IMSO 2004 Mathematics – Taiwan Second Round – Theoretical Examination

Now we are in the year 2004. The ratio of ages of my father, my mother, and my younger brother is $12:9:1$. Five years from now, my father will be 41 years old. In what year was my younger brother born?

Laila's savings in a bank is 100 dollars. Tina's savings is 40 dollars. Every end of the week, Laila withdraws 3 dollars from her savings. At the same time, Tina always deposits 2.40 dollars into her savings. After how many weeks will Laila's savings be 6 dollars more than Tina's savings?

The product of two positive integers is even, but not divisible by 4. Is their sum odd or even?

On the table, there are 6 coins each of the values 5, 10, and 50 dollars. Deni takes 75 dollars. The number of coins Deni takes is more than 5, but less than 9. Does Deni take all three types of coins? If not, which type does he not take?

The weight of a small box, two medium boxes and a large box altogether is 10 kg. The weight of a small box, two medium boxes and two large boxes altogether is 15 kg. What is the total weight of two small boxes and four medium boxes?

Replace the letter $A$ with an odd digit and the letter $B$ with an even digit, so that 12 is a factor of the number $A579B$. Find all possible values of $A579B$.

Find the missing digits.

Find the total area of the shaded regions in the figure.

In the following figures, the three squares have equal areas. Determine whether the areas of the three shaded regions in each square are also equal.

If the pattern below is continued, what is the percentage of the area of the shaded regions in the third picture compared to the area of the largest square?

Four wheels $A,B,C,D$ are connected by belts as shown. Wheels $B$ and $C$ are fastened together. Their diameters are 12 cm, 36 cm, 9 cm, and 27 cm, respectively. Wheel $A$ turns at 450 rotations per minute. At what speed does wheel $D$ turn?

Three circular disks of radius 7 cm each are bound tightly with a belt as shown. What is the length of the belt?

The volume of a small balloon is 2 liters and a larger balloon is 5 liters. The small balloon is increased at a rate of 0.3 liters per second. The larger balloon is decreased at a rate of 0.12 liters per second. After how many seconds will the two balloons have the same volume?

A train travels between two stations. The train will be on time if it runs at an average speed of 60 km/hour, but will be late by 5 minutes if it runs at an average speed of 50 km/hour. What is the distance between the two stations?

Twice the number of marbles in bag $A$ is less than the number of marbles in bag $B$. The sum of the numbers of marbles in bags $A$ and $C$ is less than the number in bag $B$. There are more marbles in bag $D$ than in bag $B$. There are 6 marbles in bag $C$ and 9 marbles in bag $D$. How many marbles does bag $B$ contain?