IMSO 2012 - Mathematics - Short Answer

The radius of a circle is increased by $100\%$. Find the percentage increase in the area?

Three committees meet today. Of these three committees, one meets every $11$ days, the second meets every $15$ days, and the third meets every $21$ days. How many days until they all meet on the same day again?

A non-zero number is halved and the result is squared. This yields a number which is twice the original number. What is the original number?

The hypotenuse of a right triangle has length $10$ cm and the other two sides have lengths $y$ and $3y$ respectively. Find the area, in $\mathrm{cm}^2$, of the triangle.

In the diagram below, $ABC$ is an equilateral triangle of side length $7$ cm. The arcs $AB$, $BC$ and $CA$ are drawn with centres $C$, $A$ and $B$ respectively. Find the total length, in cm, of the three arcs. (Let $\pi=\frac{22}{7}$.)

Carmen selects four different numbers from the set $\{1,2,3,4,5,6,7\}$ whose sum is $11$. If $l$ is the largest of these four numbers, what is the value of $l$?

When my age is divided by $2$, $3$, $4$ or $6$ there is always a remainder of $1$, but when it is divided by $7$ there is no remainder. How old am I if my age is under $100$?

If $7$ is subtracted from a three-digit number $x$, the result is a multiple of $7$. If $8$ is subtracted from $x$, the result is a multiple of $8$. If $9$ is subtracted from $x$, the result is a multiple of $9$. When $x$ is divided by $10$, what is the remainder?

In a class of $25$ children, $12$ wear glasses and $11$ wear braces. If $7$ wear both glasses and braces, what is the number of those who wear neither?

IMSO, MOSI and SMIO are some arrangements of the letters $I$, $M$, $S$ and $O$. How many different arrangements are there such that the letter $I$ is not next to the letter $O$?

A boy saved $1$ dollar in the first week, $4$ dollars in the second week, and continued to save $3$ dollars more than he did the previous week. What was the total number of dollars that the boy had in the first ten weeks?

A two-digit odd number is a multiple of $9$. The product of its digits is also a multiple of $9$. What is this number?

The product of two positive integers is $1\,000\,000$. Neither of the two numbers contains the digit $0$. What is their sum?

In triangle $ABC$, $\angle A=50^\circ$ and the external bisector of $\angle B$ and $\angle C$ meet at $O$. What is the measure, in degrees, of $\angle BOC$?

In the diagram below, each of the small squares in the $4\times4$ grid measures $1$ cm by $1$ cm. Find the area, in $\mathrm{cm}^2$, of the $11$-sided polygon.

The teacher gave ten tests during the year, each carrying the same weight. If Mary scored $10$ more marks on the last test, her average would have become $92$. What was her actual average?

The faces of a cube contain the numbers $1$, $2$, $3$, $4$, $5$ and $6$ such that the sum of the numbers on each pair of opposite faces is $7$. For each of the cube’s eight corners, we multiply the three numbers on three faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is $1\times2\times3=6$.) What is the sum of the eight values assigned to the cube’s corners?

In a certain province, there are twelve highways, each joining a pair of cities. Each city is situated at the termination of three highways. What is the number of cities in this province?

A positive integer $n$ is divisible by four of the numbers $2$, $3$, $4$, $5$, $6$ and $7$, but not by the other two. If the two numbers that do not divide $n$ are consecutive, what is the smallest value of $n$?

Let $a$ and $b$ be positive integers such that $\frac57<\frac ab<\frac9{11}$. Find the value of $a+b$ when $b$ takes the smallest possible value.