In this posting, I will share IMSO (International Mathematics and Science Olympiad) that be held in 2006. Do this math problem seriously!

1. Three signal lights were set to flash every certain specified time. The first light flashes every 12 seconds, the second flashes every 30 seconds and the third one every 66 seconds. The signal lights flash simultaneously at 8:30 a.m. At what time will the signal lights next flash together?

2. Dina’s money consists of ten-thousand and five-thousand rupiah bills. The number of ten-thousand bills is three more than twice the number of five-thousand bills. If Dina has Rp355, 000, what is the number of ten-thousand bills that she has?

3. The principal of Makmur Jaya Elementary School is replaced every 4 years. At most how many principals will the school have from 2006 to 2020?

4. The area of the shaded region shown in the figure below is 98cm2. Find the length of a.

5. The sequence below is arranged by using numbers 1, 2 and 3 only: 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, . . .
What is the 100th number?

6. Every whole number larger than 7 can always be expressed as a sum of 3’s, 5’s, or both. For example, 9 = 3 + 3 + 3, 10 = 5 + 5 and 19 = 5 + 5 + 3 + 3 + 3. With the rule that 5 always comes before 3, how many ways can we express 444?

7. Consider all possible numbers between 100 and 2006 which are formed by using only the digits 0, 1, 2, 3, 4 with no digit repeated. How many of these are divisible by 6?

8. Two runners run in opposite directions from the same starting line. They run around a field which has 300 m perimeter. If the first runner runs at 150  m/minute and the second one runs at 125 m/minute, how many times will the two runners pass each other in the first 20 minutes?

9. The ratio of the number of students in Class A to Class B is 1:2. The ratio of the respective average test scores is 8:9. If the average score of class A is 72, find the average score of all the students.

10. In the following figures, the area of the biggest equilateral triangle is 16cm2. A new triangle is formed by connecting the midpoints of the sides of the previous triangle. If the pattern continues, find the area of the smallest triangle in Figure 5.

11. Each vertex of a regular pentagon is connected to the other vertices as shown in the figure below so that the pentagon is divided into 11 non-overlapping regions. How many non-overlapping regions can be obtained if we do the same procedure to a regular hexagon?

12. A 20cm×40cm×80cm wooden block is sliced into four small identical blocks. Find the largest possible sum of the surface areas of the small blocks.

13. The perimeters of a square and an equilateral triangle are equal. If the length of the side of the equilateral triangle is 8 cm, find the area of the square.

14. The faces of a cube are to be painted so that two faces with a common edge are painted with different colours. Find the minimum number of colours needed to do this.

15. How many non-congruent triangles with perimeter 11 have integer side lengths?

16. The following magic square is to be filled with numbers 17, 18, …, 24 so that the sums of numbers in every column, every row and the two diagonals are equal. Which number should be in the cell with the star (*)?

17. The faces of a dice are marked with dots from 1 to 6. The total number of dots on two opposite faces (top-bottom, left-right, front-back) is 7. Four dices are arranged as shown below. The faces of two dices that touch each other have the same number of dots. What is the total number of dots on the faces that touch each other?

18. Every edge of a cube is colored either red or green. In order to have at least one red edge on every face of the cube, find the minimum number of edges that must be colored red.

19. Let A,B,C represent three different digits such that:

Find the largest possible value of the 3-digit number ABC.

20. When 31513 and 34369 are each divided by a certain 3-digit number, the remainders are equal. Find this remainder.

21. What is the volume of the concrete foot bridge shown below? (Use phi = 22/7 )

22. The sides of a trapezoid touch the circle of radius 10 as shown in the figure below. The non parallel sides are of lengths 23 and 27 cm respectively. Find the area of trapezoid.

23. Each of the letters A,D,E,K, S,W and Y represents a different one of the digits 0, 1, 2, 3, 4, 5, 6, 7 and 8 such that

Which digit does E represent?

24. Dogol writes a sequence of five non-negative 1-digit numbers on the blackboard. He then erases two consecutive numbers and replaces them with their difference. He obtains the sequence 5, 0, 3, 5 on the board. How many possible sequences can he start with?

25. If you read the picture on the left below, it says there are 3 ones, 1 two and 1 four, which is correct. Fill in the four boxes in the picture on the right to make it correct too. Write down the four digits from left to right as a 4-digit number.

]]> http://mathandflash.com/imso-2006-short-answer/feed/ 0 http://mathandflash.com/elementary-mathematics-international-contest-imc-2008/ http://mathandflash.com/elementary-mathematics-international-contest-imc-2008/#comments Wed, 23 Nov 2011 14:54:25 +0000 admin http://mathandflash.com/?p=872 Individual Math Contest

Time limit: 90 minutes

Instructions:

-Write down your name, team name and candidate number on the answersheet.
-Write down all answers on the answer sheet. Only Arabic NUMERICALanswers are needed
- Answer all 15 problems. Each problem is worth 1 point and the total is 15points.
- For problems involving more than one answer, points are given only when
- ALL answers are corrected.
- No calculator or calculating device is allowed.
- Answer the problems with pencil, blue or black ball pen.
- All materials will be collected at the end of the competition.

1. Starting from the central circle, move between two tangent circles. What is the number of ways of covering four circles with the numbers 2, 0, 0 and 8 inside, in that order?

2. Each duck weighs the same, and each duckling weighs the same. If the total weight of 3 ducks and 2 ducklings is 32 kilograms, the total weight of 4 ducks and 3 ducklings is 44 kilograms, what is the total weight, in kilograms, of 2 ducks and 1 duckling?

3. If 25% of the people who were sitting stand up, and 25% of the people who werestanding sit down, then 70% of the people are standing. How many percent of the people were standing initially?

4. A sedan of length 3 metres is chasing a truck of length 17 metres. The sedan is
travelling at a constant speed of 110 kilometres per hour, while the truck is travelling at a constant speed of 100 kilometres per hour. From the moment when the front of the sedan is level with the back of the truck to the moment  when the front of the truck is level with the back of the sedan, how many seconds would it take?

5. Consider all six-digit numbers consisting of each of the digits ‘0’, ‘1’, ‘2’, ‘3’, ‘4’ and ‘5’ exactly once in some order. If they are arranged in ascending order, what is the 502^nd number?

6. How many seven-digit numbers are there in which every digit is ‘2’ or ‘3’, and
no two ‘3’s are adjacent?

7. How many five-digit multiples of 3 have at least one of its digits equal to ‘3’?

8. ABCD is a parallelogram. M is a point on AD such that AM=2MD, N is a point on AB such that AN=2NB. The segments BM and DN intersect at O. If the area of
ABCD is 60 cm2, what is the total area of triangles BON and DOM?

9. ABCD is a square of side length 4 cm. E is the midpoint of AD and F is the midpoint of BC. An arc with centre C and radius 4 cm cuts EF at G, and an arc with centre F and radius 2 cm cuts EF at H. The difference between the areas of the region bounded by GH and the arcs BG and BH and the region bounded by EG, DE and the arc DG is of the form m? ?n cm², where m and n are integers. What is the value of m+n?

10.  In a chess tournament, the number of boy participants is double the number of girl participants. Every two participants play exactly one game against each other. At the end of the tournament, no games were drawn. The ratio between the number of wins by the girls and the number of wins by the boys is 7:5. How many boys were there in the tournament?

11. In the puzzle every different symbol stands for a different digit.

What is the answer of this expression which is a five-digit number?

12. In the figure below, the positive numbers are arranged in the grid follow by the arrows’ direction.

For example,
“8”is placed in Row 2, Column 3.
“9” is placed in Row 3, Column 2.

Which Row and which Column that “2008” is placed?

13. As I arrived at home in the afternoon. The 24-hour digital clock shows the time as below (HH:MM:SS). I noticed instantly that the first three digits on the platform clock were the same as the last three, and in the same order. How many times in twenty four hours does this happen?

Note: The clock shows time from 00:00:00 to 23:59:59.

Instructions:

* Write down your name and country on every page.

* Answer all 6 questions.* You have 120 minutes to work on this test.

* Write down your answer on the provided answer sheets.

1. We are given a number of equilateral triangles with lateral length 1 cm. They come in two colors, yellow and blue. Three blue and one yellow triangles can be arranged to make an equilateral triangle of lateral size 2 cm (see 1st Pattern below). Six blue and three yellow triangles are arranged to form an equilateral triangle of lateral size 3 cm (see 2nd Patternbelow).

a. How many blue triangles and yellow triangles are required in the arrangement with lateral length 6 cm?

b. If you would like to make a similar arrangement to form an equilateral triangle of lateral size 10 cm, how many blue triangles and yellow triangles are needed?

c. If you would like to make equilateral triangle of lateral size 20 cm, how many blue triangles and yellow triangles are needed?

2. We define a trapezoid as a quadrilateral, which has a pair of parallel laterals; another pair of laterals are not parallel. In the rectangular arrangement below, there are exactly three noncongruent trapezoids. One of them is BCIG.

a. Find the other two non-congruent trapezoids.

b. Find the other 7 trapezoids which are congruent to BCIG.

c. What is the total number of trapezoids that can be made on the arrangement (both congruent and non-congruent, including BCIG).

3. There are 96 distinct ways an I-tromino (1 × 3 rectangular tile) can be positioned on squares of an 8 × 8 chessboard, along the lines of the chessboard. There are 48 vertical positions and 48 horizontal positions. (see picture).

a. In how many distinct ways can an V-tromino (see picture) be positioned on squares of the chessboard?

b. In how many distinct ways can a T-tetramino (see picture) be positioned on squares of the chessboard?

c. In how many distinct ways can an L-tetramino (see picture) be positioned on squares of the chessboard?

4.  Right-isosceles triangles are used to make various arrangements, so that the arrangements contain squares, as in the following illustrations:

1st illustration: Using two triangles, we can make an arrangement, which contains one square: ABCD.

2nd illustration: Using four triangles, we can make an arrangement, which contains two squares: ABEF and BCDE.

3rd illustration: Using eight triangles, we have six squares: ABEF,BCDE,EDIH, FEHG,BDHF and ACIG.

a. Using 10 of such triangles, how many squares at most can we find?

b. How about using 12 triangles?

c. How about using 18 triangles?

d. How about using 24 triangles?

5. Popon is to deliver newspapers along the streets in his neighborhood. He is paid by the distance he makes, and thus the farther he makes, the higher the pay is. While he can cross any intersection as many times as he likes, he cannot pass any street more than once ( a street is one segment between 2 adjacent points). For example, in a neighborhood which looks like this, Popon is to start from K and to finish at O. To get the highest payment, he takes the longest possible route. One possibility is indicated by 1-2-3-4-5-6-7-8 in the figure below.

If the neighborhood looks like the following picture, what is the longest possible route from A to B? Trace and indicate the route by writing numbers 1,2,3,… on the streets of the route.

The following figures show a triangular pyramid and some of its nets.

You are given 3 sheets of graph paper to work on ( you may not use them all), and one sheet of pink paper. You are asked to draw the largest cube net on the graph paper, such that the cube net fits the pink paper.

Best Regards!!!

Problems:

1. When Anura was 8 years old his father was 31 years old. Now his father is twice as old as Anura is. How old is Anura now?

2. Nelly correctly measures three sides of a rectangle and gets a total of 88 cm. Her brother Raffy correctly measures three sides of the same rectangle and gets a total of 80 cm. What is the perimeter of the rectangle, in cm?

3. Which number should be removed from: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 so that the average of the remaining numbers is 6.1?

4. The houses in a street are located in such a way that each house is directly opposite another house. The houses are numbered 1, 2, 3, … up one side, continuing down the other side of the street. If number 37 is opposite number 64, how many houses are there in the street altogether?

5. There are 6 basketball players and 14 cheerleaders. The total weight of the 6 basketball players is 540 kg. The average weight of the 14 cheerleaders is 40 kg. What is the average weight of all 20 people?

6. How many natural numbers less than 1000 are there, so that the sum of its first digit and last digit is 13?

7. Two bikers A and B were 370 km apart traveling towards each other at a constant speed. They started at the same time, meeting after 4 hours. If biker B started 1/22 hour later than biker A, they would be 20 km apart 4 hours after A started. At what speed was biker A traveling?

8. In rectangle ABCD, AB = 12 and AD = 5. Points P, Q,R and S are all on diagonal AC, so that AP = PQ = QR = RS = SC. What is the total area of the shaded region?

9. In triangle ABC, AP = AQ and BQ = BR. Determine angle PQR, in degrees.

10. In the equation below, N is a positive whole number.

A numbered card is placed in each box. If three cards numbered 1, 2, 3 are used, we get 2 different answers for N, that is 2 and 4. How many different answers for N can we get if four cards numbered 1, 2, 3, and 5 are used?

11. A mathematics exam consists of 20 problems. A student gets 5 points for a correct answer, a deduction of 1 point for an incorrect answer and no points for a blank answer. Jolie gets 31 points in the exam. What is the most number of problems she could have answered (including correct and incorrect answers)?

12. Joni and Dini work at the same factory. After every nine days of work, Joni gets one day off. After every six days of work, Dini gets one day off. Today is Joni’s day off and tomorrow will be Dini’s day off. At least how many days from today they will have the same day off?

13. In a bank, Bava, Juan and Suren hold a distinct position of director (D), manager (M) and teller (T). The teller, who is the only child in his family, earns the least. Suren, who is married to Bava’s sister, earns more than the manager. What position does Juan hold? Give your answer in terms of D, M or T.

14. The following figures show a sequence of equilateral triangles of 1 square unit. The unshaded triangle in Pattern 2 has its vertices at the midpoint of each side of the larger triangle. If the pattern is continued as indicated by Pattern 3, what is the total area of the shaded triangles in Pattern 5, in square units?

15. There are five circles with 3 different diameters. Some of the circles touch each other as shown in the figure below. If the total area of the unshaded parts is 20 cm2, find the total area of the shaded parts, in cm2.

]]> http://mathandflash.com/indonesia-elementary-mathematics-international-contest-inaemic-2006/feed/ 1 http://mathandflash.com/asean-primary-schools-mathematics-olympias-2003-essays/ http://mathandflash.com/asean-primary-schools-mathematics-olympias-2003-essays/#comments Sat, 08 Oct 2011 17:00:59 +0000 admin http://mathandflash.com/?p=832

Directions
1. Answer all 15 problems.
2. Show all works. Do not give theanswer only.
3. You have 90 minutes to work on this test.

1. Now we are in the year 2003. The ratio of ages of my father, my mother and my youngers brother is 12:9:1. Five years from now, my father will be 41 years old. In what year was my younger brother born?

2. Laila’s saving in a bank is $100. Tina’s saving is $40. Every end of week, Laila withdraw $3 from her savings. At the same time, Tina always deposit $2.40 into her savings. After how many weeks will Laila’s savings be $6 than Tina’s savings?

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

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

5. The weight of small box, two medium boxes and large box altogether is 10 kg. The weight of 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?

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

7. Find the missing digits.

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

9. In the following figure, the three squares have equal areas. Determine wheter the areas of the three shared regions in each square are also equal.

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

11. Four wheel A, B, C, D are connected by belts, see figure. The wheels B and C are fastened together. The diamters of wheels A, B, C, and D are 12 cm, 36 cm, 9 cm and 27 cm, respectively. The wheel A tursn at a speed of 450 rotations per minute. At what speed does the wheel D turn?

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

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

14. A train travels between two stations. The train will be on time if it runs at an average speed of 50 km/hour. What is the distance between two stations?

15. Twice the number of marbles in bag A less than the number of marbles in bag B. The sum of the number of marbles in bags A and C is less than hte number
of marbles 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?

Instructions:

• Write down your name, team name and candidate number on the answer sheet.
• Write down all answers on the answer sheet.
• Answer all 15 problems. Problems are in ascending order of level of difficulty. Only NUMERICAL answers are needed.
• Each problem is worth 6 points and the total is 90 points.
• or problems involving more than one answer, points are given only when ALL answers are correct.
• Take ? = 3.14 if necessary.
• No calculator or calculating device is allowed.
• Answer the problems with pencil, blue or black ball pen.
• All materials will be collected at the end of the competition.

1. The product of two three-digit numbers abc and cba is 396396, where a > c. Find the value of abc!

2. In a right-angled triangle ACD, the area of shaded region is 10 cm^2, as shown in the figure below. AD = 5 cm, AB = BC, DE = EC. Find the length of AB!

3. A wooden rectangular block, 4 cm × 5 cm × 6 cm, is painted red and then cut into several 1 cm × 1 cm × 1 cm cubes. What is the ratio of the number of cubes with two red faces to the number of cubes with three red faces?

4. Eve said to her mother, “If I reverse the two-digits of my age, I will get your age.” Her mother said, “Tomorrow is my birthday, and my age will then be twice your age.” It is known that their birthdays are not on the same day. How old is Eve?

5. Find how many three-digit numbers satisfy all the following conditions:
if it is divided by 2, the remainder is 1,
if it is divided by 3, the remainder is 2,
if it is divided by 4, the remainder is 3,
if it is divided by 5, the remainder is 4,
if it is divided by 8, the remainder is 7.

6. A giraffe lives in an area shaped in the form of a right-angled triangle. The base and the height of the triangle are 12 m and 16 m respectively. The area is surrounded by a fence. The giraffe can eat the grass outside the fence at a maximum distance of 2 m. What is the maximum area outside the fence, in which the grass can be eaten by the giraffe?

7. Mary and Peter are running around a circular track of 400 m. Mary’s speed equals 3/5 of Peter’s. They start running at the same point and the same time, but in opposite directions. 200 seconds later, they have met four times. How many metres per second does Peter run faster than Mary?

8. A, B and C are stamp-collectors. A has 18 stamps more than B. The ratio of the number of stamps of B to that of C is 7:5. The ratio of the sum of B’s and C’s stamps to that of A’s is 6:5. How many stamps does C have?

9. What is the smallest amount of numbers in the product 1× 2 × 3× 4 × …× 26 × 27 that should be removed so that the product of the remaining numbers is a perfect square?

10. Evaluate:

11. Train A and Train B travel towards each other from Town A and Town B respectively, at a constant speed. The two towns are 1320 kilometers apart. After the two trains meet, Train A takes 5 hours to reach Town B while Train B takes 7.2 hours to reach Town A. How many kilometers does Train A run per hour?

12. Balls of the same size and weight are placed in a container. There are 8 different colors and 90 balls in each color. What is the minimum number of balls that must be drawn from the container in order to get balls of 4 different colors with at least 9 balls for each color?

13. In a regular hexagon ABCDEF, two diagonals, FC and BD, intersect at G. What is the ratio of the area of ?BCG to that of quadrilateral FEDG?

14. There are three prime numbers. If the sum of their squares is 5070, what is the product of these three numbers?

15. Let ABCDEF be a regular hexagon. O is the centre of the hexagon. M and N are the mid-points of DE and OB respectively. If the sum of areas of ?FNO and ?FME is 3 cm2, find the area of the hexagon.

Clues Across
1. A square number
3. A multiple of 11
Clues Down
1. A multiple of 7
2. A cube number

2. Notice that 2^2 + 2^2 = 23 , so two squares can sum to give a cube; however, the two squares here are equal (to 4).
(a) Find two unequal squares whose sum is a cube.
(b) Show that there are infinitely many pairs of unequal squares whose sum is equal to a cube.

Note: ^ means square

3. Is it possible to find a number 11…11 that is divisible by 19?

4. The menu in the school cafeteria never changes. It consists of 10 different dishes. Peter decides to make his school lunch different everyday (at least 1 dish). For each lunch, he may eat any number of dished, but no two are identical.
(a) What is the maximum numbers of days Peter can do so?
(b) What is the total number of dishes Peter has consumed during this period?

5. A sequence of shapes is made as follows.
(1) Shape S1 is a shaded square of side 1 unit.
(2) Shape S2 is made by dividing S1 into 9 equal squares and removing four of these, so that only the central and corner squares remain.
(3) Shape S3 is made by applying the process in (2) to each of the squares of S2.
(4) Shape S4 is made by applying the process in (2) to each of the squares of S3.
And so on.

(a) Find the area and perimeter of shape S3, giving your answers as fractions.
(b) Find the least value of k for which the shape Sk has the area less than 1/30 and also has perimeter greater than 30.

1. Complete this magic triangle so that the numbers along each side give the same sum. Use each of the numbers 5, 6, 7, 8, 9 and 10 only once. (You are required to give only one solution.)

2. The height of the ground floor of a building is 4 m. The height of each of the other floors is 3 m. The total height of this building is 61 m. Inclusive of the ground floor, how many floors does the building have?

3. The composition of Scotty Cake for 6 servings is shown below. How much rice flour is needed to make Scotty cake for 10 servings?

4. If a positive whole number B is divided by 2, 3, 4, 6 or 9, the remainder is 1. Find the smallest possible value of B.

5. During the first five months of 2004, a company suffered a loss, then gained profit in the remaining seven months. The biggest loss, occurred in March, was 10 million rupiahs. The lowest profit was 9 million rupiahs in June and the highest profit was 15 million rupiahs in October. At least how much was the company’s profit during the whole year of 2004?

6. The average score of a mathematics test in a class of 48 students was 80. Changes were made to the scores of two students. One score was changed from 86 to 93. The other score was changed from 85 to 84. What is the new average score of the test?

7. Ms. Olmer pays the employees of her company every first Wednesday of the month. She goes to the bank to get the money for the salaries every first Tuesday of the month. One Wednesday morning, Ms. Olmer realized that she had to pay the employees, but she had not yet gone to the bank to get the money. What day is the fifth day of that month?

8. The figure below shows four equal circles. Each circle touches two adjacent circles. If the radius of each circle is 10 cm, find the area of the shaded region.

9. Mr. White multiplies the first one hundred prime numbers. How many consecutive zero digits can be found at the end of the resulting number?

10. A,B and C are nonnegative whole numbers less than 10 and satisfying the following multiplication: Find one set of values for A, B and C.

11. Andy multiplies the first fifty whole numbers: 1×2×3×4×· · ·×50. Counting from the right, what is the position of the first non-zero digit? For example, in 205000, the position of the first non-zero digit from the right is 4.

12. A circular bicycle path is 1 km long. Dodi rode a bicycle for two rounds at the speed of 30 kph. If he wants to average 40 kph, what should be his speed for the next four rounds?

13. Each letter represents a non-zero whole number less than 10. Different letters represent different numbers. Find the four-digit number STNA.

14. The entries to the table below are whole numbers 1, 2, 3, . . . , 9. Each number appears only once in the table. The numbers written to the right and below the table are products of numbers in the respective rows and columns. Find the number represented by “*”.

15. The ratio of an interior angle to an exterior angle of a regular polygon is 5 : 1. Find the number of sides of the polygon.

16. The figure below shows an arrow, with length 14 cm, in its starting position. The arrow is turned clockwise and makes 7 complete rounds plus 202.5o. Find the length of the path passed by the tip of the arrow. (use phi = 22/7)

17. A plane cuts a cube through vertex A into two parts. If the cross section formed by cutting the cube is an equilateral triangle, find the number of ways to cut the cube.

18. Hyde has some candies. Every day, he eats one half of the remaining candies from the previous day, plus one more candy. After five days all the candies were gone. How many candies does Hyde have originally?

19. The number N has the following properties:
(a) It consists of 4 digits, each digit is a number less than 7.
(b) It is a square of a certain number.
(c) If 3 is added to each digit, the resulting number is also a square
of a number.
Find N.

20. A square piece of paper 12 cm by 12 cm is folded, cut and unfolded as shown. If AE : EB = 5 : 3, what is the area of the shaded region?

21. A box without top cover (Figure B) is formed from a square carton size 34 cm × 34 cm (Figure A) by cutting the four shaded areas. If the sides of each shaded square are whole numbers, find the largest possible volume of the box.

22. Let N be a 6-digit number. Its first digit is 1. If the first digit is moved to become the last digit, the resulting number is three times N. Find N.

23. The following figure shows a regular hexagon ABCDEF. Each of the points P, Q, R and S is the midpoint of a side of ABCDEF. Find the ratio of the area of rectangle ABDE to the area of rectangle PQRS.

24. An ant sits at a vertex of a dodecahedron with edge length 1 meter. The ant moves along the edges of the dodecahedron and comes back to the original vertex without visiting any other vertex more than once. How many meters is the longest journey? (This dodecahedron has 12 faces and 30 equal edges.)

25. The display of a digital clock is of the form MM : DD : HH : mm, that
is, Month : Day : Hour : minute. The display ranges are
Month (MM) from 01 to 12
Day (DD) from 01 to 31
Hour (HH) from 00 to 23
Minute (mm) from 00 to 59
How many times in the year 2005 does the display show a palindrome? (A palindrome is a number which is read the same forward as backward. Examples: 12 : 31 : 13 : 21 and 01 : 02 : 20 : 10.)

Answer:

x^o+ y^o

= 70 + 50 = 120

(Hint: Use your algebra skills to factor the expression.)

Answer ?
Answer: 1/3