IMAS 2015 Junior Division — Round 1

What is the value of $\sqrt{(-2)^8}+(1+2+2^2+\cdots+2^{2015})^0+|-16|$?

Someone set the alarm clock for 1:30 pm and fell asleep at 12:35 pm. When he was awaken by the alarm clock, for how long had this person been sleeping?

In a quadrilateral $ABCD$, $AB\parallel DC$, $BC\parallel ED$, $AD=AE$ and $\angle C=110^\circ$. What is the measure of $\angle A$?

In a sale, each dress is reduced to 49% of its price and if two dresses are purchased at the same time, both are reduced to 45% of their prices. Lily buys two dresses together and pays 90 dollars for both. By doing this instead of buying them separately, how many dollars has she saved?

Sixteen points are arranged in a 3 cm by 3 cm formation. Four of them are removed, leaving behind twelve points as shown in the diagram. If we choose three of these twelve points as vertices of a triangle, what is the largest possible area of this triangle, in cm$^2$?

Class A has 17 students more than Class B, which has 15 students less than Class C. Of the following five numbers, which can be the total number of students in these three classes?

From 0, 1, 2, 3, 4 and 5, we choose two different numbers $x$ and $y$. What is the largest possible value of $2(x+y)^2+(x-y)^2$?

Divide the rectangle $ABCD$ into four isosceles right triangles and one square, as in the diagram below. If the area of square $EFGH$ is 100 cm$^2$, what is the area of rectangle $ABCD$, in cm$^2$?

A group of students are staying in a hotel. If five of them share a room, then there is no room for six of them. If six of them share a room, there are just enough rooms, one of which has less than six students. Of the following five numbers, which cannot be the number of students?

In a pentagon, one angle is $48^\circ$. The second angle is three times as large as the first. The third angle is $30^\circ$ less than the second. The fourth angle is $10^\circ$ less than the fifth. What is the measure of the fourth angle, in degrees?

There are three shirts, three pairs of trousers and three pairs of shoes. Of each type, one is red, one is black and one is white. In how many different ways can we choose one of each type so that something white is chosen?

In triangle $ABC$, $AB$ is perpendicular to $BC$. $D$ and $E$ are points on $BC$ such that $\angle BAD=\angle DAE=\angle EAC$ and $\angle ADC-\angle C=56^\circ$. What is the measure of $\angle BAC$?

If $\frac ab=a+1$ and $\frac ba=a-1$, what is the value of $\frac{b^2}{(a-1)^2}$?

$C$ and $D$ are points on $AB$ such that $AC:CD:DB=1:2:3$. Semicircles are drawn on the same side of $AB$ with respective diameters $AB$, $AC$, $CD$ and $DB$. What fraction of the area of the largest semicircle is the total area of the other three semicircles?

Each coin is worth either 1 dollar, 5 dollars or 10 dollars. Their total worth is 60 dollars. They may be divided into three, four or five piles of equal worth. What is the minimum number of coins?

From a cube of side length 10 cm, a cylinder with diameter 6 cm and depth 8 cm is hollowed out. What is the volume, in cm$^3$, of the remaining part of the cube? Take $\pi=3.14$.

If $a$, $b$ and $c$ are all positive integers, which of the following numbers can be the value of $(a+b+c)(a+b-c)(a-b+c)(-a+b+c)$?

A three-layer structure consists of 14 unit cubes. The bottom layer consists of 9 cubes in a 3 by 3 configuration. The middle layer consists of 4 cubes in a 2 by 2 configuration. The top layer consists of a single cube. The exposed surface area of this structure is painted, including the bottom. What is the total area of the unpainted surface of the individual cubes?

In an election between four candidates, they are supported respectively by 11, 12, 13 and 14 of the first 50 voters. Six more votes are to be cast, each for one of the four candidates. In how many ways can the candidate currently with 13 supporters become the uncontested winner?

Let $x$, $y$ and $z$ be distinct positive prime numbers such that $x+y+z$ and $x^2+y^2+z^2$ are also prime numbers. What is the minimum value of $x+y+z$?