Difference between revisions of "2008 iTest Problems"
m (Put the problem in) |
m (Made the links to the problems prettier.) |
||
| Line 4: | Line 4: | ||
When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego | When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego | ||
| − | Zoo. Single day passes cost | + | Zoo. Single day passes cost 33 dollars for adults (Jerry and Hannah), 22 dollars for children (Michael is |
still young enough to get the children’s rate), and family memberships (which allow the whole | still young enough to get the children’s rate), and family memberships (which allow the whole | ||
| − | family in at once) cost | + | family in at once) cost 120 dollars. How many dollars did the family save by buying a family pass |
over buying single day passes for every member of the family? | over buying single day passes for every member of the family? | ||
| − | [[2008 iTest Problems/Problem 1]] | + | [[2008 iTest Problems/Problem 1|Solution]] |
==Problem 2== | ==Problem 2== | ||
| − | [[2008 iTest Problems/Problem 2]] | + | [[2008 iTest Problems/Problem 2|Solution]] |
==Problem 3== | ==Problem 3== | ||
| − | [[2008 iTest Problems/Problem 3]] | + | [[2008 iTest Problems/Problem 3|Solution]] |
==Problem 4== | ==Problem 4== | ||
The difference between two prime numbers is 11. Find their sum. | The difference between two prime numbers is 11. Find their sum. | ||
| − | [[2008 iTest Problems/Problem 4]] | + | [[2008 iTest Problems/Problem 4|Solution]] |
==Problem 5== | ==Problem 5== | ||
| − | [[2008 iTest Problems/Problem 5]] | + | [[2008 iTest Problems/Problem 5|Solution]] |
==Problem 6== | ==Problem 6== | ||
Let <math>L</math> be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than <math>L</math>. | Let <math>L</math> be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than <math>L</math>. | ||
| − | [[2008 iTest Problems/Problem 6]] | + | [[2008 iTest Problems/Problem 6|Solution]] |
==Problem 7== | ==Problem 7== | ||
Find the number of integers <math>n</math> for which <math>n^2 + 10n < 2008</math>. | Find the number of integers <math>n</math> for which <math>n^2 + 10n < 2008</math>. | ||
| − | [[2008 iTest Problems/Problem 7]] | + | [[2008 iTest Problems/Problem 7|Solution]] |
==Problem 8== | ==Problem 8== | ||
| Line 45: | Line 45: | ||
find <math>x+y+z</math>. | find <math>x+y+z</math>. | ||
| − | [[2008 iTest Problems/Problem 8]] | + | [[2008 iTest Problems/Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
| Line 52: | Line 52: | ||
What is the units digit of <math>2008^{2008}</math>? | What is the units digit of <math>2008^{2008}</math>? | ||
| − | [[2008 iTest Problems/Problem 9]] | + | [[2008 iTest Problems/Problem 9|Solution]] |
==Problem 10== | ==Problem 10== | ||
| − | [[2008 iTest Problems/Problem 10]] | + | [[2008 iTest Problems/Problem 10|Solution]] |
==Problem 11== | ==Problem 11== | ||
| − | [[2008 iTest Problems/Problem 11]] | + | [[2008 iTest Problems/Problem 11|Solution]] |
==Problem 12== | ==Problem 12== | ||
| − | [[2008 iTest Problems/Problem 12]] | + | [[2008 iTest Problems/Problem 12|Solution]] |
==Problem 13== | ==Problem 13== | ||
| − | [[2008 iTest Problems/Problem 13]] | + | [[2008 iTest Problems/Problem 13|Solution]] |
==Problem 14== | ==Problem 14== | ||
The sum of the two perfect cubes that are closest to 500 is 343+512 = 855. Find the sum of the two perfect cubes that are closest to 2008. | The sum of the two perfect cubes that are closest to 500 is 343+512 = 855. Find the sum of the two perfect cubes that are closest to 2008. | ||
| − | [[2008 iTest Problems/Problem 14]] | + | [[2008 iTest Problems/Problem 14|Solution]] |
==Problem 15== | ==Problem 15== | ||
How many four-digit multiples of 8 are greater than 2008? | How many four-digit multiples of 8 are greater than 2008? | ||
| − | [[2008 iTest Problems/Problem 15]] | + | [[2008 iTest Problems/Problem 15|Solution]] |
==Problem 16== | ==Problem 16== | ||
| − | [[2008 iTest Problems/Problem 16]] | + | [[2008 iTest Problems/Problem 16|Solution]] |
==Problem 17== | ==Problem 17== | ||
| − | [[2008 iTest Problems/Problem 17]] | + | [[2008 iTest Problems/Problem 17|Solution]] |
==Problem 18== | ==Problem 18== | ||
Find the number of lattice points that the line <math>19x+20y = 1909</math> passes through in Quadrant I. | Find the number of lattice points that the line <math>19x+20y = 1909</math> passes through in Quadrant I. | ||
| − | [[2008 iTest Problems/Problem 18]] | + | [[2008 iTest Problems/Problem 18|Solution]] |
==Problem 19== | ==Problem 19== | ||
Let <math>A</math> be the set of positive integers that are the product of two consecutive integers. Let <math>B</math> be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of <math>A \cap B</math>. | Let <math>A</math> be the set of positive integers that are the product of two consecutive integers. Let <math>B</math> be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of <math>A \cap B</math>. | ||
| − | [[2008 iTest Problems/Problem 19]] | + | [[2008 iTest Problems/Problem 19|Solution]] |
==Problem 20== | ==Problem 20== | ||
| − | [[2008 iTest Problems/Problem 20]] | + | [[2008 iTest Problems/Problem 20|Solution]] |
==Problem 21== | ==Problem 21== | ||
| − | [[2008 iTest Problems/Problem 21]] | + | [[2008 iTest Problems/Problem 21|Solution]] |
==Problem 22== | ==Problem 22== | ||
| − | [[2008 iTest Problems/Problem 22]] | + | [[2008 iTest Problems/Problem 22|Solution]] |
==Problem 23== | ==Problem 23== | ||
Find the number of positive integers <math>n</math> that are solutions to the simultaneous system of inequalities | Find the number of positive integers <math>n</math> that are solutions to the simultaneous system of inequalities | ||
| Line 107: | Line 107: | ||
<center><math>7n + 17 > 2008</math>.</center> | <center><math>7n + 17 > 2008</math>.</center> | ||
| − | [[2008 iTest Problems/Problem 23]] | + | [[2008 iTest Problems/Problem 23|Solution]] |
==Problem 24== | ==Problem 24== | ||
| − | [[2008 iTest Problems/Problem 24]] | + | [[2008 iTest Problems/Problem 24|Solution]] |
==Problem 25== | ==Problem 25== | ||
A cube has edges of length 120 cm. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of even smaller cubes, all of equal size. If the edge length of one of those even smaller cubes is <math>n</math> cm, where <math>n</math> is an integer, find the number of possible values of <math>n</math>. | A cube has edges of length 120 cm. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of even smaller cubes, all of equal size. If the edge length of one of those even smaller cubes is <math>n</math> cm, where <math>n</math> is an integer, find the number of possible values of <math>n</math>. | ||
| − | [[2008 iTest Problems/Problem 25]] | + | [[2008 iTest Problems/Problem 25|Solution]] |
==Problem 26== | ==Problem 26== | ||
| − | [[2008 iTest Problems/Problem 26]] | + | [[2008 iTest Problems/Problem 26|Solution]] |
==Problem 27== | ==Problem 27== | ||
Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children’s Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children’s Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes? | Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children’s Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children’s Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes? | ||
| − | [[2008 iTest Problems/Problem 27]] | + | [[2008 iTest Problems/Problem 27|Solution]] |
==Problem 28== | ==Problem 28== | ||
| − | [[2008 iTest Problems/Problem 28]] | + | [[2008 iTest Problems/Problem 28|Solution]] |
==Problem 29== | ==Problem 29== | ||
| − | [[2008 iTest Problems/Problem 29]] | + | [[2008 iTest Problems/Problem 29|Solution]] |
==Problem 30== | ==Problem 30== | ||
| − | [[2008 iTest Problems/Problem 30]] | + | [[2008 iTest Problems/Problem 30|Solution]] |
==Problem 31== | ==Problem 31== | ||
| − | [[2008 iTest Problems/Problem 31]] | + | [[2008 iTest Problems/Problem 31|Solution]] |
==Problem 32== | ==Problem 32== | ||
| − | [[2008 iTest Problems/Problem 32]] | + | [[2008 iTest Problems/Problem 32|Solution]] |
==Problem 33== | ==Problem 33== | ||
| − | [[2008 iTest Problems/Problem 33]] | + | [[2008 iTest Problems/Problem 33|Solution]] |
==Problem 34== | ==Problem 34== | ||
| − | [[2008 iTest Problems/Problem 34]] | + | [[2008 iTest Problems/Problem 34|Solution]] |
==Problem 35== | ==Problem 35== | ||
| − | [[2008 iTest Problems/Problem 35]] | + | [[2008 iTest Problems/Problem 35|Solution]] |
==Problem 36== | ==Problem 36== | ||
| − | [[2008 iTest Problems/Problem 36]] | + | [[2008 iTest Problems/Problem 36|Solution]] |
==Problem 37== | ==Problem 37== | ||
| − | [[2008 iTest Problems/Problem 37]] | + | [[2008 iTest Problems/Problem 37|Solution]] |
==Problem 38== | ==Problem 38== | ||
| − | [[2008 iTest Problems/Problem 38]] | + | [[2008 iTest Problems/Problem 38|Solution]] |
==Problem 39== | ==Problem 39== | ||
| − | [[2008 iTest Problems/Problem 39]] | + | [[2008 iTest Problems/Problem 39|Solution]] |
==Problem 40== | ==Problem 40== | ||
| − | [[2008 iTest Problems/Problem 40]] | + | [[2008 iTest Problems/Problem 40|Solution]] |
==Problem 41== | ==Problem 41== | ||
| − | [[2008 iTest Problems/Problem 41]] | + | [[2008 iTest Problems/Problem 41|Solution]] |
==Problem 42== | ==Problem 42== | ||
| − | [[2008 iTest Problems/Problem 42]] | + | [[2008 iTest Problems/Problem 42|Solution]] |
==Problem 43== | ==Problem 43== | ||
| − | [[2008 iTest Problems/Problem 43]] | + | [[2008 iTest Problems/Problem 43|Solution]] |
==Problem 44== | ==Problem 44== | ||
| − | [[2008 iTest Problems/Problem 44]] | + | [[2008 iTest Problems/Problem 44|Solution]] |
==Problem 45== | ==Problem 45== | ||
| − | [[2008 iTest Problems/Problem 45]] | + | [[2008 iTest Problems/Problem 45|Solution]] |
==Problem 46== | ==Problem 46== | ||
| − | [[2008 iTest Problems/Problem 46]] | + | [[2008 iTest Problems/Problem 46|Solution]] |
==Problem 47== | ==Problem 47== | ||
Find <math>a + b + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the hundreds, tens, and units digits of the six-digit integer | Find <math>a + b + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the hundreds, tens, and units digits of the six-digit integer | ||
<math>123abc</math>, which is a multiple of 990. | <math>123abc</math>, which is a multiple of 990. | ||
| − | [[2008 iTest Problems/Problem 47]] | + | [[2008 iTest Problems/Problem 47|Solution]] |
==Problem 48== | ==Problem 48== | ||
| − | [[2008 iTest Problems/Problem 48]] | + | [[2008 iTest Problems/Problem 48|Solution]] |
==Problem 49== | ==Problem 49== | ||
Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy’s friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarns would intersect if possible. If all possible positions of the students around the table are equally likely, let <math>m/n</math> be the probability that the yarns intersect, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy’s friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarns would intersect if possible. If all possible positions of the students around the table are equally likely, let <math>m/n</math> be the probability that the yarns intersect, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | ||
| − | [[2008 iTest Problems/Problem 49]] | + | [[2008 iTest Problems/Problem 49|Solution]] |
==Problem 50== | ==Problem 50== | ||
| − | [[2008 iTest Problems/Problem 50]] | + | [[2008 iTest Problems/Problem 50|Solution]] |
==Problem 51== | ==Problem 51== | ||
| − | [[2008 iTest Problems/Problem 51]] | + | [[2008 iTest Problems/Problem 51|Solution]] |
==Problem 52== | ==Problem 52== | ||
A triangle has sides of length 48, 55, and 73. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If <math>c</math> and <math>d</math> are relatively prime positive integers such that <math>c/d</math> is the length of a side of the square, find the value of <math>c + d</math>. | A triangle has sides of length 48, 55, and 73. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If <math>c</math> and <math>d</math> are relatively prime positive integers such that <math>c/d</math> is the length of a side of the square, find the value of <math>c + d</math>. | ||
| − | [[2008 iTest Problems/Problem 52]] | + | [[2008 iTest Problems/Problem 52|Solution]] |
==Problem 53== | ==Problem 53== | ||
| − | [[2008 iTest Problems/Problem 53]] | + | [[2008 iTest Problems/Problem 53|Solution]] |
==Problem 54== | ==Problem 54== | ||
One of Michael’s responsibilities in organizing the family vacation is to call around and find room rates for hotels along the route the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael’s eye. Michael notices that the first four digits of 987-1234 descend (9-8-7-1) and that the last four ascend in order (1-2-3-4). This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. “Wow,” he thinks, “that’s good marketing strategy.” Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many 7-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order? | One of Michael’s responsibilities in organizing the family vacation is to call around and find room rates for hotels along the route the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael’s eye. Michael notices that the first four digits of 987-1234 descend (9-8-7-1) and that the last four ascend in order (1-2-3-4). This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. “Wow,” he thinks, “that’s good marketing strategy.” Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many 7-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order? | ||
| − | [[2008 iTest Problems/Problem 54]] | + | [[2008 iTest Problems/Problem 54|Solution]] |
==Problem 55== | ==Problem 55== | ||
| − | [[2008 iTest Problems/Problem 55]] | + | [[2008 iTest Problems/Problem 55|Solution]] |
==Problem 56== | ==Problem 56== | ||
| − | [[2008 iTest Problems/Problem 56]] | + | [[2008 iTest Problems/Problem 56|Solution]] |
==Problem 57== | ==Problem 57== | ||
| − | [[2008 iTest Problems/Problem 57]] | + | [[2008 iTest Problems/Problem 57|Solution]] |
==Problem 58== | ==Problem 58== | ||
| − | [[2008 iTest Problems/Problem 58]] | + | [[2008 iTest Problems/Problem 58|Solution]] |
==Problem 59== | ==Problem 59== | ||
| − | [[2008 iTest Problems/Problem 59]] | + | [[2008 iTest Problems/Problem 59|Solution]] |
==Problem 60== | ==Problem 60== | ||
| − | [[2008 iTest Problems/Problem 60]] | + | [[2008 iTest Problems/Problem 60|Solution]] |
==Problem 61== | ==Problem 61== | ||
| − | [[2008 iTest Problems/Problem 61]] | + | [[2008 iTest Problems/Problem 61|Solution]] |
==Problem 62== | ==Problem 62== | ||
Find the number of values of <math>x</math> such that the number of square units in the area of the isosceles triangle with sides <math>x</math>, 65, and 65 is a positive integer. | Find the number of values of <math>x</math> such that the number of square units in the area of the isosceles triangle with sides <math>x</math>, 65, and 65 is a positive integer. | ||
| − | [[2008 iTest Problems/Problem 62]] | + | [[2008 iTest Problems/Problem 62|Solution]] |
==Problem 63== | ==Problem 63== | ||
Looking for a little time alone, Michael takes a jog along the beach. The crashing of waves reminds him of the hydroelectric plant his father helped maintain before the family moved to Jupiter Falls. Michael was in elementary school at the time. He thinks for a moment about how much his life has changed in just a few years. Michael looks forward to finishing high school, but isn’t sure what he wants to do next. He thinks about whether he wants to study engineering in college, like both his parents did, or pursue an education in business. His aunt Jessica studied business and appraises budding technology companies for a venture capital firm. Other possibilities also tug a little at Michael for different reasons. Michael stops and watches a group of girls who seem to be around Tony’s age play a game around an ellipse drawn in the sand. There are two softball bats stuck in the sand. Michael recognizes these as the foci of the ellipse. The bats are 24 feet apart. Two children stand on opposite ends of the ellipse where the ellipse intersects the line on which the bats lie. These two children are 40 feet apart. Five other children stand on different points on the ellipse. One of them blows a whistle and all seven children run screaming toward one bat or the other. Each child runs as fast as she can, touching one bat, then the next, and finally returning to the spot on which she started. When the first girl gets back to her place, she declares, “I win this time! I win!” Another of the girls pats her on the back, and the winning girl speaks again, “This time I found the place where I’d have to run the shortest distance.” Michael thinks for a moment, draws some notes in the sand, then compute the shortest possible distance one of the girls could run from her starting point on the ellipse, to one of the bats, to the other bat, then back to her starting point. He smiles for a moment, then keeps jogging. If Michael’s work is correct, what distance did he compute as the shortest possible distance one of the girls could run during the game? | Looking for a little time alone, Michael takes a jog along the beach. The crashing of waves reminds him of the hydroelectric plant his father helped maintain before the family moved to Jupiter Falls. Michael was in elementary school at the time. He thinks for a moment about how much his life has changed in just a few years. Michael looks forward to finishing high school, but isn’t sure what he wants to do next. He thinks about whether he wants to study engineering in college, like both his parents did, or pursue an education in business. His aunt Jessica studied business and appraises budding technology companies for a venture capital firm. Other possibilities also tug a little at Michael for different reasons. Michael stops and watches a group of girls who seem to be around Tony’s age play a game around an ellipse drawn in the sand. There are two softball bats stuck in the sand. Michael recognizes these as the foci of the ellipse. The bats are 24 feet apart. Two children stand on opposite ends of the ellipse where the ellipse intersects the line on which the bats lie. These two children are 40 feet apart. Five other children stand on different points on the ellipse. One of them blows a whistle and all seven children run screaming toward one bat or the other. Each child runs as fast as she can, touching one bat, then the next, and finally returning to the spot on which she started. When the first girl gets back to her place, she declares, “I win this time! I win!” Another of the girls pats her on the back, and the winning girl speaks again, “This time I found the place where I’d have to run the shortest distance.” Michael thinks for a moment, draws some notes in the sand, then compute the shortest possible distance one of the girls could run from her starting point on the ellipse, to one of the bats, to the other bat, then back to her starting point. He smiles for a moment, then keeps jogging. If Michael’s work is correct, what distance did he compute as the shortest possible distance one of the girls could run during the game? | ||
| − | [[2008 iTest Problems/Problem 63]] | + | [[2008 iTest Problems/Problem 63|Solution]] |
==Problem 64== | ==Problem 64== | ||
| − | [[2008 iTest Problems/Problem 64]] | + | [[2008 iTest Problems/Problem 64|Solution]] |
==Problem 65== | ==Problem 65== | ||
| − | [[2008 iTest Problems/Problem 65]] | + | [[2008 iTest Problems/Problem 65|Solution]] |
==Problem 66== | ==Problem 66== | ||
| − | [[2008 iTest Problems/Problem 66]] | + | [[2008 iTest Problems/Problem 66|Solution]] |
==Problem 67== | ==Problem 67== | ||
At lunch, the seven members of the Kubik family sits down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if lexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.) | At lunch, the seven members of the Kubik family sits down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if lexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.) | ||
| − | [[2008 iTest Problems/Problem 67]] | + | [[2008 iTest Problems/Problem 67|Solution]] |
==Problem 68== | ==Problem 68== | ||
| − | [[2008 iTest Problems/Problem 68]] | + | [[2008 iTest Problems/Problem 68|Solution]] |
==Problem 69== | ==Problem 69== | ||
| − | [[2008 iTest Problems/Problem 69]] | + | [[2008 iTest Problems/Problem 69|Solution]] |
==Problem 70== | ==Problem 70== | ||
| − | [[2008 iTest Problems/Problem 70]] | + | [[2008 iTest Problems/Problem 70|Solution]] |
==Problem 71== | ==Problem 71== | ||
| − | [[2008 iTest Problems/Problem 71]] | + | [[2008 iTest Problems/Problem 71|Solution]] |
==Problem 72== | ==Problem 72== | ||
| − | [[2008 iTest Problems/Problem 72]] | + | [[2008 iTest Problems/Problem 72|Solution]] |
==Problem 73== | ==Problem 73== | ||
| − | [[2008 iTest Problems/Problem 73]] | + | [[2008 iTest Problems/Problem 73|Solution]] |
==Problem 74== | ==Problem 74== | ||
| − | [[2008 iTest Problems/Problem 74]] | + | [[2008 iTest Problems/Problem 74|Solution]] |
==Problem 75== | ==Problem 75== | ||
| − | [[2008 iTest Problems/Problem 75]] | + | [[2008 iTest Problems/Problem 75|Solution]] |
==Problem 76== | ==Problem 76== | ||
| − | [[2008 iTest Problems/Problem 76]] | + | [[2008 iTest Problems/Problem 76|Solution]] |
==Problem 77== | ==Problem 77== | ||
| − | [[2008 iTest Problems/Problem 77]] | + | [[2008 iTest Problems/Problem 77|Solution]] |
==Problem 78== | ==Problem 78== | ||
| − | [[2008 iTest Problems/Problem 78]] | + | [[2008 iTest Problems/Problem 78|Solution]] |
==Problem 79== | ==Problem 79== | ||
| − | [[2008 iTest Problems/Problem 79]] | + | [[2008 iTest Problems/Problem 79|Solution]] |
==Problem 80== | ==Problem 80== | ||
Let | Let | ||
| Line 294: | Line 294: | ||
the remainder when <math>|r(2008)|</math> is divided by <math>1000</math>. | the remainder when <math>|r(2008)|</math> is divided by <math>1000</math>. | ||
| − | [[2008 iTest Problems/Problem 80]] | + | [[2008 iTest Problems/Problem 80|Solution]] |
==Problem 81== | ==Problem 81== | ||
| − | [[2008 iTest Problems/Problem 81]] | + | [[2008 iTest Problems/Problem 81|Solution]] |
==Problem 82== | ==Problem 82== | ||
Tony’s favorite “sport” is a spectator event known as the Super Mega Ultra Galactic Thumbwrestling Championship (SMUG TWC). During the 2008 SMUG TWC, 2008 professional thumbwrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of Thumbzilla. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOXTM television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let m be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute . | Tony’s favorite “sport” is a spectator event known as the Super Mega Ultra Galactic Thumbwrestling Championship (SMUG TWC). During the 2008 SMUG TWC, 2008 professional thumbwrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of Thumbzilla. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOXTM television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let m be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute . | ||
| − | [[2008 iTest Problems/Problem 82]] | + | [[2008 iTest Problems/Problem 82|Solution]] |
==Problem 83== | ==Problem 83== | ||
| − | [[2008 iTest Problems/Problem 83]] | + | [[2008 iTest Problems/Problem 83|Solution]] |
==Problem 84== | ==Problem 84== | ||
| − | [[2008 iTest Problems/Problem 84]] | + | [[2008 iTest Problems/Problem 84|Solution]] |
==Problem 85== | ==Problem 85== | ||
| − | [[2008 iTest Problems/Problem 85]] | + | [[2008 iTest Problems/Problem 85|Solution]] |
==Problem 86== | ==Problem 86== | ||
| − | [[2008 iTest Problems/Problem 86]] | + | [[2008 iTest Problems/Problem 86|Solution]] |
==Problem 87== | ==Problem 87== | ||
| − | [[2008 iTest Problems/Problem 87]] | + | [[2008 iTest Problems/Problem 87|Solution]] |
==Problem 88== | ==Problem 88== | ||
| − | [[2008 iTest Problems/Problem 88]] | + | [[2008 iTest Problems/Problem 88|Solution]] |
==Problem 89== | ==Problem 89== | ||
| − | [[2008 iTest Problems/Problem 89]] | + | [[2008 iTest Problems/Problem 89|Solution]] |
==Problem 90== | ==Problem 90== | ||
| − | [[2008 iTest Problems/Problem 90]] | + | [[2008 iTest Problems/Problem 90|Solution]] |
==Problem 91== | ==Problem 91== | ||
| − | [[2008 iTest Problems/Problem 91]] | + | [[2008 iTest Problems/Problem 91|Solution]] |
==Problem 92== | ==Problem 92== | ||
| − | [[2008 iTest Problems/Problem 92]] | + | [[2008 iTest Problems/Problem 92|Solution]] |
==Problem 93== | ==Problem 93== | ||
| − | [[2008 iTest Problems/Problem 93]] | + | [[2008 iTest Problems/Problem 93|Solution]] |
==Problem 94== | ==Problem 94== | ||
| − | [[2008 iTest Problems/Problem 94]] | + | [[2008 iTest Problems/Problem 94|Solution]] |
==Problem 95== | ==Problem 95== | ||
| − | [[2008 iTest Problems/Problem 95]] | + | [[2008 iTest Problems/Problem 95|Solution]] |
==Problem 96== | ==Problem 96== | ||
| − | [[2008 iTest Problems/Problem 96]] | + | [[2008 iTest Problems/Problem 96|Solution]] |
==Problem 97== | ==Problem 97== | ||
| − | [[2008 iTest Problems/Problem 97]] | + | [[2008 iTest Problems/Problem 97|Solution]] |
==Problem 98== | ==Problem 98== | ||
| − | [[2008 iTest Problems/Problem 98]] | + | [[2008 iTest Problems/Problem 98|Solution]] |
==Problem 99== | ==Problem 99== | ||
Given a convex, <math>n</math>-sided polygon <math>P</math>, form a <math>2n</math>-sided polygon <math>\text{clip}(P)</math> by cutting off each corner of <math>P</math> at the edges’ trisection points. In other words, <math>\text{clip}(P)</math> is the polygon whose vertices are the <math>2n</math> edge trisection points of <math>P</math>, connected in order around the boundary of <math>P</math>. Let <math>P_1</math> be an isosceles trapezoid with side lengths <math>13, 13, 13</math>, and <math>3</math>, and for each <math>i > 2</math>, let <math>P_i = \text{clip}(P_{i-1})</math>. This iterative clipping process approaches a limiting shape <math>P_1 = \lim_{i \rightarrow \infty} P_i</math>. If the difference of the areas of <math>P_{10}</math> and <math>P_{1}</math> is written as a fraction <math>\frac{x}{y}</math> in lowest terms, calculate the number of positive integer factors of <math>x \cdot y</math>. | Given a convex, <math>n</math>-sided polygon <math>P</math>, form a <math>2n</math>-sided polygon <math>\text{clip}(P)</math> by cutting off each corner of <math>P</math> at the edges’ trisection points. In other words, <math>\text{clip}(P)</math> is the polygon whose vertices are the <math>2n</math> edge trisection points of <math>P</math>, connected in order around the boundary of <math>P</math>. Let <math>P_1</math> be an isosceles trapezoid with side lengths <math>13, 13, 13</math>, and <math>3</math>, and for each <math>i > 2</math>, let <math>P_i = \text{clip}(P_{i-1})</math>. This iterative clipping process approaches a limiting shape <math>P_1 = \lim_{i \rightarrow \infty} P_i</math>. If the difference of the areas of <math>P_{10}</math> and <math>P_{1}</math> is written as a fraction <math>\frac{x}{y}</math> in lowest terms, calculate the number of positive integer factors of <math>x \cdot y</math>. | ||
| − | [[2008 iTest Problems/Problem 99]] | + | [[2008 iTest Problems/Problem 99|Solution]] |
==Problem 100== | ==Problem 100== | ||
| − | [[2008 iTest Problems/Problem 100]] | + | [[2008 iTest Problems/Problem 100|Solution]] |
Revision as of 14:00, 22 March 2012
This is an empty template page which needs to be filled. You can help us out by finding the needed content and editing it in. Thanks.
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 Problem 51
- 52 Problem 52
- 53 Problem 53
- 54 Problem 54
- 55 Problem 55
- 56 Problem 56
- 57 Problem 57
- 58 Problem 58
- 59 Problem 59
- 60 Problem 60
- 61 Problem 61
- 62 Problem 62
- 63 Problem 63
- 64 Problem 64
- 65 Problem 65
- 66 Problem 66
- 67 Problem 67
- 68 Problem 68
- 69 Problem 69
- 70 Problem 70
- 71 Problem 71
- 72 Problem 72
- 73 Problem 73
- 74 Problem 74
- 75 Problem 75
- 76 Problem 76
- 77 Problem 77
- 78 Problem 78
- 79 Problem 79
- 80 Problem 80
- 81 Problem 81
- 82 Problem 82
- 83 Problem 83
- 84 Problem 84
- 85 Problem 85
- 86 Problem 86
- 87 Problem 87
- 88 Problem 88
- 89 Problem 89
- 90 Problem 90
- 91 Problem 91
- 92 Problem 92
- 93 Problem 93
- 94 Problem 94
- 95 Problem 95
- 96 Problem 96
- 97 Problem 97
- 98 Problem 98
- 99 Problem 99
- 100 Problem 100
Problem 1
(story eliminated)
When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. Single day passes cost 33 dollars for adults (Jerry and Hannah), 22 dollars for children (Michael is still young enough to get the children’s rate), and family memberships (which allow the whole family in at once) cost 120 dollars. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?
Problem 2
Problem 3
Problem 4
The difference between two prime numbers is 11. Find their sum.
Problem 5
Problem 6
Let 
be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than .
Problem 7
Find the number of integers 
for which 
.
Problem 8
(story eliminated)
Given the system of equations
,
,
,find 
.
Problem 9
(story eliminated)
What is the units digit of 
?
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
The sum of the two perfect cubes that are closest to 500 is 343+512 = 855. Find the sum of the two perfect cubes that are closest to 2008.
Problem 15
How many four-digit multiples of 8 are greater than 2008?
Problem 16
Problem 17
Problem 18
Find the number of lattice points that the line 
passes through in Quadrant I.
Problem 19
Let 
be the set of positive integers that are the product of two consecutive integers. Let 
be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of 
.
Problem 20
Problem 21
Problem 22
Problem 23
Find the number of positive integers that are solutions to the simultaneous system of inequalities
,
.Problem 24
Problem 25
A cube has edges of length 120 cm. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of even smaller cubes, all of equal size. If the edge length of one of those even smaller cubes is cm, where is an integer, find the number of possible values of .
Problem 26
Problem 27
Hannah Kubik leads a local volunteer group of thirteen adults that takes turns holding classes for patients at the Children’s Hospital. At the end of August, Hannah took a tour of the hospital and talked with some members of the staff. Dr. Yang told Hannah that it looked like there would be more girls than boys in the hospital during September. The next day Hannah brought the volunteers together and it was decided that three women and two men would volunteer to run the September classes at the Children’s Hospital. If there are exactly six women in the volunteer group, how many combinations of three women and two men could Hannah choose from the volunteer group to run the classes?
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Find 
, where 
, 
, and 
are the hundreds, tens, and units digits of the six-digit integer
, which is a multiple of 990.
Problem 48
Problem 49
Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy’s friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarns would intersect if possible. If all possible positions of the students around the table are equally likely, let 
be the probability that the yarns intersect, where 
and are relatively prime positive integers. Compute 
.
Problem 50
Problem 51
Problem 52
A triangle has sides of length 48, 55, and 73. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If and 
are relatively prime positive integers such that 
is the length of a side of the square, find the value of 
.
Problem 53
Problem 54
One of Michael’s responsibilities in organizing the family vacation is to call around and find room rates for hotels along the route the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael’s eye. Michael notices that the first four digits of 987-1234 descend (9-8-7-1) and that the last four ascend in order (1-2-3-4). This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. “Wow,” he thinks, “that’s good marketing strategy.” Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many 7-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order?
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Find the number of values of 
such that the number of square units in the area of the isosceles triangle with sides , 65, and 65 is a positive integer.
Problem 63
Looking for a little time alone, Michael takes a jog along the beach. The crashing of waves reminds him of the hydroelectric plant his father helped maintain before the family moved to Jupiter Falls. Michael was in elementary school at the time. He thinks for a moment about how much his life has changed in just a few years. Michael looks forward to finishing high school, but isn’t sure what he wants to do next. He thinks about whether he wants to study engineering in college, like both his parents did, or pursue an education in business. His aunt Jessica studied business and appraises budding technology companies for a venture capital firm. Other possibilities also tug a little at Michael for different reasons. Michael stops and watches a group of girls who seem to be around Tony’s age play a game around an ellipse drawn in the sand. There are two softball bats stuck in the sand. Michael recognizes these as the foci of the ellipse. The bats are 24 feet apart. Two children stand on opposite ends of the ellipse where the ellipse intersects the line on which the bats lie. These two children are 40 feet apart. Five other children stand on different points on the ellipse. One of them blows a whistle and all seven children run screaming toward one bat or the other. Each child runs as fast as she can, touching one bat, then the next, and finally returning to the spot on which she started. When the first girl gets back to her place, she declares, “I win this time! I win!” Another of the girls pats her on the back, and the winning girl speaks again, “This time I found the place where I’d have to run the shortest distance.” Michael thinks for a moment, draws some notes in the sand, then compute the shortest possible distance one of the girls could run from her starting point on the ellipse, to one of the bats, to the other bat, then back to her starting point. He smiles for a moment, then keeps jogging. If Michael’s work is correct, what distance did he compute as the shortest possible distance one of the girls could run during the game?
Problem 64
Problem 65
Problem 66
Problem 67
At lunch, the seven members of the Kubik family sits down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if lexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.)
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Let
and let 
be the polynomial remainder when 
is divided by 
. Find
the remainder when 
is divided by 
.
Problem 81
Problem 82
Tony’s favorite “sport” is a spectator event known as the Super Mega Ultra Galactic Thumbwrestling Championship (SMUG TWC). During the 2008 SMUG TWC, 2008 professional thumbwrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of Thumbzilla. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOXTM television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let m be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute .
Problem 83
Problem 84
Problem 85
Problem 86
Problem 87
Problem 88
Problem 89
Problem 90
Problem 91
Problem 92
Problem 93
Problem 94
Problem 95
Problem 96
Problem 97
Problem 98
Problem 99
Given a convex, -sided polygon 
, form a 
-sided polygon 
by cutting off each corner of at the edges’ trisection points. In other words, is the polygon whose vertices are the edge trisection points of , connected in order around the boundary of . Let 
be an isosceles trapezoid with side lengths 
, and 
, and for each 
, let 
. This iterative clipping process approaches a limiting shape 
. If the difference of the areas of 
and 
is written as a fraction 
in lowest terms, calculate the number of positive integer factors of 
.
