AMO 2016 Secondary 2 (Grade 8)

Fully simplify using the order of operations: $17-18\div9\times3+8$.

The number $20,000,100$ may be expressed in expanded form as $(2\times10^x)+(1\times10^y)$. Evaluate $x(100+y)+1$.

The pattern of six individual figures shown below repeats indefinitely to form a sequence. What is the $2020$th figure in the sequence?

How much greater is the least 3-digit prime than the greatest 2-digit prime?

Each of the nine digits $1,2,3,4,5,6,7,8,9$ is placed, one to a box, to form the two-digit and three-digit arithmetic problem shown. Find the least integer $N$ such that $N=\square\square+\square\square\square-(\square\square+\square\square)+1000$.

The first five pentagonal numbers are $1,5,12,22,35$, as shown by the numbers of dots in the diagrams. What is the ninth pentagonal number?

An ant travels from vertex A to vertex B of the cube shown, always walking along an edge. How many shortest paths are there from A to B?

Milk is poured from a full pitcher into an empty glass until both pitcher and glass are $\frac67$ full. If the pitcher holds $84$ ounces, how many more ounces of milk are needed to fill the glass?

Suppose $x=+\frac13$ and $y=-\frac19$. Select the number of the expression having the numerically greatest result.

The pyramid shown has 7 vertices, 12 edges, and 7 faces, one of which is a hexagon. At least one edge on each face is to be colored red. What is the least number of edges that must be colored red?

Yevgeny was born on August 24, ____. On August 25, 2015, he was in the sixth year of his seventh decade of life. What is the sum of the digits of his birth year?

Find the whole number $N$ when $14^2+2\times14\times N+N^2=19^2$.

An equilateral triangular rug has sides of length 20 feet. It is entirely composed of 100 equilateral triangular patches with sides of length 2 feet. How many smaller patches have one or more sides on the perimeter of the rug?

Find the sum of the digits in $1001\times1001\times1001$.

A ski resort offers three pricing options: 60 dollars for a lift ticket only, 20 dollars for a ski lesson only, or 70 dollars for both. On Sunday, 60 people each bought exactly one option. If 42 lift tickets and 23 lessons were sold, what was the mean purchase amount in dollars? Round to the nearest whole number.

Sixteen players enter a chess tournament with one prize for the winner of an upper section and one for the winner of a lower section. Winners in round one enter the upper section; losers enter the lower section. Thereafter players only play opponents in their own section and lose eligibility for a prize after their next loss. If there are no ties, how many matches are played?

Find the least whole-number value of $B+C+D$ such that $\frac{17}{18}=\frac B3+\frac C9+\frac D{18}$, where $B,C,D$ are positive whole numbers and each of the three fractions is in lowest terms.

Suppose $3x-4y=10$ and $6x-7y+z=53$. Find the whole-number value of $y+z$.

A rectangular box measures $7\text{ cm}\times10\text{ cm}\times10\text{ cm}$. Point $A$ is a vertex and $B$ is the center of the front face, as shown. Find the shortest distance, in centimeters, from $A$ to $B$ along the surface of the box.

The squares of two consecutive whole numbers differ by 79. Find the lesser whole number.