IMAS 2016 — Junior — Round 1

What is the value of $\sqrt{(-20)^2}+16^2-15^2$?

The table below summarizes the results of a test in a certain class. What is the total score of this class?

A three-digit number is not divisible by 24. When divided by 24, the quotient is $a$ and the remainder is $b$. What is the minimum value of $a+b$?

In the trapezium $ABCD$, $AB$ is parallel to $CD$. $E$ and $F$ are points on $AD$ and $BC$ respectively such that $EF$ is also parallel to $AB$. The area, in cm$^2$, of triangles $BAF$, $CDF$ and $BCE$ are 8, 7 and 18 respectively. What is the area, in cm$^2$, of $ABCD$?

What is the value of the negative number $x$ which satisfies $|x-3|=|3x|+1$?

The radius of each wheel of Rick's bicycle is 25 cm. He rides to school at a constant speed and arrives after 10 minutes. During this time, each wheel makes 160 revolutions per minite. Of the following five distances, which is closest to that between Rick's home and school?

How many two-digit numbers are there such that at least one digit is divisible by 3?

The chart below shows the sale figures of a certain merchandise in 2014 and 2015 by the season. How many more items were sold in 2015 than in 2014?

$ABC$ is an equilateral triangle. $D$ is a point inside such that $BCD$ is a right isosceles triangle. The altitude $BE$ of $ABC$ intersects $CD$ at $F$. What is the measure, in degrees, of $\angle CFE$?

In how many ways can 36 be expressed as the sum of two prime numbers, the first larger than the second?

Every student in a class is either in the mathematics club or the language club, and one third of them are in both. If there are 22 students in the language club, 4 less than the number of students in the mathematics club, how many students are there in this class?

The average of a group of numbers is 5. A second group contains twice as many numbers and its average is 11. What is the average when the two groups are combined?

What is the value of $x^y$ if $\sqrt{x-1}+\sqrt{1-x}+y=2016$?

Each of A and B goes to the gymnasium 3 or 4 times a week. After $n$ weeks, A has been there 57 times while B has been there only 47 times. What is the value of $n$?

$D$ is a point on $AB$ such that $AD=1$ and $BD=2$. How many points $C$ are there in the plane such that both $ACD$ and $BCD$ are isosceles triangles?

From a $5\times5$ square piece of paper, two $2\times4$ rectangles are cut off along the grid lines. In how many different ways can this be done?

The number $a$ is 5 more than its reciprocal. What is the value of $(a^2-1)^2-125a$?

With each vertex of a parallelogram $ABCD$ as centre, a circle is drawn. Exterior common tangents are then drawn, as shown in the diagram below. If the perimeter of $ABCD$ is 36 cm and the radius of each circle is 2 cm, what is the maximum area, in cm$^2$, of the figure enclosed by the circular arcs and tangents?

What is the smallest positive integer with 12 positive divisors such that it is relatively prime to $(2016^3-2016)$?

At most how many right triangles can be formed by five lines on the plane?