**Printed examination instructions.** Answer the following 5 questions and show your detailed solution in the answer sheet. Write down the question number on each paper. Each question is worth 8 points. Time limit: 60 minutes.
The solution to each clue of this crossnumber is a two-digit number. None of these numbers begins with zero. Complete the crossnumber, stating the order in which you solved the clues and explaining why there is only one solution.
**Across:** 1. A square number; 3. A multiple of 11.
The menu in the school cafeteria never changes. It consists of 10 different dishes. Peter decides to make his school lunch different every day, with at least 1 dish. For each lunch, he may eat any number of dishes, but no two are identical.
(a) What is the maximum number of days Peter can do so?
(b) What is the total number of dishes Peter has consumed during this period?
1. Shape $S_1$ is a shaded square of side 1 unit.
2. Shape $S_2$ is made by dividing $S_1$ into 9 equal squares and removing four of these, so that only the central and corner squares remain.
3. Shape $S_3$ is made by applying the process in step 2 to each of the squares of $S_2$.
4. Shape $S_4$ is made by applying the process in step 2 to each of the squares of $S_3$.
And so on.
(a) Find the area and perimeter of shape $S_3$, giving your answers as fractions.
(b) Find the least value of $k$ for which shape $S_k$ has area less than $1/30$ and also perimeter greater than 30.
Chien-Ming Wang is a Taiwanese pitcher for the New York Yankees, and is scheduled to pitch in a game on Wednesday starting at 6:00 pm New York time. If Taipei time is 13 hours ahead, when will a live broadcast on local television begin?
In Pingtung, the average minimum temperature during a week was 30°C. The minimum temperatures on the first six days were 27°C, 29°C, 32°C, 33°C, 29°C and 32°C. What was the minimum temperature on the last day?
The Student Council has 36 members, and the ratio of the number of boys to the number of girls is $3:1$. How many more girls should be added so that the ratio of boys to girls becomes $9:5$?
The first digit of a 6-digit number is 1. We move this 1 so that it becomes the last digit, without disturbing the order of the other five digits. For example, 123456 becomes 234561. If the new number is exactly 3 times the original number, what is the original number?
The net of a cube and four candidate cubes are shown below. Which of the four cubes may be folded from the net? (The original page places the net on the right and the cubes on the left.)
Two polygons overlap so that $\frac23$ of the smaller polygon lies outside the larger, and $\frac89$ of the larger lies outside the smaller. When not overlapping, their total area is 120 cm². What is the area of the overlapping part?
A square piece of paper is divided into 16 identical boxes in a $4\times4$ array, each filled with a different letter as shown. The paper is folded four times:
1. The upper half covers the lower half.
2. The left half covers the right half.
3. The lower half covers the upper half.
4. The right half covers the left half.
Which letter is in the box on the 10th layer from the top?
The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34,... . Each term after the first two is obtained by adding the last two terms. What is the greatest value of the ratio of two neighboring terms?
