IMSO 2013 — Mathematics — Short Answer

Square pieces of sides $0.5$ cm are cut from a sheet which is $11$ cm long and $2$ cm wide. What is the total number of squares that can be cut?

Study the following pattern: $$\frac1{1\times2}=\frac12,\quad \frac1{1\times2}+\frac1{2\times3}=\frac23,\quad \frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}=\frac34.$$ Given that $$\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+\cdots+\frac1{2013\times2014}=\frac{a+2}{a+3},$$ where $a$ is a positive integer. Find the value of $a$.

Thirty girls joined a mathematics contest. The first girl scored 70 and the second girl scored 80. The teacher then announced that the score of every girl after the first two is equal to the average of the scores of all the girls before her. What was the score of the last girl?

Five boys, A, B, C, D and E, attended a meeting. In this meeting,

a. A shook hands with one boy.

b. B shook hands with two boys.

c. C shook hands with three boys.

d. D shook hands with four boys.

How many boys did E shake hands with?

What is the simplified value of $$2\times1\frac12+3\times1\frac13+4\times1\frac14+5\times1\frac15+6\times1\frac16+7\times1\frac17+8\times1\frac18+9\times1\frac19?$$

The sum of the digits of a two-digit number $\overline{ab}$ is $6$. By reversing its digits, one will obtain another two-digit number $\overline{ba}$. If $\overline{ab}-\overline{ba}=18$, find the original two-digit number.

In the diagram below, the side length of the largest square is $10$ cm long. If the total shaded regions inside formed by the two diagonals inside the circle and the two smaller squares is $26$ cm², what is the side length, in cm, of the smallest square?

The product of $1110$, $1111$, $1112$ and $1113$ is the thirteen-digit number $\overline{152628x755760}$, with one digit replaced by $x$. What is the value of $x$?

Each of A, B, C and D either always tells the truth or always tells lies. A says C always tells lies, B says A always tells lies, C says D always tells the truth and D says either A or C always tells lies. Who always tells lies?

In the diagram below, each of the interior angles of hexagon $PQRSTU$ is $120^\circ$. Given that $PQ=1$ cm, $QR=RS=4$ cm and $ST=3$ cm. Find the perimeter, in cm, of the hexagon $PQRSTU$.

In the diagram below, $PQRSTU$ is a regular hexagon with side length $2$ cm. The polygon $ABCDEFGHIJKL$ is obtained by drawing the equilateral triangles of side length $4$ cm, from the sides of the hexagon. Find $\frac{\text{area of }ABCDEFGHIJKL}{\text{area of }PQRSTU}$.

Nine lines, parallel to the base of a triangle, divide each of the other sides into 10 equal segments and the area into 10 distinct parts. Find the area, in cm², of the original triangle, if the area of the largest of these parts is 76 cm².

The dates of three Sundays in a month are all even numbers. What day is the 28th day of the month?

The Coco Company has a number of operational cars. The tax for the first car is 2000 dollars, the tax for second car is 5% more than the tax for the first car, the tax for third car is 10% more than the tax for the first car, and the tax for the other cars are 15% more than the tax for the first car. If the company pays a tax of 15500 dollars for all its cars. How many cars does the company have?

There are 1500 red dots and 513 white dots on a circle. We write 1 between two red dots, −1 between two white dots, and 0 between two dots that have different colours. What is the sum of the 2013 numbers we have written on this circle?

A paper cutter will cut a stack of up to 500 sheets of paper in one operation. If no piece of paper is ever folded, what is the minimum number of operations to get 2013 sheets of paper, starting with one sheet?

On Monday, Austin kept half of his cows and divided the other half equally between Dustin and Justin. On Tuesday, Dustin kept half of his cows and divided the other half equally between Justin and Austin. On Wednesday, Justin kept half of his cows and divided the other half equally between Austin and Dustin. Each one of them now has 8 cows. How many cows did Dustin have originally?

In a $4\times4$ table, there is a symbol in each square: a $\circ$, a $\bullet$, a $\triangle$ or a $\triangledown$. There is exactly one symbol of each kind in each row, each column and each of the two long diagonals. The symbols in four of the squares are shown in the diagram. What symbol should be placed in the square marked with a question mark?

In the diagram below, $AB=5$ cm, $AC=12$ cm, $DB=1$ cm, $ED=4$ cm and $CE=8$ cm. What is the angle measure, in degrees, of $\angle EAD$?

Find the smallest multiple of 9 which is not divisible by the sum of its digits.