AMO 2018 Secondary 2 (Grade 8)

What is the greatest three-digit number divisible by $5$, $6$ and $7$?

Determine the integer closest to $\sqrt[3]{\frac{64}{0.03}}+2$.

The point $(-3,5)$ is reflected over the $x$-axis. Its image is then reflected over the $y$-axis to $(a,b)$. Find $a-b$.

Solve for $N$: $\frac{2}{3+\frac1N}=\frac{2N}{13}$.

Different letters represent different digits in the cryptarithm. Find the first three digits of the least possible value of $\mathrm{PINT}$. Assume $P$ and $C$ are not both $0$.

Clarification: PINT is a four-digit number with no leading zero. The source states only that P and C are not both zero.

How many prime numbers are between $50$ and $100$?

Consider $10$ points, no three of which lie on a straight line. How many different triangles can be formed using three of these points as vertices?

Thirty-six points form a unit-square array as shown. Find the area of the eight-sided figure in square units.

A sequence begins with $789$. Each subsequent term is the sum of the cubes of the digits of the preceding term. Find the $2018$th term.

The diagonals of a kite are perpendicular. If the perimeter of kite $KITE$ shown is $2(\sqrt{x}+\sqrt{y})$ units, find $x+y$.

Clarification of the intended interpretation: $\sqrt{x}$ and $\sqrt{y}$ denote the two distinct side lengths of the kite.

Jacob initially had $5$ biscuits for every $3$ Sheila had. After Jacob gave Sheila $1$ biscuit, their ratio became $3:2$. How many biscuits did Jacob have initially?

The sum of the last four page numbers in a puzzle book is $786$. Find the last page number.

How many numbers less than $1000$ are perfect squares that are also three times an odd integer?

Two primes have product $9991$. Find the larger prime.

Thirty-six points form a unit-square array. Find the sum of the interior angles of polygon $ABCDEF$, in degrees.

Define $n!=n(n-1)\cdots2\cdot1$; for example, $4!=24$ and $3!=6$. Simplify $\frac{41!+40!}{3\times39!+2\times39!+39!}$.

Find the greatest whole number less than $1000$ that can be expressed as each of: a sum of two consecutive whole numbers; a sum of three consecutive whole numbers; and a sum of five consecutive whole numbers.

When the shown pattern is continued indefinitely, what number appears directly below $526$?

The largest circle completely inside an equilateral triangle has radius $16$ cm. What is the altitude of the triangle, in cm?

Find the number of distinct positive divisors of $2020\times2016$.