AMO 2016 Secondary 1 (Grade 7)

Fully simplify using the order of operations: $15-16\div8\times2+6$.

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

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

The pattern of 6 individual figures shown below repeats indefinitely to form a sequence.



What is the 2015th figure in the sequence? Indicate the number of the shape.

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

Each of the seven digits $1,2,4,5,6,7,9$ is placed, one to a box, to form the two-digit and three-digit arithmetic problem shown. Find the least value of integer $N$ such that $N=83+\square\square\square-(\square\square+\square\square)$.

Milk is poured from a full pitcher into an empty glass until both pitcher and glass are $\frac45$ full. If the capacity of the pitcher is $40$ ounces of liquid, how many more ounces of milk are needed to fill the glass?

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

A rug is in the shape of an equilateral triangle with sides of length 10 feet. The rug is entirely composed of 100 equilateral triangular patches with sides of length 1 foot. How many of the smaller patches have one or more of their sides on the perimeter of the large rug?

Suppose $x=+\frac12$ and $y=-\frac14$. Select the number of the expression having the numerically greatest result: (1) $x+y$; (2) $x^2-y$; (3) $x^2+y$; (4) $x-y$; (5) $y-x$. Enter its original number, from 1 to 5.

The first five “pentagonal numbers” are 1, 5, 12, 22 and 35 as shown by the number of dots in the diagrams. What is the 9th pentagonal number?

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

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 digits of his year of birth?

A ski resort has three pricing options: 60 dollars for just a lift ticket, 20 dollars for just a ski lesson, or 70 dollars when both are bought together. On Sunday, 50 people bought exactly one of the three options. If 42 lift tickets and 23 lessons were sold, what was the mean purchase amount, in dollars, for these 50 people? Round your answer to the nearest whole number.

In the illustration, 42, 23 and 50 are numbers of people; 60, 70 and 20 are prices in dollars, not numbers of people.

Find the whole number value of $N$ when $13^2+2\times13\times N+N^2=18^2$.

The squares of two consecutive whole numbers differ by forty-five. Find the lesser whole number.

Sixteen players enter a chess tournament which offers a prize to the winner in each of an upper section and a lower section. Winners of the matches in round one are placed in the upper section. The players who lose matches in round one are placed in the lower section. From this point onward, players only play opponents in their own section and are eliminated from receiving a prize after their next loss. If there are no ties, how many matches are played in the tournament?

A rectangular box is $7\text{ cm}\times10\text{ cm}\times10\text{ cm}$. Point $A$ is a vertex, and point $B$ 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.

Suppose $2x-3y=10$ and $4x-5y+z=37$. Find the whole number value of $y+z$.

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 $\frac B3$, $\frac C9$, and $\frac D{18}$ is in lowest terms.

The bars illustrate the denominators using equal divisions; they do not show the amounts of the fractions.