$. Find the area of triangle $XYC$ in square centimetres if the rectangle has area $880$ square centimetres.

How many four-digit positive integers have distinct nonzero digits whose sum is 14?

Two squares with integer side lengths overlap so that two sides of the smaller square lie along two sides of the larger square, as shown. The L-shaped shaded area is 27 square units. Find the sum of the two possible side lengths of the smaller square.

Jethro wants to tile a $10\text{ cm}\times15\text{ cm}$ rectangular floor using the shown J-shaped tiles. Each tile consists of six $1\text{ cm}\times1\text{ cm}$ squares and has area $6$ square centimetres. What is the smallest area, in square centimetres, left uncovered? Tiles cannot be cut or overlap.