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1 Notice: Individual students, nonprofit libraries, or schools are permitted to make fair use of the papers and its solutions. Republication, systematic copying, or multiple reproduction of any part of this material is permitted only under license from the hiuchang Mathematics Foundation. Requests for such permission should be made by ing Mr. Wen-Hsien SUN

2 International Mathematics and Science Olympiad 0 SHORT NSWER PROBLEMS () lex and Benito make 880 pies in 8 hours working together. lex makes 0 more pies in one hour than Benito. Find the number of pies made by lex in one hour. Submitted by Philippines Solution In 8 hours lex makes 0 8 = 80 more pies than Benito. Hence if Benito made same pies as lex in one hour, they can make = 90 pies in 8 hours. So lex made 90 8 = 0 pies per hour. nswer: 0 pies () ivide 08 students into four groups such that two times the number of students in group is (i) half of the number of students in group, (ii) less than the number of students in group 3. (iii) more than the number of students in group. Find the number of students in group. Submitted by Philippines Solution If we move two students from Group Three to Group Four, then each of them has twice as many students as Group One, while Group Two has four times as many. Hence there are 08 ( ) = students in Group One. nswer: students (3) In the diagram below,, and E are points on the line B. Given B = 9. cm and E =.7 cm, find the sum of the lengths of all ten line segments determined by these five points. Submitted by Thailand Solution There are 0 line segments, the sum of lengths of which is: B+ ( + B) + ( + B) + ( E+ EB) + E+ ( + E) = 9.+.7=.cm. nswer:. cm () Four cube with edge length m are cut up into cubes each with edge length cm. If all these cubes were placed one on the right of the other to form a line, find the length of the line, in m. Submitted by Jury Solution Since m = 00 cm, there are 00 = of the cm cubes along each edge of the m cube. So there are of the cm cubes. So, when they are placed one on the right of the other, the height is = 0000 cm = 00 m. E B

3 Solution Observe that the sum of the volumes of the cubes with length cm is equal to the sum of the volumes of the four cubes with length m, which is m 3 = cm 3. Since the area of each face of a cube with length cm is = cm, the length of the line is = 0000cm = 00 m. nswer: 00 m () Michael wanted to tie 0 ropes. The length of each rope was 0 cm. cm of one end of a rope was tied to cm of one end of another rope. Each of the resulting knots was cm long. What was the length, in cm, of the new rope? Submitted by Brunei Solution Since there were 0 ropes, the total length of these ropes was 0 0 = 000 cm. However, some parts of the ropes were used to tie the knots. So the new rope was shorter than the total length of these ropes. Since there were 0 pieces of ropes altogether and the two ends of the rope were not tied, the number of knots tied was 0 = 9. + = 0cm of rope was used to tie each knot but the knot was only cm long. Thus, the new rope would be 9 = 9cm shorter than the total length of the 0 ropes. Therefore, the new rope was = 90cm long. nswer: 90 cm () lass and lass B have the same number of students. The number of students in class who took part in a mathematics competition is of the students in class B who did not take part. 3 The number of students in class B who took part in a mathematics competition is of the students in class who did not take part. Find the ratio of the number of students in class who did not take part in this competition to the number of students in class B who did not take part. Submitted by Malaysia Solution Let a and b be the number did not take part in class and B respectively. Then the number of students in class is a+ b, the number of students in class B is b+ a. 3 Since lass and B have the same number of students, we have a+ b= b+ a, 3 and hence a= b, i.e. a:b = :. 3 nswer: :

4 (7) What number can be added to both 70 and 30 so that the sums are in the ratio 3:? Submitted by Philippines Solution Note that = 80. The unknown number is added to 90 three times but added to 70 only once. So twice this number is 80 and this number is = nswer: (8) Two different shirts at a shop were sold at the same price. While one shirt made a profit of 30%, the shop had incurred a 30% loss for the other one. id the shop record a profit or loss from these two transactionsnd by how many %? Submitted by Thailand Solution Suppose each shirt is sold for ( )(00 30) = 900 dollars. Then the cost of the first shirt is900 (00% + 30%) = 7000 dollars and the cost of the second shirt is 900 (00% 30%) = 3000 dollars. The total cost is 0000 dollarsnd the total take is 900 = 800 dollars. Hence the net loss is = = 9% nswer: loss 9 % (9) television show has 83 episodes. If the show starts on Saturday and broadcasts everyday with three episodes each day, on what day will the last episode be broadcasted? Submitted by Philippines Solution Since there are three episodes broadcasted each day, it will take 83 3 = days to broadcast 83 episodes. Since Saturday is counted as the first day, Sunday will be counted as the second daynd so on: Saturday Sunday Monday Tuesday Wednesday Thursday Friday 3 7 = 3 7, so the last episode will be broadcasted on Friday. nswer: Friday (0) Find the area, in cm, of the isosceles trapezoid B, given that = cm, B = 8 cm, B= and = =. Submitted by Vietnam B

5 Solution Let E and F be the points on so that BE and F, respectivelys the diagram shown below. B E F Observe that BE = F and EF = B = 8cm. Since = =, BE and F, we know BE = F = and hence BE F. So E = F = ( 8) = cm. Thus BE = F = cm and we can conclude that the area of isosceles trapezoid B is ( + 8) = 8 cm. Solution B onstruct 3 additional trapezoids identical to trapezoid B to create a large square, inside which there is a small square. Then the area of trapezoid B is one fourth the difference between the two squares and is equal to ( 8 ) = ( ) = 8cm. nswer: 8 cm () On her 0 th birthday, Mrs. Sharma makes gifts to her two sons whose ages are prime numbers. She gives to one son a number of dollars equal to the square of his agend to the other son a number of dollars equal to his age. She gives 300 dollars in total. Find the sum of the ages of Mrs. Sharma s two sons. Submitted by India Solution Since 89 = 7 < 300 < 9 = 3, hence must have a son younger than 9. If a son s age is 7, then another son s age is =, which is also a prime. If a son s age is less than 7, then another son s age is no less than = 3, which is impossible since the ages of his son must be less than the mother's. So the sum of the ages of Mrs. Sharma s two sons is 7 + = 8. nswer: 8

6 () The numbers,, 7, 8, 9, 0 are to be filled in the squares so that the sum of the numbers in the row is equal to the sum of the numbers in the column. How many different possible values of are there? Submitted by Malaysia Solution The sum of the two numbers in the row other than is equal to the sum of the three numbers in the column other than. We have = and = lready < Hence there are only two possible cases, with = 7 in the first and = 9 in the second. nswer: (3) farmer harvested 0 apples. He wishes to pack them as many boxes as possible, not necessarily packing all the apples, with each box a whole number of apples. The second box must be 0 more than the first, the third 0 more than the second and so on. What is the smallest number of apples left unpacked? Submitted by Jury Solution Start with a first box of apple. The numbers will be,, nd so onnd we are looking for a sum of numbers to be less than 0. The sum is = 90 and there are 0 boxes. This leaves 9 apples to be divided as much as possible into 0 equal parts which can be added to each box. The largest such number is the integer part of 9 0 which is. So if we start with a box with apple, we get the sum of the numbers = 000. So the smallest number of apples left is. nswer: apples () Three containers, Bnd contain a total of 8 apples. First, apples are taken from and are put into B. Second, 9 apples are taken from B and are put into. Now, each container has the same number of apples. What is the original number of apples in container? Submitted by Philippines Solution Since there are a total of 8 apples and each container has the same number of apples now, each container has 8 3 = apples. learly there were + = apples in container. nswer: apples

7 () The square PQRS has area of 00 cm. The points X and Y divide PQ into 3 parts. P X Z Y Q If the perimeter of triangle XYZ is S R of the perimeter of triangle SRZ, find the area, in cm, of XYZ. Submitted by Jury Solution The triangles XYZ and RSZ are similar. Since 00 = 0, the side length of square PQRS is 0 cm. s the perimeter of triangle XYZ is of the perimeter of triangle SRZ, XY = SR = cm and the height of XYZ is of the height of SRZ or of the side of the square, so the area of XYZ is 0= 0 cm. nswer: 0 cm () In the diagram, line B and line E meet in O and OF = 88. Given that OE is the angle bisector of OF and OB is the angle bisector of OF. Find the measure, in degrees, of O. Submitted by Philippines O 88 B Solution Since OB is the angle bisector of OF = 80 = 3. Since OE is the angle bisector of OF, we have OB = BOF = and OF Hence O = =. E, we have F 3 EOF = = 8. nswer:

8 (7) 3a88 3b = 7, find the value for a b. Submitted by Indonesia Solution When 3088 is divided by 7, the remainder is 0. Therefore0 is a multiple of 7nd we have 80 = 7. Hence a = 8. ividing 3888 by 7, the quotient is 39. Hence b = 9 and a b= 7. nswer: 7 (8) Find the area of the cross made of five identical squares in the figure below, given that the length of is cm. Submitted by Vietnam Solution onstruct additional lines and replace the yellow right triangles with the red right triangles. B Then the area of the cross equals the area of square B which is 7 = cm. Solution Let the side length of the squares is a, then the area of the cross is a. From Pythagoras Theorem, we have (3 a) + a = 0a = =, hence the area of the cross is a = = 7cm. nswer: 7 cm (9) Three positive two-digit integers and 3 are arranged in a table. For each row and column of the table, the product of the two numbers in this row or column is calculated. When all four such products are added together, the result is 0. What is the largest possible number in the square of the table? Submitted by Jury 3

9 Solution Let the three positive two-digit integers be a, b and c as shown. a b c 3 We have ab + 3c + ac + 3b = 0, so ( a+ 3)( b+ c) = 0 = 3 7. Since b+ c 0, the smallest possible value of b+ c is 7 3 =. Thus the largest possible value of a + 3 is 3= 9, i.e. the largest value of a is 9 3 = 33. n example as following: nswer: 33 (0) li has consecutive numbers while Ben has 7 consecutive numbers, none of the li s number is in the group of Ben s numbers. If the second number of li s number is and the sum of li s and Ben s numbers are 8. What is the largest number of Ben s number? Submitted by Indonesia Solution If the second number of li s number is, so the li s number are,,, 7, 8 and their sum is 30. From this we got the sum of Ben s number is 8 30 = 98. Since Ben has 7 consecutive numbers, hence its middle number is 98 7 = nd the largest number is + 3 = 7. nswer: 7 () Sam, Tom and Una are three chefs of a restaurant. One day, they cooked 30 plates of spaghetti and in this day, Sam cooked for hours, Tom cooked for 8 hours and Una cooked for hours. They also cook spaghetti at different speeds, with Sam cooking plates for every 3 plates Tom cooks and every plates Una cooks. How many plates of spaghetti did Sam cook this day? Submitted by Jury Solution If we multiply the rates of work by the days worked for each chef, we get the ratio of their effective total workload. Sam : Tom : Una = 30 : : 0. So for every plates, Sam cooks 30, Tom cooks and Una cooks =, so Sam cooks 30= 0 plates of spaghetti. nswer: 0 plates () How many equilateral triangles are in the figure below, in all possible sizes and directions? Submitted by Vietnam

10 Solution The number of triangles with area equal is:. The number of triangles with area equal is:. The number of triangles with area equal 9 is:. Hence the total number of triangles is ++=38. NS:38 equilateral triangles (3) With the appropriate order of the digits,, 3,,,, 7, 8nd 9, find the smallest 9-digit number that is divisible by 99. Submitted by Vietnam Solution Let the number formed be aaaaaaaaa , where ( a ) is a permutation of (,, 3,,,, 7, 8, 9). Because a + a + a3 + a + a + a + a7 + a8 + a9 = = is divisible by 9aaaaaaaa is divisible by 9. In order for aaaaaaaaa to be divisible by 99, the following difference must be divisible by : ( a + a3 + a + a7 + a9) ( a + a + a + a8) = ( a + a + a + a8) In order for the above difference to be divisible by + a + a + a8 must leave a remainder of when divided by. Since 0 a + a + a + a a + a + a8 equals 7 or 8. (a) onsider a + a + a + a 8 = 8, then a 8. In order for aaaaaaaaa to be as small as possible, let a = =, and the smallest number possible is (b) onsider a + a + a + a 8 = 7. In order for aaaaaaaaa to be as small as possible, let a =. Then ( a 8 ) is a permutation of (, 3,, 8) or (,,, ). Thus, the smallest number possible in this case is From (a) and (b), we have the smallest number that can be formed is Solution Let the number formed be aaaaaaaaa , where ( a ) is a permutation of (,, 3,,,, 7, 8, 9). Because a+ a + a3 + a + a + a + a7 + a8 + a9 = aaaaaaaa is divisible by 9. In order for aaaaaaaaa to be divisible by 99, the following difference must be divisible by : = ( a+ a3 + a + a7 + a9) ( a + a + a + a8) Observe that 0 since a + a + a 3 + a + a + a + a 7 + a 8 + a 9 =. Since we want to find the smallest 9-digit number, we can take a = first. Then ( ) ( ) = 7. So = and hence a + a + a + a 8 = = 7. Now we can try a =, a 3 = 3 and a =. Thus a + a8 = 7 =. Since all of,, 3 and can t be the

11 value of a, the possible values of a are and. Thus we can take a = and hence a 8 =. Now a + a7 + a9 = and the sum of unused digits is =, so we can take a = 7 7 = 8 and a 9 = 9 to get the smallest number that can be formed is nswer: 3789 () In the diagram shown below, B, GH and EFI are isosceles right triangles. Given G= GF = = E = cm and FE = cm. Find the ratio of area of shaded region to the area of triangle B. Submitted by Indonesia Solution Note that = 8cm and G = cm. Make another picture and combine it so we can make a G squarend point T is the intersection of diagonal of the F squares the figure shown. Now we can conclude: Two times of the area of shaded region equal to the T different of the area of middle square and smallest square, I E that is = 0cm. Hence the ratio of area of shaded region to the area of H 0 B triangle B is =. 8 8 nswer: () Whenever Sam reads a date like 0//0, he incorrectly interprets it as two divisions, with the second one evaluated before the first one: ( 0) = = 3 For some dates, like this one, he does not get an integer, while for others, like 0/8/0, he gets 0 (8 0) = 00 n integer. How many dates this year (day/month/year) give him a non-integer? Submitted by Jury Solution m 0d date d/m/0 will give the fraction d = which is a whole number 0 m whenever 0d is a multiple of m. Now, 0 = 3 7, so all days in months,, 3,,, 7, 8, 9 and give integersnd in months 0, the number d needs to be a multiple of. In all other months, d needs to be a multiple of m. In summary: month m integer days d number of integer days, 0,, 0,, 30 0, 0,, 0,, 30, Totally (3 ) + (3 ) + (30 ) = 78 dates give him a non-integer. nswer: 78 dates

y = y = f ( x ) intersects the line ! 4 + 2" 3 is equal to twice x, and is also equal to half of y. Give the value of x + 2y & 5' !

y = y = f ( x ) intersects the line ! 4 + 2 3 is equal to twice x, and is also equal to half of y. Give the value of x + 2y & 5' ! FMT 009 Page of NO CLCULTORS are allowed on this test The abbreviation "NOT" means "None of These nswers" Diagrams are not necessarily drawn to scale 4 If f ( x) = x! x, then the graph of y = f ( x ) intersects