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Difference between revisions of "2008 iTest Problems"

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{{empty}}
 
 
==Problem 1==
 
==Problem 1==
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(story eliminated)
 
  
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When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego
+
Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are <math>12</math> years old.
−
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
+
When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.  
−
family in at once) cost 120 dollars. How many dollars did the family save by buying a family pass
+
Single day passes cost <math>\textdollar{33} for adults (Jerry and Hannah), </math>\textdollar{22} for children
−
over buying single day passes for every member of the family?
+
(Michael is still young enough to get the children's rate), and family memberships  
 +
(which allow the whole family in at once) cost <math>\textdollar{120}. How many dollars did the family save by buying a family pass  
 +
over buying single day passes for every member of the family?  
  
 
[[2008 iTest Problems/Problem 1|Solution]]
 
[[2008 iTest Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
 +
 +
One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly </math>12<math> feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around </math>15<math> feet. He then measures </math>2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?
 +
 +
<asy>
 +
size(400);
 +
defaultpen(linewidth(0.8));
 +
draw((0,-3)--(0,3));
 +
int d=8;
 +
for(int i=0;i<=5;i=i+1)
 +
{
 +
draw(circle(7/2+d*i,3/2));
 +
}
 +
draw((5*d+7,-3)--(5*d+7,3));
 +
draw((0,0)--(2,0),Arrows(size=7));
 +
draw((5,0)--(2+d,0),Arrows(size=7));
 +
draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7));
 +
label("$2$",(1,0),S);
 +
label("$12$",((7+d)/2,0),S);
 +
label("$15$",((7+3*d)/2,0),S);</asy>
  
 
[[2008 iTest Problems/Problem 2|Solution]]
 
[[2008 iTest Problems/Problem 2|Solution]]
 
==Problem 3==
 
==Problem 3==
 +
 +
Michael plays catcher for his school's baseball team. He has always been a great player behind the plate, but this year as a junior, Michael's offense is really improving.
 +
His batting average is <math>.417</math> after six games, and the team is <math>6-0</math> (six wins and no losses). They are off to their best start in years.
 +
 +
On the way home from their sixth game, Michael notes to his father that the attendance seems to be increasing due to the team's great start, "There were <math>181</math>
 +
people at the first game, then <math>197</math> at the second, <math>203</math> the third, <math>204</math> the fourth, <math>212</math> at the fifth, and there were <math>227</math> at today's game." Just then, Michael's
 +
genius younger brother Tony, just seven-years-old, computes the average attendance of the six games. What is their average? 
  
 
[[2008 iTest Problems/Problem 3|Solution]]
 
[[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 <math>11</math>. Find their sum.
  
 
[[2008 iTest Problems/Problem 4|Solution]]
 
[[2008 iTest Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
 +
 +
Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment.
 +
Factory A previously filtered <math>80</math>% of pollutants and Factory B previously filled <math>72</math>% of pollutants. After installing the new scrubber system, both factories now
 +
filter <math>99.5</math>% of pollutants.
 +
 +
Jerry explains the level of pollution reduction to Michael, "Factory <math>A</math> is the much larger factory. It's four times as large as Factory <math>B</math>. Without any filters at all,
 +
it would pollute four times as much as Factory <math>B</math>. Even with the better pollution filtration system, Factory A was polluting nearly three times as much as Factory B."
 +
 +
Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find <math>a+b</math> where <math>a/b</math> is the ratio in lowest
 +
terms of volume of pollutants unfiltered from both factories <math>\textit{after}</math> installation of the new scrubber system to the volume of pollutants unfiltered
 +
from both factories <math>\textit{before}</math> installation of the new scrubber system.
  
 
[[2008 iTest Problems/Problem 5|Solution]]
 
[[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>.
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==Problem 8==
 
==Problem 8==
−
(story eliminated)
 
  
−
Given the system of equations
+
The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations:
  
 
<center><math>2x + 3y + 3z = 8</math>,
 
<center><math>2x + 3y + 3z = 8</math>,
Line 48: Line 86:
  
 
==Problem 9==
 
==Problem 9==
−
(story eliminated)
 
  
−
What is the units digit of <math>2008^{2008}</math>?
+
Joshua likes to play with numbers and patterns. Joshua's favorite number is <math>6</math> because it is the units digit of his birth year, <math>1996</math>.
 +
Part of the reason Joshua likes the number 6 so much is that the powers of <math>6</math> all have the same units digit as they grow from <math>6^1</math>:
 +
<cmath>\begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*}</cmath>
 +
However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are
 +
simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds,
 +
"You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers
 +
and see what he can discover. "See if you can find the units digit of <math>2008^{2008}</math>," Michael challenges. After a little while,
 +
Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of <math>2008^{2008}</math>)?  
 +
 
  
 
[[2008 iTest Problems/Problem 9|Solution]]
 
[[2008 iTest Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
 +
Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around <math>9</math> o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is 18 feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?
  
 
[[2008 iTest Problems/Problem 10|Solution]]
 
[[2008 iTest Problems/Problem 10|Solution]]
 
==Problem 11==
 
==Problem 11==
 +
 +
After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are <math>2000</math> gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)
  
 
[[2008 iTest Problems/Problem 11|Solution]]
 
[[2008 iTest Problems/Problem 11|Solution]]
 
==Problem 12==
 
==Problem 12==
 +
 +
One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates
 +
Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and
 +
challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is <math>3</math> higher than Tony's. "Let's play once more," says Tony. Let <math>a/b</math> be the probability
 +
that the difference between the outcomes of the two dice is again exactly <math>3</math> (regardless of which of the boys rolls higher), where a and b are relatively prime positive
 +
integers. Find <math>a+b</math>.
  
 
[[2008 iTest Problems/Problem 12|Solution]]
 
[[2008 iTest Problems/Problem 12|Solution]]
 
==Problem 13==
 
==Problem 13==
 +
 +
In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag.
 +
Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly
 +
beans in the next row, five jelly beans in the row after that, and so on:
 +
 +
<cmath> \begin{array}{ccccccccc}&&&&*&&&&\\&&&*&*&*&&&\\&&*&*&*&*&*&&\\&*&*&*&*&*&*&*&\\ *&*&*&*&*&*&*&*&*\\&&&&\vdots&&&&\end{array} </cmath>
 +
 +
Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly <math>25</math> jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between <math>10</math> and <math>20</math> rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no marble gets put inside more than one bag.)
  
 
[[2008 iTest Problems/Problem 13|Solution]]
 
[[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 <math>500</math> is <math>343+512 = 855</math>. Find the sum of the two perfect cubes that are closest to <math>2008</math>.
  
 
[[2008 iTest Problems/Problem 14|Solution]]
 
[[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 <math>8</math> are greater than <math>2008</math>?
  
 
[[2008 iTest Problems/Problem 15|Solution]]
 
[[2008 iTest Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
 +
In order to encourage the kids to straighten up their closets and the storage shed, Jerry offers his kids some extra spending money for their upcoming vacation. "I don't care what you do, I just want to see everything look clean and organized."
 +
 +
While going through his closet, Joshua finds an old bag of marbles that are either blue or red. The ratio of blue to red marbles in the bag is <math>17:7</math>. Alexis also has some marbles of the same colors, but hasn't used them for anything in years. She decides to give Joshua her marbles to put in his marble bag so that all the marbles are in one place. Alexis has twice as many red marbles as blue marbles, and when the twins get all their marbles in one bag, there are exactly as many red marbles and blue marbles, and the total number of marbles is between <math>200</math> and <math>250</math>. How many total marbles do the twins have together?
  
 
[[2008 iTest Problems/Problem 16|Solution]]
 
[[2008 iTest Problems/Problem 16|Solution]]
 
==Problem 17==
 
==Problem 17==
 +
 +
One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy counts a total of <math>28</math> heads belonging to the people, horses, and dog. She counts a total of <math>92</math> legs belonging to everyone, and notes that nobody is missing any legs.
 +
 +
Upon returning home Wendy gives Alexis a little problem solving practice, "I saw <math>28</math> heads and <math>92</math> legs belonging to people, horses, and dogs. Assuming two legs per person and four for the other animals, how many people did I see?" Alexis scribbles out some algebra and answers correctly. What is her answer?
  
 
[[2008 iTest Problems/Problem 17|Solution]]
 
[[2008 iTest Problems/Problem 17|Solution]]
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==Problem 20==
 
==Problem 20==
 +
 +
In order to earn a little spending money for the family vacation, Joshua and Wendy offer to clean up the storage shed. After clearing away some trash, Joshua and Wendy set aside give boxes that belong to the two of them that they decide to take up to their bedrooms. Each is in the shape of a cube. The four smaller boxes are all of equal size, and when stacked up, reach the exact height of the large box. If the volume of one of the smaller boxes is <math>216</math> cubic inches, find the sum of the volumes of all five boxes (in cubic inches).
  
 
[[2008 iTest Problems/Problem 20|Solution]]
 
[[2008 iTest Problems/Problem 20|Solution]]
 
==Problem 21==
 
==Problem 21==
 +
 +
One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called <math>\textit{Risk}</math> that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Michael explains a bit of strategy.
 +
 +
"You have the first move and you occupy three of the four territories in the Australian continent. You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn."
 +
 +
"Don't tell her <math>\textit{that!}</math>" complains Joshua.
 +
 +
Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia. Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack. Finally, Wendy is left with one chance. Wendy and Joshua each roll just one six-sided die. Wendy wins if her roll is <math>\textit{higher}</math> than Joshua's roll. Let a and b be relatively prime positive integers so that <math>a/b</math> is the probability that Wendy rolls higher, giving her control over the continent of Australia. Find the value of <math>a+b</math>.
  
 
[[2008 iTest Problems/Problem 21|Solution]]
 
[[2008 iTest Problems/Problem 21|Solution]]
 
==Problem 22==
 
==Problem 22==
 +
 +
Tony plays a game in which he takes <math>40</math> nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself <math>5</math> points for each nickel that lands in the jar, and takes away <math>2</math> points from his score for each nickel that hits the ground. After Tony is done tossing all <math>40</math> nickels, he computes <math>88</math> as his score. Find the greatest number of nickels he could have successfully tossed into the jar.
  
 
[[2008 iTest Problems/Problem 22|Solution]]
 
[[2008 iTest Problems/Problem 22|Solution]]
Line 104: Line 187:
 
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
  
−
<center><math>4n-18 < 2008</math>,</center>
+
<math></math><center><math>4n-18 < 2008</math>,</center>
−
<center><math>7n + 17 > 2008</math>.</center>
+
<center><math>7n + 17 > 2008</math>.</center><math></math>
  
 
[[2008 iTest Problems/Problem 23|Solution]]
 
[[2008 iTest Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
 +
In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes.
 +
 +
On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?"
 +
 +
"Shut up. I'm \textit{not} sniffing the carpet. I'm <math>\textit{doing something}</math>."
 +
 +
"Sure, if <math>\textit{sniffing the carpet}</math> counts as <math>\textit{doing something}</math>." At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel.
 +
 +
Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm <math>\textit{designing}</math> something."
 +
 +
Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?"
 +
 +
"Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome."
 +
 +
"And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project.
 +
 +
"I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be <math>9</math> inches from one of the corners, <math>15</math> inches from another, 16 inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days.
 +
 +
Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).
  
 
[[2008 iTest Problems/Problem 24|Solution]]
 
[[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>.
Line 121: Line 225:
 
[[2008 iTest Problems/Problem 26|Solution]]
 
[[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|Solution]]
 
[[2008 iTest Problems/Problem 27|Solution]]
 
==Problem 28==
 
==Problem 28==
 +
 +
Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital <math>\textit{every}</math> month, and having half the women work while only <math>2</math> of the <math>7</math> men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs.
 +
 +
Hannah comes up with a plan that the committee likes. Beginning in October, the committee of five volunteer teachers will consist of any five members of the volunteer group,
 +
so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that <math>\textit{must}</math> go by
 +
(including October when the first set of five teachers is selected, but not September) such that some five-member committee <math>\textit{must have}</math> taught together twice
 +
(all five members are the same during two different months)? 
  
 
[[2008 iTest Problems/Problem 28|Solution]]
 
[[2008 iTest Problems/Problem 28|Solution]]
 
==Problem 29==
 
==Problem 29==
 +
 +
Find the number of ordered triplets <math>(a,b,c)</math> of positive integers such that <math>abc=2008</math> (the product of <math>a, b</math>, and <math>c</math> is <math>2008</math>).
  
 
[[2008 iTest Problems/Problem 29|Solution]]
 
[[2008 iTest Problems/Problem 29|Solution]]
 
==Problem 30==
 
==Problem 30==
 +
 +
Find the number of ordered triplets <math>(a,b,c)</math> of positive integers such that <math>a<b<c</math> and <math>abc=2008</math>.
  
 
[[2008 iTest Problems/Problem 30|Solution]]
 
[[2008 iTest Problems/Problem 30|Solution]]
 
==Problem 31==
 
==Problem 31==
 +
 +
The <math>n^\text{th}</math> tern of a sequence is <math>a_n=(-1)^n(4n+3)</math>. Compute the sum <math>a_1+a_2+a_3+\cdots+a_{2008}</math>.
  
 
[[2008 iTest Problems/Problem 31|Solution]]
 
[[2008 iTest Problems/Problem 31|Solution]]
 
==Problem 32==
 
==Problem 32==
 +
 +
A right triangle has perimeter <math>2008</math>, and the area of a circle inscribed in the triangle is <math>100\pi^3</math>. Let A be the area of the triangle. Compute <math>\lfloor A\rfloor</math>.
  
 
[[2008 iTest Problems/Problem 32|Solution]]
 
[[2008 iTest Problems/Problem 32|Solution]]
 
==Problem 33==
 
==Problem 33==
 +
 +
One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel <math>50</math> miles per hour while heading to our final vacation destination..."
 +
Hannah teases her husband, "You <math>\textit{would}</math> drive that <math>\textit{slowly}\text{!}</math>"
 +
 +
Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we
 +
travel at a constant rate of 50 miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same rout.
 +
If our faster return rate is an integer number of miles per hour, and our average speed for the <math>\textit{whole round trip}</math> is <math>\textit{also}</math> an integer number of
 +
miles per hour, what must be our speed during the return trip?"
 +
Michael pipes up, "How about <math>4950</math> miles per hour?!"
 +
Wendy smiles, "For the sake of your <math>\textit{other}</math> children, please don't let <math>\textit{Michael}</math> drive."
 +
Jerry adds, "How about we assume that we never <math>\textit{ever}</math> drive more than <math>100 </math> miles per hour. Michael and Wendy, let Josh and Alexis try this one."
 +
Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate.
 +
But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?
  
 
[[2008 iTest Problems/Problem 33|Solution]]
 
[[2008 iTest Problems/Problem 33|Solution]]
 
==Problem 34==
 
==Problem 34==
 +
 +
While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let a be the probability that the cards are of different ranks. Compute <math>\lfloor 1000a\rfloor</math>.
  
 
[[2008 iTest Problems/Problem 34|Solution]]
 
[[2008 iTest Problems/Problem 34|Solution]]
 
==Problem 35==
 
==Problem 35==
 +
 +
Let b be the probability that the cards are from different suits. Compute <math>\lfloor1000b\rfloor</math>.
  
 
[[2008 iTest Problems/Problem 35|Solution]]
 
[[2008 iTest Problems/Problem 35|Solution]]
 
==Problem 36==
 
==Problem 36==
 +
 +
Let c be the probability that the cards are neither from the same suit or the same rank. Compute <math>\lfloor 1000c\rfloor</math>.
  
 
[[2008 iTest Problems/Problem 36|Solution]]
 
[[2008 iTest Problems/Problem 36|Solution]]
 
==Problem 37==
 
==Problem 37==
 +
 +
A triangle has sides of length <math>48, 55</math>, and <math>73</math>. Let a and b be relatively prime positive integers such that <math>a/b</math> is the length of the shortest altitude of the triangle.
 +
Find the value of <math>a+b</math>.
  
 
[[2008 iTest Problems/Problem 37|Solution]]
 
[[2008 iTest Problems/Problem 37|Solution]]
 
==Problem 38==
 
==Problem 38==
 +
 +
The volume of a certain rectangular solidis <math>216\text{ cm}^3</math>, its total surface area is <math>288\text{ cm}^2</math>, and its three dimensions are in
 +
geometric progression. Find the sum of the lengths in cm of all the edges of this solid.
  
 
[[2008 iTest Problems/Problem 38|Solution]]
 
[[2008 iTest Problems/Problem 38|Solution]]
 
==Problem 39==
 
==Problem 39==
 +
 +
Let <math>\phi(n)</math> denote <math>\textit{Euler's phi function}</math>, the number of integers <math>1\leq i\leq n</math> that are relatively prime to <math>n</math>. (For example, <math>\phi(6)=2</math> and <math>\phi(10)=4</math>.)
 +
Let <math>S=\sum_{d|2008}\phi(d)</math>, in which <math>d</math> ranges through all positive divisors of <math>2008</math>, including <math>1</math> and <math>2008</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.
  
 
[[2008 iTest Problems/Problem 39|Solution]]
 
[[2008 iTest Problems/Problem 39|Solution]]
 
==Problem 40==
 
==Problem 40==
 +
 +
Find the number of integers n that satisfy <math>\textit{both}</math> of the following conditions:
 +
<math>208<n<2008</math>,
 +
n has the same remainder when divided by <math>24</math> or by <math>30</math>.
 +
  
 
[[2008 iTest Problems/Problem 40|Solution]]
 
[[2008 iTest Problems/Problem 40|Solution]]
 
==Problem 41==
 
==Problem 41==
 +
 +
Suppose that <math>x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009</math>.
 +
Find the value of <math>\left\lfloor|S|\right\rfloor, where S=\displaystyle\sum_{n=1}^{2008}x_n</math>.
  
 
[[2008 iTest Problems/Problem 41|Solution]]
 
[[2008 iTest Problems/Problem 41|Solution]]
 
==Problem 42==
 
==Problem 42==
 +
 +
Joshua's physics teacher, Dr. Lisi, lives next door to the Kubiks and is a long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing
 +
and raising chickens as he does trying to map out a <math>\textit{Theory of Everything}</math>. Dr. Lisi often poses problems to the Kubik children to challenge them to
 +
think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first <math>2008</math> terms of an arithmetic
 +
progression that begins <math>-1776,-1765,-1754,\ldots</math>. Joshua then computes the (positive) difference between the <math>1980^\text{th}</math> term in the sequence,
 +
and the <math>1977^\text{th}</math> term in the sequence. What number does Joshua compute?
  
 
[[2008 iTest Problems/Problem 42|Solution]]
 
[[2008 iTest Problems/Problem 42|Solution]]
 
==Problem 43==
 
==Problem 43==
 +
 +
Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all <math>2008</math> terms in the sequence.
 +
Alexis thinks about the problem and remembers a story one of her teachers at school taught her about how a young Karl Gauss quickly computed the sum
 +
<math>1+2+3+\cdots+98+99+100</math> in elementary school. Using Gauss's method, Alexis correctly finds the sum of the <math>2008</math> terms in Dr. Lisi's sequence. What is this sum?
  
 
[[2008 iTest Problems/Problem 43|Solution]]
 
[[2008 iTest Problems/Problem 43|Solution]]

Revision as of 00:03, 12 October 2014

Problem 1

Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are $12$ years old.

When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. Single day passes cost $\textdollar{33} for adults (Jerry and Hannah),$\textdollar{22} 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 $\textdollar{120}. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?

[[2008 iTest Problems/Problem 1|Solution]]

==Problem 2==

One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly$ (Error compiling LaTeX. Unknown error_msg)12$feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around$15$feet. He then measures$2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?

[asy] size(400); defaultpen(linewidth(0.8)); draw((0,-3)--(0,3)); int d=8; for(int i=0;i<=5;i=i+1) { draw(circle(7/2+d*i,3/2)); } draw((5*d+7,-3)--(5*d+7,3)); draw((0,0)--(2,0),Arrows(size=7)); draw((5,0)--(2+d,0),Arrows(size=7)); draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7)); label("$2$",(1,0),S); label("$12$",((7+d)/2,0),S); label("$15$",((7+3*d)/2,0),S);[/asy]

Solution

Problem 3

Michael plays catcher for his school's baseball team. He has always been a great player behind the plate, but this year as a junior, Michael's offense is really improving. His batting average is $.417$ after six games, and the team is $6-0$ (six wins and no losses). They are off to their best start in years.

On the way home from their sixth game, Michael notes to his father that the attendance seems to be increasing due to the team's great start, "There were $181$ people at the first game, then $197$ at the second, $203$ the third, $204$ the fourth, $212$ at the fifth, and there were $227$ at today's game." Just then, Michael's genius younger brother Tony, just seven-years-old, computes the average attendance of the six games. What is their average?

Solution

Problem 4

The difference between two prime numbers is $11$. Find their sum.

Solution

Problem 5

Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment. Factory A previously filtered $80$% of pollutants and Factory B previously filled $72$% of pollutants. After installing the new scrubber system, both factories now filter $99.5$% of pollutants.

Jerry explains the level of pollution reduction to Michael, "Factory $A$ is the much larger factory. It's four times as large as Factory $B$. Without any filters at all, it would pollute four times as much as Factory $B$. Even with the better pollution filtration system, Factory A was polluting nearly three times as much as Factory B."

Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find $a+b$ where $a/b$ is the ratio in lowest terms of volume of pollutants unfiltered from both factories $\textit{after}$ installation of the new scrubber system to the volume of pollutants unfiltered from both factories $\textit{before}$ installation of the new scrubber system.

Solution

Problem 6

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than $L$.

Solution

Problem 7

Find the number of integers $n$ for which $n^2 + 10n < 2008$.

Solution

Problem 8

The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations:

$2x + 3y + 3z = 8$,

$3x + 2y + 3z = 808$,

$3x + 3y + 2z = 80808$,

find $x+y+z$.

Solution

Problem 9

Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$. Part of the reason Joshua likes the number 6 so much is that the powers of $6$ all have the same units digit as they grow from $6^1$: \begin{align*}6^1&=6,\\6^2&=36,\\6^3&=216,\\6^4&=1296,\\6^5&=7776,\\6^6&=46656,\\\vdots\end{align*} However, not all units digits remain constant when exponentiated in this way. One day Joshua asks Michael if there are simple patterns for the units digits when each one-digit integer is exponentiated in the manner above. Michael responds, "You tell me!" Joshua gives a disappointed look, but then Michael suggests that Joshua play around with some numbers and see what he can discover. "See if you can find the units digit of $2008^{2008}$," Michael challenges. After a little while, Joshua finds an answer which Michael confirms is correct. What is Joshua's correct answer (the units digit of $2008^{2008}$)?


Solution

Problem 10

Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is 18 feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?

Solution

Problem 11

After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)

Solution

Problem 12

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where a and b are relatively prime positive integers. Find $a+b$.

Solution

Problem 13

In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[\begin{array}{ccccccccc}&&&&*&&&&\\&&&*&*&*&&&\\&&*&*&*&*&*&&\\&*&*&*&*&*&*&*&\\ *&*&*&*&*&*&*&*&*\\&&&&\vdots&&&&\end{array}\]

Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly $25$ jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between $10$ and $20$ rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no marble gets put inside more than one bag.)

Solution

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 $2008$.

Solution

Problem 15

How many four-digit multiples of $8$ are greater than $2008$?

Solution

Problem 16

In order to encourage the kids to straighten up their closets and the storage shed, Jerry offers his kids some extra spending money for their upcoming vacation. "I don't care what you do, I just want to see everything look clean and organized."

While going through his closet, Joshua finds an old bag of marbles that are either blue or red. The ratio of blue to red marbles in the bag is $17:7$. Alexis also has some marbles of the same colors, but hasn't used them for anything in years. She decides to give Joshua her marbles to put in his marble bag so that all the marbles are in one place. Alexis has twice as many red marbles as blue marbles, and when the twins get all their marbles in one bag, there are exactly as many red marbles and blue marbles, and the total number of marbles is between $200$ and $250$. How many total marbles do the twins have together?

Solution

Problem 17

One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy counts a total of $28$ heads belonging to the people, horses, and dog. She counts a total of $92$ legs belonging to everyone, and notes that nobody is missing any legs.

Upon returning home Wendy gives Alexis a little problem solving practice, "I saw $28$ heads and $92$ legs belonging to people, horses, and dogs. Assuming two legs per person and four for the other animals, how many people did I see?" Alexis scribbles out some algebra and answers correctly. What is her answer?

Solution

Problem 18

Find the number of lattice points that the line $19x+20y = 1909$ passes through in Quadrant I.

Solution

Problem 19

Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A \cap B$.

Solution

Problem 20

In order to earn a little spending money for the family vacation, Joshua and Wendy offer to clean up the storage shed. After clearing away some trash, Joshua and Wendy set aside give boxes that belong to the two of them that they decide to take up to their bedrooms. Each is in the shape of a cube. The four smaller boxes are all of equal size, and when stacked up, reach the exact height of the large box. If the volume of one of the smaller boxes is $216$ cubic inches, find the sum of the volumes of all five boxes (in cubic inches).

Solution

Problem 21

One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Michael explains a bit of strategy.

"You have the first move and you occupy three of the four territories in the Australian continent. You'll want to attack Joshua in Indonesia so that you can claim the Australian continent which will give you bonus armies on your next turn."

"Don't tell her $\textit{that!}$" complains Joshua.

Wendy and Joshua begin rolling dice to determine the outcome of their struggle over Indonesia. Joshua rolls extremely well, overcoming longshot odds to hold off Wendy's attack. Finally, Wendy is left with one chance. Wendy and Joshua each roll just one six-sided die. Wendy wins if her roll is $\textit{higher}$ than Joshua's roll. Let a and b be relatively prime positive integers so that $a/b$ is the probability that Wendy rolls higher, giving her control over the continent of Australia. Find the value of $a+b$.

Solution

Problem 22

Tony plays a game in which he takes $40$ nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself $5$ points for each nickel that lands in the jar, and takes away $2$ points from his score for each nickel that hits the ground. After Tony is done tossing all $40$ nickels, he computes $88$ as his score. Find the greatest number of nickels he could have successfully tossed into the jar.

Solution

Problem 23

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities

$$ (Error compiling LaTeX. Unknown error_msg)

$4n-18 < 2008$,
$7n + 17 > 2008$.

$$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 24

In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes.

On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?"

"Shut up. I'm \textit{not} sniffing the carpet. I'm $\textit{doing something}$."

"Sure, if $\textit{sniffing the carpet}$ counts as $\textit{doing something}$." At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel.

Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm $\textit{designing}$ something."

Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?"

"Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome."

"And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project.

"I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be $9$ inches from one of the corners, $15$ inches from another, 16 inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days.

Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).

Solution

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 $n$ cm, where $n$ is an integer, find the number of possible values of $n$.

Solution

Problem 26

Solution

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?

Solution

Problem 28

Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes. Beginning in October, the committee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member committee $\textit{must have}$ taught together twice (all five members are the same during two different months)?

Solution

Problem 29

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a, b$, and $c$ is $2008$).

Solution

Problem 30

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $a<b<c$ and $abc=2008$.

Solution

Problem 31

The $n^\text{th}$ tern of a sequence is $a_n=(-1)^n(4n+3)$. Compute the sum $a_1+a_2+a_3+\cdots+a_{2008}$.

Solution

Problem 32

A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let A be the area of the triangle. Compute $\lfloor A\rfloor$.

Solution

Problem 33

One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel $50$ miles per hour while heading to our final vacation destination..." Hannah teases her husband, "You $\textit{would}$ drive that $\textit{slowly}\text{!}$"

Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of 50 miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same rout. If our faster return rate is an integer number of miles per hour, and our average speed for the $\textit{whole round trip}$ is $\textit{also}$ an integer number of miles per hour, what must be our speed during the return trip?" Michael pipes up, "How about $4950$ miles per hour?!" Wendy smiles, "For the sake of your $\textit{other}$ children, please don't let $\textit{Michael}$ drive." Jerry adds, "How about we assume that we never $\textit{ever}$ drive more than $100$ miles per hour. Michael and Wendy, let Josh and Alexis try this one." Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?

Solution

Problem 34

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let a be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$.

Solution

Problem 35

Let b be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.

Solution

Problem 36

Let c be the probability that the cards are neither from the same suit or the same rank. Compute $\lfloor 1000c\rfloor$.

Solution

Problem 37

A triangle has sides of length $48, 55$, and $73$. Let a and b be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.

Solution

Problem 38

The volume of a certain rectangular solidis $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.

Solution

Problem 39

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let $S=\sum_{d|2008}\phi(d)$, in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

Solution

Problem 40

Find the number of integers n that satisfy $\textit{both}$ of the following conditions: $208<n<2008$, n has the same remainder when divided by $24$ or by $30$.


Solution

Problem 41

Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor, where S=\displaystyle\sum_{n=1}^{2008}x_n$.

Solution

Problem 42

Joshua's physics teacher, Dr. Lisi, lives next door to the Kubiks and is a long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing and raising chickens as he does trying to map out a $\textit{Theory of Everything}$. Dr. Lisi often poses problems to the Kubik children to challenge them to think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776,-1765,-1754,\ldots$. Joshua then computes the (positive) difference between the $1980^\text{th}$ term in the sequence, and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?

Solution

Problem 43

Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teachers at school taught her about how a young Karl Gauss quickly computed the sum $1+2+3+\cdots+98+99+100$ in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?

Solution

Problem 44

Solution

Problem 45

Solution

Problem 46

Solution

Problem 47

Find $a + b + c$, where $a$, $b$, and $c$ are the hundreds, tens, and units digits of the six-digit integer $123abc$, which is a multiple of 990.

Solution

Problem 48

A repunit is a natural number whose digits are all $1$. For instance,

\[1,11,111,1111, \ldots\]

are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97$?

Solution

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 $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Solution

Problem 50

Solution

Problem 51

Solution

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 $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c + d$.

Solution

Problem 53

Solution

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?

Solution

Problem 55

Solution

Problem 56

Solution

Problem 57

Solution

Problem 58

Solution

Problem 59

Solution

Problem 60

Solution

Problem 61

Solution

Problem 62

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, 65, and 65 is a positive integer.

Solution

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?

Solution

Problem 64

Solution

Problem 65

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Problem 66

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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.)

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Problem 68

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Problem 69

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Problem 70

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Problem 71

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Problem 72

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Problem 73

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Problem 74

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Problem 75

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Problem 76

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Problem 77

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Problem 78

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Problem 79

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Problem 80

Let

$p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$

and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.

Solution

Problem 81

Solution

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 .

Solution

Problem 83

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Problem 84

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Problem 85

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Problem 86

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Problem 87

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Problem 88

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Problem 89

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Problem 90

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Problem 91

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Problem 92

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Problem 93

For how many positive integers $n$, $1 \le n \le 2008$, can the set

${1, 2, 3, . . . , 4n}$

be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?

Solution

Problem 94

Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence

\[\left\lfloor \frac{2008}{1} \right\rfloor, \left\lfloor \frac{2008^2}{2} \right\rfloor, \left\lfloor \frac{2008^3}{3}\right\rfloor, \left\lfloor \frac{2008^4}{4} \right\rfloor, \cdots\]

Solution

Problem 95

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Problem 96

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Problem 97

(storyline deleted) Let $k$ be the number of students in a circle. Then let $m$ be the number of ways they can rearrange ourselves so that each of them is in the same spot or within one spot of where they started, and no two people are ever on the same spot. If $m$ leaves a remainder of $1$ when divided by $5$, how many possible values are there of $k$, where $k$ is at least $3$ and at most $2008$?

Solution

Problem 98

Solution

Problem 99

Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges’ trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13, 13, 13$, and $3$, and for each $i > 2$, let $P_i = \text{clip}(P_{i-1})$. This iterative clipping process approaches a limiting shape $P_1 = \lim_{i \rightarrow \infty} P_i$. If the difference of the areas of $P_{10}$ and $P_{1}$ is written as a fraction $\frac{x}{y}$ in lowest terms, calculate the number of positive integer factors of $x \cdot y$.

Solution

Problem 100

Solution

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