AMO 2018 – Secondary 3 (Grade 9)

How many integers from $1$ to $50$ are multiples of $3$ or $5$?

Which integer is closest to $\sqrt[3]{215}-\sqrt[4]{80}$?

The points $(0,1)$ and $(1,2)$ are adjacent vertices of a square. Find the sum of the $y$-coordinates of all possible vertices diagonally opposite $(0,1)$.

Source discrepancy: the supplied answer includes a square in which the given vertices are opposite, contrary to the word “adjacent”. Answer this question with the printed adjacency condition.

Emily and Hannah initially had the same number of tickets. Emily gave Hannah $5$ tickets, then Hannah bought $2$ more. Hannah ended with four times as many tickets as Emily. How many tickets did Emily have initially?

How many prime numbers are between $30$ and $100$?

Different letters represent different digits in $CUP+CUP=PINT$. Neither $P$ nor $C$ is zero. Find the last three digits of the smallest possible value of $PINT$.

The diagonals of quadrilateral $KITE$ are perpendicular. Given $KI=5$, $IT=3\sqrt5$, $ET=10$, and the distance from $K$ to the intersection of the diagonals is $4$, find its area.

Source clarification: the question paper calls the shape a kite, but its lengths are incompatible with that classification. As in the supplied solution, use a quadrilateral with perpendicular diagonals.

Solve for $N$: $\frac1{N-6}=\frac4{N^2-4N-8}$.

The sum of the last four page numbers of a puzzle book is $1186$. Find the first of these four page numbers.

The fractions $4/A$ and $5/B$ are in lowest terms and $\frac4A+\frac5B=\frac{41}{60}$. If $A$ and $B$ are positive integers, find $A+B$.

A sequence starts with $789$. Each later term is the sum of the cubes of the digits of the previous term. Find the $3000$th term.

Fifteen points are given, with no three on a straight line. How many different triangles can be formed using three of these points as vertices?

A three-letter palindrome reads the same forwards and backwards, for example BQB. How many three-letter palindromes can be made using only the vowels A, E, I, O, U?

Triangle $ABC$ is right-angled at $B$, and $BD$ is perpendicular to hypotenuse $AC$. Given $AD=8$ cm and $DC=18$ cm, find $BD$ in cm.

How many perfect squares less than $2000$ are also three times an odd integer?

How many integers $x$ satisfy $|2x+5|-18\le-6$?

Find the sum of the two primes whose product is $9991$.

$N$ is a whole number and exactly three statements below are true.
(1) $N$ is prime.
(2) $N+3$ is prime.
(3) $1<\sqrt N<9$.
(4) $N+5$ has an odd number of positive factors.
How many values of $N$ are possible?

How many distinct positive divisors does $4040\times2016$ have?

Define $n!=n(n-1)\cdots2\cdot1$; for example $4!=24$ and $3!=6$. Simplify:

$$\frac{81!+80!\times70-80!}{20\times79!+15\times79!+10\times79!+5\times79!}$$