IMSO 2015 — Math Division — Short Answer

Anne asks her teacher his age. He replied: ‘My age now is a square number but after my birthday it will be a prime number’. If his age is greater than 20 but less than 65, how old is he now?

Given that $\overline{240a84}\times234=\overline{56b90256}$. What is the value of $a+b$?

In the expression $5\_\_\_4\_\_\_6\_\_\_3$, place each of the three operational symbols $+,-$ and $\times$ exactly once into the three blanks and list down all its possible values. What is the largest possible value of the resulting expression?

In the diagram below, regular octagon $ABCDEFGH$ and regular hexagon $IJKLMN$ are both centered at point $O$ such that $AB\parallel IJ$. If the measure of $\angle CBJ=56^\circ$, what is the angle measure, in degrees, of $\angle BJK$?

A boy has a cup of tea and a girl has an empty glass having the same volume as the cup. In the first step, the boy pours $\frac12$ of the tea from the cup into the glass. In the second step, the girl pours $\frac13$ of the tea from the glass into the cup. In the third step, the boy pours $\frac14$ of the tea from the cup into the glass. In the fourth step, the girl pours $\frac15$ of the tea from the glass into the cup. This alternate pouring continues such that in each step, the denominator increases by $1$. What fraction of the tea is in the cup after the thirteenth step?

The chairman of a 5-member committee is seated permanently on a chair at the round table. In how many ways can the other 4 members be seated at the same table if there are exactly 8 chairs?

In the arrangement below, each number is the non-negative difference of the two numbers above it. What is the average of the eight possible values of $z$?

How many positive integers from 1 to 2015 which are not divisible by any of the following numbers: 2, 20, 201 and 2015?

In a survey of 100 students, 84 said they like playing tennis, 74 said they like skiing and 62 students said they both like playing tennis and skiing. How many students dislike playing both tennis and skiing?

We know that 0, 2, 4, 6 and 8 are even digits. How many even digits are used from 1 to 100?

Two numbers are called mirror numbers if one is obtained from the other by reversing the order of its digits. For example, 123 and 321 are considered as such. If the product of a pair of mirror numbers is 146047, then what is the sum of this pair of mirror numbers?

In the hexagon below, $1$ is placed in the top triangle as shown. In how many different ways can we place the numbers $2,3,4,5$ and $6$ in the remaining empty triangles, such that the sum of the numbers in opposite triangles are exactly $5,7$ and $9$?

Six identical pyramids with square bases are assembled to make a cube which has a volume of $2744$ cm³ (Refer to Diagram 2). What is the length of $OH$, in cm, which is the height from the vertex to the base square $ABCD$ of each pyramid? (Refer to Diagram 1)?

Consider the following numbers: $1\times2,2\times3,3\times4,\ldots,29\times30$. How many of them are divisible by $3$ or $5$?

A fox sees a rabbit grazing 21 meters from his place. In one second, the rabbit was able to run 5 steps while the fox only 3 steps. It is also known that the distance travelled by the fox in 4 steps will take the rabbit 9 steps to completely cover. If the distance travelled by the rabbit in every step is 0.6 meter, how many seconds will it take the fox to catch the rabbit?

Consider all 3-digit numbers such that its digits are all different and there is no digit “0” used. Find the sum of all such 3-digit numbers.

In the diagram below, $ABC$ is a right triangle with $\angle C=90^\circ$. The bisector of $\angle A$ intersects $CB$ at $D$. If $CD=3$ cm and $BD=5$ cm, what is the length, in cm, of $AB$?

The diagram below is composed of $15$ unit squares. If $AB$ divides the area of the given diagram into two equal parts. Find the value of $\frac{MB}{BN}$.

Square $ABCD$ is drawn with points $E,F,G$ and $H$ as the midpoints of each side as shown in the diagram below. If the total area of the shaded region is $15$ cm², what is the area, in cm², of the square $ABCD$?

The letters I, M, S and O, as well as digits 1 and 5 are written in a row in some order.

(1) M is located either on the first or the last from the left.
(2) S is the located on the fourth position from the left.
(3) S is to the left, not necessarily immediately, of O.
(4) I is to the right, not necessarily immediately, of M.
(5) Letter O and digit 1 are next to each other.
(6) There is exactly one other letter or digit between I and 1.

What must be written in the second position from the left?