Difference between revisions of "2007 iTest Problems"
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===Problem 21=== | ===Problem 21=== | ||
+ | James writes down fifteen 1's in a row and randomly writes <math>+</math> or <math>-</math> between each pair of consecutive 1's. | ||
+ | One such example is | ||
+ | <cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath> | ||
+ | What is the probability that the value of the expression James wrote down is <math>7</math>? | ||
+ | |||
+ | <math>\text{(A) }0\qquad | ||
+ | \text{(B) }\frac{6435 }{2^{14}}\qquad | ||
+ | \text{(C) }\frac{6435 }{2^{13}}\qquad | ||
+ | \text{(D) }\frac{429}{2^{12}}\qquad | ||
+ | \text{(E) }\frac{429}{2^{11}}\qquad | ||
+ | \text{(F) }\frac{429}{2^{10}}\qquad | ||
+ | \text{(G) }\frac{1}{15}\qquad | ||
+ | \text{(H) }\frac{1}{31}\qquad \ </math> | ||
+ | <math>\text{(I) }\frac{1}{30}\qquad | ||
+ | \text{(J) }\frac{1}{29}\qquad | ||
+ | \text{(K) }\frac{6435 }{2^{15}}\qquad | ||
+ | \text{(L) }\frac{6435 }{2^{14}}\qquad | ||
+ | \text{(M) }\frac{6435 }{2^{13}}\qquad | ||
+ | \text{(N) }\frac{1}{2^{7}}\qquad | ||
+ | \text{(O) }\frac{1}{2^{14}}\qquad | ||
+ | \text{(P) }\frac{1}{2^{15}}\qquad \ </math> | ||
+ | <math>\text{(P) }\frac{2007}{2^{14}}\qquad | ||
+ | \text{(P) }\frac{2007}{2^{15}}\qquad | ||
+ | \text{(P) }\frac{2007}{2^{2007}}\qquad | ||
+ | \text{(P) }\frac{1}{2007}\qquad | ||
+ | \text{(P) }\frac{-2007}{2^{14}}\qquad</math> | ||
===Problem 22=== | ===Problem 22=== |
Revision as of 00:42, 6 October 2014
Contents
[hide]- 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 Question
- 4 Tiebreaker Questions
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" $2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}$ (Error compiling LaTeX. Unknown error_msg), 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
Problem 16
How many lattice points lie within or on the border of the circle in the -plane defined by the equation
Problem 17
Problem 18
Problem 19
Problem 20
Find the largest integer such that is divisible by
$\text{(A) } 1\qquad \text{(B) } 2\qquad \text{(C) } 3\qquad \text{(D) } 4\qquad \text{(E) } 5\qquad \text{(F) } 6\qquad \text{(G) } 7\qquad \text{(H) } 8\qquad $ (Error compiling LaTeX. Unknown error_msg) $\text{(I) } 9\qquad \text{(J) } 10\qquad \text{(K) } 11\qquad \text{(L) } 12\qquad \text{(M) } 13\qquad \text{(N) } 14\qquad \text{(O) } 15\qquad \text{(P) } 16\qquad $ (Error compiling LaTeX. Unknown error_msg)
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 ?
$\text{(A) }0\qquad \text{(B) }\frac{6435 }{2^{14}}\qquad \text{(C) }\frac{6435 }{2^{13}}\qquad \text{(D) }\frac{429}{2^{12}}\qquad \text{(E) }\frac{429}{2^{11}}\qquad \text{(F) }\frac{429}{2^{10}}\qquad \text{(G) }\frac{1}{15}\qquad \text{(H) }\frac{1}{31}\qquad $ (Error compiling LaTeX. Unknown error_msg) $\text{(I) }\frac{1}{30}\qquad \text{(J) }\frac{1}{29}\qquad \text{(K) }\frac{6435 }{2^{15}}\qquad \text{(L) }\frac{6435 }{2^{14}}\qquad \text{(M) }\frac{6435 }{2^{13}}\qquad \text{(N) }\frac{1}{2^{7}}\qquad \text{(O) }\frac{1}{2^{14}}\qquad \text{(P) }\frac{1}{2^{15}}\qquad $ (Error compiling LaTeX. Unknown error_msg)
Problem 22
Problem 23
Problem 24
Problem 25
Ted's favorite number is equal to
Find the remainder when Ted's favorite number is divided by 25.
$\text{(A) } 0\qquad \text{(B) } 1\qquad \text{(C) } 2\qquad \text{(D) } 3\qquad \text{(E) } 4\qquad \text{(F) } 5\qquad \text{(G) } 6\qquad \text{(H) } 7\qquad \text{(I) } 8\qquad $ (Error compiling LaTeX. Unknown error_msg)
$\text{(J) } 9\qquad \text{(K) } 10\qquad \text{(L) } 11\qquad \text{(M) } 12\qquad \text{(N) } 13\qquad \text{(O) } 14\qquad \text{(P) } 15\qquad \text{(Q) } 16\qquad $ (Error compiling LaTeX. Unknown error_msg)
Short Answer Section
Problem 26
Problem 27
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
Problem 48
Problem 49
Problem 50
Ultimate Question
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
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.