AMO 2019 Secondary 2 (Grade 8)

Compute $(75\times5\div3)+(75\div5\times3)$.

Simplify $\sqrt{2^8}-\sqrt[4]{2^8}-\sqrt[8]{2^8}$.

$56\%$ of $57$ is the same as $76\%$ of what integer?



Diagram note: the shaded quantities are equal. Each bar is normalized independently; the two bars do not use a common length scale.

$\overline{XYX}$ and $\overline{YXY}$ are three-digit integers, where $X,Y$ are distinct nonzero digits. What is the least possible value of $|\overline{XYX}-\overline{YXY}|$?

Mr. and Mrs. Hansen and their three children sit around a circular table. In how many different ways can they sit if the parents sit next to each other? Rotations count as the same seating; reflections count as different. In the figure, Cases 1 and 2 are rotations, but Case 3 is different.

Find $1-2+3-4+5-6+\cdots+99-100+101$.

Let $G$ be the greatest value of $(x-y)/x$, where $x$ is a positive divisor of $131$ and $y$ is a positive divisor of $197$. Find $524G$.

$P,Q,R$ are prime numbers satisfying $P+Q+R=65$ and $2Q+3R=53$. Find $P$.

A dog barks every $b$ minutes and a cat meows every $m$ minutes. The numbers $b,m$ are prime, with $1\le b\le10$ and $10\le m\le20$. What is the greatest number of simultaneous sounds in a two-hour period? Clarification: choose the timing to maximize the count, and include a simultaneous sound at the start of the period.

How many points on the line segment from $X(-4,4)$ to $Y(12,-8)$ have integer coordinates?

Compute $\frac{123456789^2-123455678^2-1111^2}{1111\times123455678}$.

What is the greatest number of consecutive integers whose sum is $-33$?

Three faces of a rectangular box have areas $42$, $72$, and $84$ square centimetres. Find its volume in cubic centimetres. Clarification: these are three mutually perpendicular faces.

Thirty unit squares form the staircase arrangement shown. How many squares with integer side lengths can be found in it? Count squares bounded by the drawn grid segments.

Andy and Brett start at opposite points of a circular track of circumference $600$ feet and run in the directions 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 $\overline{ABBBA}$, $A$ and $B$ are different nonzero digits. How many such integers are divisible by $33$?

Exactly one of the following statements about Betsy is false: she has at least $11$ sports posters; at least $20$; at least $24$; at least $28$. What is the greatest possible number of posters?

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

Find the greatest positive integer that divides $377$, $127$, and $852$ with the same remainder in each case.

Stacy lies on Tuesdays, Wednesdays and Thursdays and tells the truth on other days. Anthony lies on Saturdays, Sundays and Mondays and tells the truth on other days. On which day could both say, “Tomorrow, I will lie”? Use Monday $=001$, Tuesday $=002$, …, Sunday $=007$.