IMSO 2012 - Mathematics - Essay

The sum of the numbers $A$, $B$ and $C$ is $390$. Given that $A$ is $3$ times of $B$ and $A$ is one third of $C$, find the value of $C$.

A palindrome is a number which reads the same backward as forward. A car odometer read $26962$ km. After two hours driving, the odometer showed the next palindrome. What was the average speed, in km per hour, of the car?

Class A has $10$ students and class B has $15$ students. In a test, the average score for class A is $60$, while the average score for class B is $66$. A new student answers the same test in the office. If he is put in class A, its new class average will become $62$. If he is put in class B, what will its new class average be?

Helen has a string of black beads and white beads which follows a certain pattern. She puts a portion of the string beads inside the box as shown in the diagram below. How many black beads are there in the portion of the string inside the box?



BOX means box.

Two years ago, Steve was three times as old as Bill, and in three years he will be twice as old as Bill. Find the sum of their ages now.

Dad walks at $6$ km per hour when alone and mom walks at $4$ km per hour when alone. When they walk together, they compromise at $5$ km per hour. They leave home to go to the store which is $1$ km away. Six minutes after leaving home, dad has to return for the shopping list while mom goes on. How long does mom have to wait in the store, in minutes, before dad arrives with the shopping list?

A says: “I ate it.”
B says: “The one who ate it was either C or D.”
C says: “Exactly one of A and B is lying.”
D says: “C did not eat it.”
If exactly two of them are lying, who ate it?

In the diagram below, $\angle ABC=\angle BDC=90^\circ$. If $\frac{AD}{DC}=\frac94$, then what is the value of $\frac{BD}{AC}$?

A three-digit number is multiplied by a two-digit number whose tens digit is $9$. The product is a four-digit number whose hundreds digit is $2$. How many three-digit numbers satisfy this condition?

In the diagram below, $ABC$ is a triangle with a right angle at $C$. Point $E$ is on $AC$ and point $D$ is on the extension of $CB$ such that triangle $DEC$ is similar to triangle $ABC$. If $AB$ cuts $DE$ at $F$, and $AE=EF$, then what is the measure, in degrees, of $\angle ABC$.

Points $P$ and $Q$ are on sides $AB$ and $BC$ of triangle $ABC$, respectively, such that $BP=3PA$, $QC=2BQ$ and point $K$ is the midpoint of segment $PQ$. If the area of triangle $ABC$ is $120\ \mathrm{cm}^2$, find the area, in $\mathrm{cm}^2$, of triangle $AKC$.

A triangle is divided into seven triangles. Four of them have areas $420\ \mathrm{cm}^2$, $80\ \mathrm{cm}^2$, $60\ \mathrm{cm}^2$ and $30\ \mathrm{cm}^2$ as shown in the diagram below. Find the area, in $\mathrm{cm}^2$, of triangle $AEF$.

The diagram below shows six distinct positive integers in a ring and the sum of any two neighboring numbers is a perfect square.



The below diagram is to be filled with six different positive integers such that it has the same property. If $X\le20$, find all possible values of $X$.