IMSO 2005 Mathematics · Taiwan Short Answer (20 Questions)

Source instructions: 20 questions; each correct answer is worth 2 points. Write answers on the answer sheet. Time limit: 60 minutes.

A foundation has allocated a certain amount of money for 1st, 2nd and 3rd prizes in a competition. The money is divided in the ratio of 3:2 where the larger amount is for the 1st prize and the smaller amount is divided again in the ratio of 3:2 for the 2nd and 3rd prizes respectively. It becomes known that the 3rd prize is 3300 dollars less than the first prize. How much is the 2nd prize?

Three men and three children arrive at the river where there is a small boat that will hold one adult or two children. What is the minimum number of trips across the river in either direction to get the family across?

Mr. Sun has a broken calculator. When just turned on, it displays 0. If the + key is pressed, it adds 35. If the − key is pressed, it subtracts 35. If the × key is pressed, it adds 91. If the ÷ key is pressed, it subtracts 91. The other keys do not function. Mr. Sun turns the calculator on. What is the number closest to 2005 that he can get using this calculator?

There are 500 unit cubes. As many of these cubes as needed are glued together to form the largest possible cube which looks solid from any point on the outside but is hollow inside. What is the side length of the largest cube?

What is the ratio of the shaded square to that of the largest square shown in the diagram?

A three-digit number N leaves remainder 3 when divided by 7, remainder 5 when divided by 11 and remainder 8 when divided by 17. What is N?

How many rectangles are there in this grid, where vertices are points of the grid and the edges are lines of the grid?

A six-digit number is $\overline{abcdef}$, where the letters are its digits. Multiplying it by 6 gives $\overline{defabc}$. What is the number?

If $a,b,c,d$ are positive integers such that $a+\dfrac1{b+d/c}=\dfrac{2005}{101}$, what is $a+b+c+d$?

Alan has a stride of 75 cm. He walks 5 steps forward and one step back repeatedly. What is the least number of steps needed to reach a spot 24 metres away?

$N$ is a positive integer such that $N$ and $N+97$ are both perfect squares. What is $N$?

Five students sit an exam with maximum score 100. Their average score is 89. What is the smallest possible score of one student?

In the figure, $OX=OY=10$ are radii of a circular quadrant. A semicircle is drawn on $XY$ as shown. $T,S,C$ denote the resulting triangle, segment and crescent. What is the area of $C$?

A watermelon weighs 12 kg, with 97% of its weight being water. Some water evaporates in the sun, leaving 90% of its weight as water. What does it now weigh?

What is the number of lines of symmetry in the plane of the diagram?

What is the value of $x$ in the diagram?

When $10^{2005}-2005$ is expressed as a single number, what is the sum of its digits?

A triangle $PQR$ has $PQ=5,QR=3,RP=4$. The bisectors of angles $P$ and $Q$ meet at $I$. What is the area of triangle $PQI$?

A cube with edge length 10 is painted and then divided into 1000 unit cubes. How many small cubes have one or two painted faces?

In the 5 × 5 square the numbers 1,2,3,4 and 5 are arranged so that every number occurs precisely once in each column. In the square shown, what is the entry marked ☆?