IMAS 2017 — Junior — Round 1

What is the simplified value of $\sqrt{(-18)^2}-1^{2016}-(-1)^{2017}$?

Which of the following numbers is the sum of four consecutive positive integers?

In a supermarket, 3 kg of pear costs 16.26 dollars, while 2 kg of apple costs 13.62 dollars. How much does 1 kg of apple cost more than 1 kg of pear?

The value of the faction $\frac{m}{n}$ increases by 1 when the numerator increases by 2017. Find the value of $n$.

In the figure below, triangle $ABC$ is an isosceles triangle with $AB=AC$. Point $D$ is on $AC$ with $BD=BC$. If $\angle ABD=21^\circ$, what is the measure, in degrees, of $\angle BAC$?

What is the sum of all the prime divisors in the final result of $2^3+0^3+1^3+7^3$ (repeated prime divisors are counted only once)?

Let $a$, $b$, $c$, $d$, $e$ and $f$ are distinct digits such that the expression $\overline{ab}+\overline{cd}=\overline{ef}$. What is the least possible value of $\overline{ef}$?

How many integers $x$ satisfy $|2x+1|\le8$?

In the figure below, $AQBC$ is a convex quadrilateral with $QA=QB$ and $\angle C=60^\circ$. Triangle $QAB$ is folded along $AB$ to form triangle $PAB$. It is also known that $\angle PBC=30^\circ$ and $\angle PAC=20^\circ$. What is the measure, in degrees, of $\angle AQB$?

If all the edges of a rectangle are integers, then which of the following CANNOT be the length of its diagonals?

The length of 2 altitudes on adjacent sides of a parallelogram are 2 cm and 3 cm respectively with perimeter of the parallelogram is 18 cm. What is the area, in cm$^2$, of the parallelogram?

Three students are to participate in four games. Each student participates in at least one game and each game has exactly one student participating. In how many ways can this be done?

The sum of $a$, $b$ and $c$ is negative, while the product of these three numbers is positive. If $x=\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}$, what is the value of $x^{2017}-2017x^2+36$?

Given two positive integers, $\frac{4}{7}$ of the first is exactly $\frac{2}{5}$ of the second. What is the minimum sum of these two integers?

Tom starts working at 9:00 in the morning and finishes at 5:00 in the afternoon. How many more degrees does the minute hand rotates than the hour hand does on the clock during this period?

The price criteria of the subway ticket of a city is as follows: 2 dollars for within 4 km, 1 dollar more per 4 km for distances between 4 km and 12 km, 1 dollar more per 6 km for distances over 12 km. It costs 8 dollars to take subway from station A to station B. Which of the following is closest to the distance between A and B?

Consider a 3-digit number $\overline{abc}$, where $a$, $b$ and $c$ are distinct digits, so that their sum is 7. How many such three-digit numbers are there?

If positive numbers $x$ and $y$ satisfy $x^2-y^2=2$. What is the simplified value of $x\sqrt{2+y^2}-y\sqrt{x^2-2}$?

In the figure below, $ABCD$ is a quadrilateral where $AB=4$ cm, $BC=6$ cm, $CD=5$ cm, $DA=3$ cm and $\angle BAD=90^\circ$. Find the length, in cm, of $AC$.

If $a=\frac13$ and $b=\frac14$, find the numerical value of $a^3+b^3-a^2b-ab^2$.