AMO 2019 – Secondary 3 (Grade 9)

Simplify $\sqrt[5]{9^{10}}+\sqrt{(-2)^6}$.

Find the integer closest to $\sqrt{\dfrac{33}{0.499}-17.1}$.

$\overline{ABC}$ and $\overline{CBA}$ are three-digit integers. The digits $A,B,C$ are distinct and nonzero. Find the least possible sum $\overline{ABC}+\overline{CBA}$.

Mr. and Mrs. Hansen and their three children sit randomly around a circular table. The probability that children A and B sit next to each other is $m/n$ in lowest terms. Find $m+n$. Rotations count as the same arrangement; reflections count as different. In the figure, Cases 1 and 2 agree by rotation, whereas Case 3 is different.

Evaluate $101-102+103-104+105-106+\cdots+199-200+201$.

$x$ is a positive divisor of $20$ and $y$ is a positive divisor of $12$. The probability that $(x-y)/x$ is negative is $m/n$ in lowest terms. Find $m+n$. Clarification: choose each divisor independently and uniformly.

A dog barks every $b$ minutes and a cat meows every $m$ minutes, where $b,m$ are primes satisfying $1\le b\le10$ and $10\le m\le20$. What is the greatest number of simultaneous calls in two hours? Timing clarification, as intended by the supplied solution: both start together at time $0$; count calls at $0<t\le120$, excluding the starting call.

The sequence $1,2,3,\ldots,7,8,10,\ldots$ contains all positive integers except those containing the digit $9$. Find its $300$th term.

$M(x,y)$ lies on the perpendicular bisector of $AB$, where $A(-6,5)$ and $B(10,-7)$. If $x,y$ are integers, find the least possible value of $|x+y|$.

What is the greatest number of consecutive integers whose sum is $-100$?

A rectangular box has volume $70\text{ cm}^3$. Its length, width and height are distinct prime numbers in centimetres. Find its surface area in square centimetres.

Andy and Brett start at opposite points on a circular track of circumference $900$ feet, running in the same direction shown. Andy runs at $4.3$ feet per second and Brett at $5.5$ feet per second. After how many seconds do they first meet?

In the five-digit integer $\overline{XYYYX}$, $X,Y$ are distinct nonzero digits. How many such integers are divisible by $44$?

Joe and Tracy have $26$ pieces of candy altogether. Joe gives Tracy $n$ pieces so that she then has twice as many as Joe had initially. How many different values of $n$ are possible?

Exactly one statement about Betsy is false: she has at least $122$ sports posters; at least $131$; at least $135$; at least $139$. What is the greatest possible number of posters she has?

How many positive four-digit integers have exactly three positive divisors?

Find the greatest positive integer that divides $298$, $348$ and $873$ with the same remainder.

Jessica lies on Wednesday, Thursday and Friday and tells the truth on all other days. Tim lies on Sunday, Monday and Tuesday and tells the truth on all other days. On which day can both say, “Tomorrow I will lie”? Enter the day code: Monday $001$, Tuesday $002$, Wednesday $003$, Thursday $004$, Friday $005$, Saturday $006$, Sunday $007$.

Two circles of radii $4$ cm and $3$ cm overlap in a crosshatched region of area $6\pi\text{ cm}^2$. A dart lands inside their union. Its probability of hitting the overlap is $m/n$ in lowest terms. Find $m+n$. Clarification: the landing point is uniform over the union.

For a positive integer $n$, $n!=n(n-1)\cdots1$; for example, $4!=4\cdot3\cdot2\cdot1$. Find the greatest prime factor of $16!+18!$.