IMSO 2007 Mathematics Contest in Taiwan – Short Answer

The fraction $\frac{2007}{7000}$ is written as a decimal. What digit is in the 2007th place? (For example, in the decimal 0.23456 the digit 4 is in the 3rd place.)

Rectangle $PQRS$ measures 24 cm by 16 cm. Points $T,U,V,W$ lie on its sides with measurements, in centimeters, as shown. Find the area, in square centimeters, of the shaded portion.

Mary's brother and grandmother both died young. The sum of their lifespans equaled 78 years. Mary's brother died 99 years after their grandmother was born. How many years after their grandmother died was Mary's brother born?

What would be the third number from the left of the 75th row of the accompanying triangular number pattern?

Nine squares are arranged as shown. If square $A$ has area 4 cm² and square $B$ has area 324 cm², what is the area of square $I$, in square centimeters?

Suzanne has 20 coins in her purse. They are 10-dollar, 20-dollar and 50-dollar coins, with total value 500 dollars. If she has more 50-dollar coins than 10-dollar coins, how many 10-dollar coins does she have?

Each face of a regular octahedron is numbered with a different integer. Each vertex receives a “vertex number”, the sum of the numbers on the faces meeting there. The vertex numbers are then added. What is the highest number that must divide this sum for every possible numbering of the faces?

The number 119 is very curious: when divided by 2, 3, 4, 5, 6, it leaves remainders 1, 2, 3, 4, 5 respectively. How many four-digit numbers have this property?

In the multiplication example, all digits 1 through 9 are used once and only once. Three digits are given. Fill in the rest.

A $5\times5$ table has A,B,C,D,E in the top row and A,B,C in the central three cells of its fourth row. Fill the remaining cells with A,B,C,D,E so that no row, column or diagonal contains a repeated symbol. Which symbol must be in the shaded cell?

A goat is tied by a rope to a corner of a rectangular shed as shown. The shed is 8 metres long and 5 metres wide, and the rope is 10 metres long. Grass surrounds the shed. What area, in square metres, can the goat graze? The ratio of the circumference of a circle to its diameter is given as $\pi=3.14$.

The diagram is a road plan of a city. All roads go east–west or north–south except one short diagonal road. A road is impassable at X because of repairs. Among all routes from P to Q, how many are shortest routes?