IMAS 2012 Junior First Round

What is the value of $2012^0+(-1)^2+|-2012|$?

In the diagram below, $ABCD$ is a square and $ACE$ is an equilateral triangle. What is the measure, in degrees, of $\angle BCE$?

The smallest interior angle of a triangle is $50^\circ$. Which of the following statements about this triangle is correct?

The diagram to the below shows three squares $EFGH$, $KLMN$ and $PQRS$ inside a rectangle $ABCD$. The areas of the three squares are $1\text{ cm}^2$, $9\text{ cm}^2$ and $4\text{ cm}^2$ respectively. What is the sum of areas of the shaded regions in $\text{cm}^2$?

A triangle is formed with 10 matchsticks of equal length connected end to end. No matchsticks are bent or broken. How many different triangles can be formed?

A piece of paper in the shape of parallelogram is folded into two with the crease bisecting the area of parallelogram. How many different kinds of origami methods are possible?

A TV company plans to broadcast a series with 48 episodes. One episode is aired each day except on Saturday and Sunday. If the first episode is aired on Thursday, on what day of the week will the last episode be aired?

Cups labelled 1, 2, 3, 4 and 5 with mouth upwards line in row, as shown below. Initially a ball is put into cup #3. In each move, the ball is transferred to an adjacent cup. If the ball is in cup #1, it can only be moved to cup #2. If the ball is in cup #5, it can only be moved to cup #4. After $2^{10}+3^8$ moves, which of the following statements about the ball is correct?

During the holidays, Dick worked part-time washing bowls in a restaurant. He got paid 3 dollars for washing one bowl. If he broke a bowl, he got no pay for washing it, and must pay 9 dollars to the owner. In one week, Dick washed 500 bowls and earned 1368 dollars. How many bowls did he break?

The diagram below shows four squares with numbers which exhibit a certain pattern. What number should be inside the fourth box?

The diagram below shows a square network of roads, $A_1$, $A_2$, $A_3$ and $A_4$ are four intersections on the same diagonal. We want to go from $A_1$ to $A_4$ by going only to the east or to the south, without passing through $A_3$. How many different paths are there?

Each row in a cinema has 80 seats, and row 13 to row 24 are reserved for students from a secondary school. There are 15 empty seats in these rows when all the students have taken their seats. How many secondary school students went to the cinema?

The total weight of 3 apples is equal to that of 4 bananas, and the total weight of 5 bananas is equal to that of 6 oranges. How many apples have the same total weight as 16 oranges?

The diagram below shows a cube with three of its faces labelled A, B and C, and a $3\times3$ square with six of its squares labelled 1, 2, 3, 4, 5 and 6. The cube is tipped over so that face C lies on square 1, tipped over again so that face B lies on square 2, and so on until the cube lies on square 6. What is the sum of the numbers of the squares on which the cube has laid with face B on top?

A deck of 54 cards has 2 jokers, and 13 cards of each of spades, hearts, clubs and diamonds. At least how many cards should be drawn at random so that there are at least 4 cards of the same suit?

The number $5^n+7^n$ is divided by 100 where $n$ is any non-negative integer. How many different values of the remainder are there?

For any positive integers $a$ and $b$, define a new operation $a\odot b$ which yields the remainder when the larger of $a$ and $b$ is divided by the smaller one. For example, $5\odot12=12\odot5=2$. Given that $(19\odot x)\odot19=5$, what value below is not possible for $x$?

What is the total number of positive integers consisting of three different digits in which the tens digit is equal to the units digit of the sum of the other two digits?

The diagram below shows seven squares resting on a straight line. The areas of three tilted squares are 1, 2 and 3. What is the total area of the other four squares?

On a $4\times4$ chessboard shown in the diagram below on the left, we wish to place a minimum number of copies of the shape shown in the diagram below on the right, so that no more copies of this shape can be placed. Copies may be rotated. What is this minimum number of copies?