2007 iTest Problems
Contents
- 1 Multiple Choice Section
- 1.1 Problem 1
- 1.2 Problem 2
- 1.3 Problem 3
- 1.4 Problem 4
- 1.5 Problem 5
- 1.6 Problem 6
- 1.7 Problem 7
- 1.8 Problem 8
- 1.9 Problem 9
- 1.10 Problem 10
- 1.11 Problem 11
- 1.12 Problem 12
- 1.13 Problem 13
- 1.14 Problem 14
- 1.15 Problem 15
- 1.16 Problem 16
- 1.17 Problem 17
- 1.18 Problem 18
- 1.19 Problem 19
- 1.20 Problem 20
- 1.21 Problem 21
- 1.22 Problem 22
- 1.23 Problem 23
- 1.24 Problem 24
- 1.25 Problem 25
- 2 Short Answer Section
- 2.1 Problem 26
- 2.2 Problem 27
- 2.3 Problem 28
- 2.4 Problem 29
- 2.5 Problem 30
- 2.6 Problem 31
- 2.7 Problem 32
- 2.8 Problem 33
- 2.9 Problem 34
- 2.10 Problem 35
- 2.11 Problem 36
- 2.12 Problem 37
- 2.13 Problem 38
- 2.14 Problem 39
- 2.15 Problem 40
- 2.16 Problem 41
- 2.17 Problem 42
- 2.18 Problem 43
- 2.19 Problem 44
- 2.20 Problem 45
- 2.21 Problem 46
- 2.22 Problem 47
- 2.23 Problem 48
- 2.24 Problem 49
- 2.25 Problem 50
- 3 Ultimate Questions
- 4 Tiebreaker Questions
- 5 See Also
Multiple Choice Section
Problem 1
A twin prime pair is a set of two primes such that is greater than . What is the arithmetic mean of the two primes in the smallest twin prime pair?
Problem 2
Find if and satisfy and .
Problem 3
An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?
Problem 4
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
Problem 5
Compute the sum of all twenty-one terms of the geometric series .
Problem 6
Find the units digit of the sum
Problem 7
An equilateral triangle with side length has the same area as a square with side length . Find .
Problem 8
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?
Problem 9
Suppose that and are positive integers such that , the geometric mean of and is greater than , and the arithmetic mean of and is less than . How many pairs satisfy these conditions?
Problem 10
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?
Problem 11
Consider the "tower of power" , where there are 2007 twos including the base. What is the last (units digit) of this number?
Problem 12
My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?
Problem 13
What is the smallest positive integer such that the number ends in two zeros?
Problem 14
Let be the number of positive integers which are relatively prime to . For how many distinct values of is ?
Problem 15
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?
Problem 16
How many lattice points lie within or on the border of the circle in the -plane defined by the equation
Problem 17
If and are acute angles such that and , find the value of .
Problem 18
Suppose that is a cubic with a double root at and another root at b, where and are real numbers. If and , what is ?
Problem 19
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin. If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins What is the expected number of gold coins Jason wins at this game?
Problem 20
Find the largest integer such that is divisible by
Problem 21
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. One such example is What is the probability that the value of the expression James wrote down is ?
Problem 22
Find the value of such that the system of equations has exactly two solutions in real numbers.
Problem 23
Find the product of the non-real roots of the equation
Problem 24
Let be the smallest positive integer such that is a perfect square and is a perfect cube. Find the remainder when is divided by .
Problem 25
Ted's favorite number is equal to
Find the remainder when Ted's favorite number is divided by 25.
Short Answer Section
Problem 26
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?
Problem 27
The face diagonal of a cube has length . Find the value of n given that is the of the cube.
Problem 28
The space diagonal (interior diagonal) of a cube has length 6. Find the of the cube.
Problem 29
Let be equal to the sum . Find the remainder when is divided by .
Problem 30
While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers , and , and recalled that their product is , but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than with fewer than divisors. Help James by computing .
Problem 31
Let be the length of one side of a triangle and let y be the height to that side. If , find the maximum possible of the area of the triangle.
Problem 32
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio to . How many integer values of k are there such that and the area between the parabola and the -axis is an integer?
Problem 33
How many four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?
Problem 34
Let be the probability that a randomly selected divisor of is a multiple of . If and are relatively prime positive integers, find .
Problem 35
Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits.
Problem 36
Let b be a real number randomly selected from the interval . Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation has two distinct real solutions. Find the value of .
Problem 37
Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are , and respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.
Problem 38
Find the largest positive integer that is equal to the cube of the sum of its digits.
Problem 39
Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation . Find .
Problem 40
Let be the sum of all such that and . Compute .
Problem 41
The sequence of digits is obtained by writing the positive integers in order. If the digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define to be . For example, because the digit enters the sequence in the placement of the two-digit integer . Find the value of .
Problem 42
During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a foot by foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as , where all four variables are positive integers, is a multple of no perfect square greater than , a is coprime with , and is coprime with . Find the value of .
Problem 43
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following -digit integers:
She notes that two of them have exactly positive divisors each. Find the common prime divisor of those two integers.
Problem 44
A positive integer between and inclusive is selected at random. If and are natural numbers such that is the probability that and are relatively prime, find the value of .
Problem 45
Find the sum of all positive integers such that , where represent distinct base digits, .
Problem 46
Let be an ordered triplet of real numbers that satisfies the following system of equations: If is the minimum possible value of , find the modulo residue of .
Problem 47
Let and be sequences defined as follows: ,
Let be the largest integer that satisfies all of the following conditions: , for some positive integer ; , for some positive integer ; and . Find the remainder when is divided by .
Problem 48
Let a and b be relatively prime positive integers such that is the maximum possible value of , where, for is a nonnegative real number, and . Find the value of .
Problem 49
How many -element subsets of are there, the sum of whose elements is divisible by ?
Problem 50
A block is formed by gluing one face of a solid cube with side length onto one of the circular faces of a right circular cylinder with radius and height so that the centers of the square and circle coincide. If is the smallest convex region that contains , calculate (the greatest integer less than or equal to the volume of ).
Ultimate Questions
In the next 10 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem.
For those who want to try these problems without having to find the T-values of the previous problem, a link will be here. Also, all solutions will have the T-values substituted.
Problem 51
Find the highest point (largest possible -coordinate) on the parabola
Problem 52
Let . Let be the region consisting of points of the Cartesian plane satisfying both and . Find the area of region .
Problem 53
Let . Three distinct positive Fibonacci numbers, all greater than , are in arithmetic progression. Let be the smallest possible value of their sum. Find the remainder when is divided by .
Problem 54
Let . Consider the sequence . Inserting the difference between and between them, we get the sequence . Repeating the process of inserting differences between numbers, we get the sequence . A third iteration of this process results in . A total of iterations produces a sequence with terms. If the integer (that is, times the integer ) appears a total of times among these terms, find the remainder when gets divided by .
Problem 55
Let , and let . Let be the smallest real solution of . Find the value of .
Problem 56
Let . In the binary expansion of , how many of the first digits to the right of the radix point are 's?
Problem 57
Let . How many positive integers are within of exactly perfect squares? (Note: is considered a perfect square.)
Problem 58
Let . For natural numbers , we define Compute the value of .
Problem 59
Let . Fermi and Feynman play the game in which Fermi wins with probability , where and are relatively prime positive integers such that . The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is . Find the value of .
Problem 60
Let . Triangle has and . Point is on so that bisects angle . The circle through , and has center and intersects line again at , and likewise the circle through , and has center and intersects line again at . If the four points , and lie on a circle, find the length of .
Tiebreaker Questions
Problem TB1
The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.
Problem TB2
Factor completely over integer coefficients the polynomial . Demonstrate that your factorization is complete.
Problem TB3
4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.
Problem TB4
Circle is the circumcircle of non-isosceles triangle . The tangent lines to circle at points and intersect at , and the tangents at and intersect at . The external angle bisectors of triangle at and meet at and the external bisectors at and intersect at . Prove that lines , , and are concurrent.
See Also
2007 iTest (Problems, Answer Key) | ||
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Followed by: 2008 iTest | |
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