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− | ==Multiple Choice Section==
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− | ===Problem 1===
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− | A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair?
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− | <math>\mathrm{(A)}\, 4</math>
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− | [[2007 iTest Problems/Problem 1|Solution]]
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− |
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− | ===Problem 2===
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− | Find <math>a + b</math> if <math>a</math> and <math>b</math> satisfy
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− | <math>3a + 7b = 1977</math> and <math>5a + b = 2007</math>.
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− | <math>\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498</math>
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− |
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− | [[2007 iTest Problems/Problem 2|Solution]]
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− |
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− | ===Problem 3===
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− | 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?
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− | <math>\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56</math>
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− | [[2007 iTest Problems/Problem 3|Solution]]
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− |
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− | ===Problem 4===
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− | Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
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− | <math>\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}</math>
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− | [[2007 iTest Problems/Problem 4|Solution]]
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− |
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− | ===Problem 5===
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− | Compute the sum of all twenty-one terms of the geometric series
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− | <cmath>1 + 2 + 4 + 8 + \ldots + 1048576</cmath>.
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− | <math>\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161</math>
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− | [[2007 iTest Problems/Problem 5|Solution]]
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− | ===Problem 6===
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− | Find the units digit of the sum
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− |
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− | <cmath>\sum_{i=1}^{100}(i!)^{2}</cmath>
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− | <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math>
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− |
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− | [[2007 iTest Problems/Problem 6|Solution]]
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− | ===Problem 7===
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− | An equilateral triangle with side length <math>1</math> has the same area as a square with side length <math>s</math>.
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− | Find <math>s</math>.
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− | <math>\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}</math>
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− | [[2007 iTest Problems/Problem 7|Solution]]
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− | ===Problem 8===
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− | 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?
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− | <math>\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789</math>
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− | [[2007 iTest Problems/Problem 8|Solution]]
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− |
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− | ===Problem 9===
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− | Suppose that <math>m</math> and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>, and the arithmetic mean of <math>m</math> and <math>n</math> is less than <math>2007</math>. How many pairs <math>(m, n)</math> satisfy these conditions?
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− | <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007</math>
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− |
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− | [[2007 iTest Problems/Problem 9|Solution]]
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− |
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− | ===Problem 10===
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− | My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only <math>4</math> 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?
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− | <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007</math>
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− |
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− | [[2007 iTest Problems/Problem 10|Solution]]
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− |
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− | ===Problem 11===
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− | Consider the "tower of power" <math>2^{2^2\cdot \cdot \cdot^2}</math>, where there are 2007 twos including the base. What is the last (units digit) of this number?
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− | <math>\text{(A) }0\qquad
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− | \text{(B) }1\qquad
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− | \text{(C) }2\qquad
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− | \text{(D) }3\qquad
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− | \text{(E) }4\qquad
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− | \text{(F) }5\qquad
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− | \text{(G) }6\qquad
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− | \text{(H) }7\qquad
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− | \text{(I) }8\qquad
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− | \text{(J) }9\qquad
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− | \text{(K) }2007\qquad</math>
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− |
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− | [[2007 iTest Problems/Problem 11|Solution]]
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− |
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− | ===Problem 12===
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− | My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score.
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− | The game ends after one of the two teams scores three points (total, not necessarily consecutive).
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− | If every possible sequence of scores is equally likely, what is the expected score of the losing team.
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− | <math>\text{(A) }\frac{2}{3}\qquad
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− | \text{(B) }1\qquad
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− | \text{(C) }\frac{3}{2}\qquad
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− | \text{(D) }\frac{8}{5}\qquad
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− | \text{(E) }\frac{5}{8}\qquad
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− | \text{(F) }2\qquad\\ \\
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− | \text{(G) }0\qquad
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− | \text{(H) }\frac{5}{2}\qquad
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− | \text{(I) }\frac{2}{5}\qquad
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− | \text{(J) }\frac{3}{4}\qquad
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− | \text{(K) }\frac{4}{3}\qquad
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− | \text{(L) }2007\qquad</math>
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− | [[2007 iTest Problems/Problem 12|Solution]]
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− |
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− | ===Problem 13===
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− | What is the smallest positive integer <math>k</math> such that the number <math>{{2k}\choose k}</math> ends in two zeros?
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− | <math>\text{(A) } 3 \quad
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− | \text{(B) } 4 \quad
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− | \text{(C) } 5 \quad
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− | \text{(D) } 6 \quad
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− | \text{(E) } 7 \quad
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− | \text{(F) } 8 \quad
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− | \text{(G) } 9 \quad
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− | \text{(H) } 10 \quad
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− | \text{(I) } 11 \quad
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− | \text{(J) } 12 \quad
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− | \text{(K) } 13 \quad
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− | \text{(L) } 14 \quad
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− | \text{(M) } 2007\quad </math>
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− |
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− | [[2007 iTest Problems/Problem 13|Solution]]
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− |
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− | ===Problem 14===
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− | Let <math>\phi(n)</math> be the number of positive integers <math>k< n</math> which are relatively prime to <math>n</math>. For how many distinct values of <math>n</math> is <math>\phi(n)=12</math>?
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− | <math>\text{(A) } 0 \quad
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− | \text{(B) } 1 \quad
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− | \text{(C) } 2 \quad
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− | \text{(D) } 3 \quad
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− | \text{(E) } 4 \quad
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− | \text{(F) } 5 \quad
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− | \text{(G) } 6 \quad
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− | \text{(H) } 7 \quad
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− | \text{(I) } 8 \quad
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− | \text{(J) } 9 \quad
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− | \text{(K) } 10 \quad
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− | \text{(L) } 11 \quad
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− | \text{(M) } 12\quad
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− | \text{(N) } 13\quad </math>
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− |
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− | [[2007 iTest Problems/Problem 14|Solution]]
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− |
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− | ===Problem 15===
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− |
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− | Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and
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− | placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the
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− | pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the
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− | equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?
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− | <math>\text{(A) }\frac{2}{3}\qquad
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− | \text{(B) }\frac{3}{4}\qquad
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− | \text{(C) }1\qquad
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− | \text{(D) }\frac{5}{4}\qquad
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− | \text{(E) }\frac{4}{3}\qquad
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− | \text{(F) }\frac{\sqrt{2}}{2}\qquad
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− | \text{(G) }\frac{\sqrt{3}}{2}\qquad
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− | \text{(H) }\sqrt{2}\qquad</math>
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− | <math>\text{(I) }\sqrt{3}\qquad
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− | \text{(J) }\frac{1+\sqrt{3}}{2}\qquad
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− | \text{(K) }\frac{2+\sqrt{6}}{2}\qquad
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− | \text{(L) }\frac{7}{6}\qquad
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− | \text{(M) }\frac{2+\sqrt{6}}{4}\qquad
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− | \text{(N) }\frac{4}{5}\qquad
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− | \text{(O) }2007\qquad</math>
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− | [[2007 iTest Problems/Problem 15|Solution]]
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− |
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− | ===Problem 16===
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− | How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation
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− | <cmath>x^2+y^2=100</cmath>
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− | <math>\text{(A) }1\qquad
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− | \text{(B) }2\qquad
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− | \text{(C) }4\qquad
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− | \text{(D) }5\qquad
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− | \text{(E) }41\qquad
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− | \text{(F) }42\qquad
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− | \text{(G) }69\qquad
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− | \text{(H) }76\qquad
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− | \text{(I) }130\qquad \\
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− | \text{(J) }133\qquad
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− | \text{(K) }233\qquad
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− | \text{(L) }311\qquad
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− | \text{(M) }317\qquad
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− | \text{(N) }420\qquad
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− | \text{(O) }520\qquad
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− | \text{(P) }2007</math>
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− | [[2007 iTest Problems/Problem 16|Solution]]
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− |
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− | ===Problem 17===
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− | If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.
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− | <math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad
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− | \text{(B) }\frac{35\sqrt{2}-6}{71}\qquad
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− | \text{(C) }\frac{35\sqrt{3}+12}{33}\qquad
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− | \text{(D) }\frac{37\sqrt{3}+24}{33}\qquad
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− | \text{(E) }1\qquad</math>
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− | <math>\text{(F) }\frac{5}{7}\qquad
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− | \text{(G) }\frac{3}{7}\qquad
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− | \text{(H) }6\qquad
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− | \text{(I) }\frac{1}{6}\qquad
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− | \text{(J) }\frac{1}{2}\qquad
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− | \text{(K) }\frac{6}{7}\qquad
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− | \text{(L) }\frac{4}{7}\qquad</math>
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− | <math>\text{(M) }\sqrt{3}\qquad
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− | \text{(N) }\frac{\sqrt{3}}{3}\qquad
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− | \text{(O) }\frac{5}{6}\qquad
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− | \text{(P) }\frac{2}{3}\qquad
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− | \text{(Q) }\frac{1}{2007}\qquad</math>
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− | [[2007 iTest Problems/Problem 17|Solution]]
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− |
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− | ===Problem 18===
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− | Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers.
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− | If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?
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− | <math>\text{(A) }0\qquad
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− | \text{(B) }4\qquad
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− | \text{(C) }108\qquad
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− | \text{(D) It could be }0 \text{ or } 4\qquad
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− | \text{(E) It could be }0 \text{ or } 108</math>
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− | <math>\text{(F) }18\qquad
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− | \text{(G) }-4\qquad
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− | \text{(H) }-108\qquad
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− | \text{(I) It could be } 0 \text{ or } -4</math>
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− |
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− | <math>\text{(J) It could be } 0 \text{ or } {-108} \qquad
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− | \text{(K) It could be } 4 \text{ or } {-4}\qquad
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− | \text{(L) There is no such value of } r\qquad</math>
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− | <math>\text{(M) } 1 \qquad
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− | \text{(N) } {-2} \qquad
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− | \text{(O) It could be } 4 \text{ or } -4 \qquad
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− | \text{(P) It could be } 0 \text{ or } -2 \qquad</math>
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− |
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− | <math>\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad
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− | \text{(R) } 2007</math>
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− | [[2007 iTest Problems/Problem 18|Solution]]
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− |
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− | ===Problem 19===
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− | 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!"
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− | 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.
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− | 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 <math>\textit{ten gold coins.}</math> What is the expected number of gold coins Jason wins at this game?
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− | <math>\textbf{(A) }0\qquad
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− | \textbf{(B) }\dfrac1{10}\qquad
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− | \textbf{(C) }\dfrac18\qquad
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− | \textbf{(D) }\dfrac15\qquad
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− | \textbf{(E) }\dfrac14\qquad
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− | \textbf{(F) }\dfrac13\qquad
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− | \textbf{(G) }\dfrac25\qquad</math>
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− | <math>\textbf{(H) }\dfrac12\qquad
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− | \textbf{(I) }\dfrac35\qquad
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− | \textbf{(J) }\dfrac23\qquad
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− | \textbf{(K) }\dfrac45\qquad
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− | \textbf{(L) }1\qquad
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− | \textbf{(M) }\dfrac54\qquad</math>
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− | <math>\textbf{(N) }\dfrac43\qquad
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− | \textbf{(O) }\dfrac32\qquad
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− | \textbf{(P) }2\qquad
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− | \textbf{(Q) }3\qquad
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− | \textbf{(R) }4\qquad
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− | \textbf{(S) }2007</math>
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− | [[2007 iTest Problems/Problem 19|Solution]]
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− |
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− | ===Problem 20===
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− |
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− | Find the largest integer <math>n</math> such that <math>2007^{1024}-1</math> is divisible by <math>2^n</math>
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− | <math>\text{(A) } 1\qquad
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− | \text{(B) } 2\qquad
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− | \text{(C) } 3\qquad
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− | \text{(D) } 4\qquad
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− | \text{(E) } 5\qquad
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− | \text{(F) } 6\qquad
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− | \text{(G) } 7\qquad
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− | \text{(H) } 8\qquad</math>
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− | <math>\text{(I) } 9\qquad
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− | \text{(J) } 10\qquad
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− | \text{(K) } 11\qquad
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− | \text{(L) } 12\qquad
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− | \text{(M) } 13\qquad
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− | \text{(N) } 14\qquad
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− | \text{(O) } 15\qquad
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− | \text{(P) } 16\qquad</math>
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− | <math>\text{(Q) } 55\qquad
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− | \text{(R) } 63\qquad
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− | \text{(S) } 64\qquad
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− | \text{(T) } 2007\qquad</math>
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− |
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− |
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− | [[2007 iTest Problems/Problem 20|Solution]]
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− |
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− | ===Problem 21===
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− | James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's.
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− | One such example is
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− | <cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath>
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− | What is the probability that the value of the expression James wrote down is <math>7</math>?
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− |
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− | <math>\text{(A) }0\qquad
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− | \text{(B) }\frac{6435 }{2^{14}}\qquad
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− | \text{(C) }\frac{6435 }{2^{13}}\qquad
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− | \text{(D) }\frac{429}{2^{12}}\qquad
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− | \text{(E) }\frac{429}{2^{11}}\qquad
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− | \text{(F) }\frac{429}{2^{10}}\qquad
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− | \text{(G) }\frac{1}{15}\qquad
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− | \text{(H) }\frac{1}{31}\qquad</math>
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− |
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− | <math>\text{(I) }\frac{1}{30}\qquad
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− | \text{(J) }\frac{1}{29}\qquad
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− | \text{(K) }\frac{1001 }{2^{15}}\qquad
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− | \text{(L) }\frac{1001 }{2^{14}}\qquad
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− | \text{(M) }\frac{1001 }{2^{13}}\qquad
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− | \text{(N) }\frac{1}{2^{7}}\qquad
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− | \text{(O) }\frac{1}{2^{14}}\qquad
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− | \text{(P) }\frac{1}{2^{15}}\qquad</math>
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− |
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− | <math>\text{(Q) }\frac{2007}{2^{14}}\qquad
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− | \text{(R) }\frac{2007}{2^{15}}\qquad
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− | \text{(S) }\frac{2007}{2^{2007}}\qquad
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− | \text{(T) }\frac{1}{2007}\qquad
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− | \text{(U) }\frac{-2007}{2^{14}}\qquad</math>
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− |
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− |
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− | [[2007 iTest Problems/Problem 21|Solution]]
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− |
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− | ===Problem 22===
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− | Find the value of <math>c</math> such that the system of equations
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− | <cmath> |x+y|=2007</cmath>
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− | <cmath>|x-y|=c</cmath>
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− | has exactly two solutions <math>(x,y)</math> in real numbers.
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− |
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− |
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− | <math>\text{(A) } 0 \quad
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− | \text{(B) } 1 \quad
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− | \text{(C) } 2 \quad
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− | \text{(D) } 3 \quad
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− | \text{(E) } 4 \quad
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− | \text{(F) } 5 \quad
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− | \text{(G) } 6 \quad
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− | \text{(H) } 7 \quad
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− | \text{(I) } 8 \quad
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− | \text{(J) } 9 \quad
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− | \text{(K) } 10 \quad
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− | \text{(L) } 11 \quad
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− | \text{(M) } 12 \quad</math>
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− |
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− | <math>\text{(N) } 13 \quad
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− | \text{(O) } 14 \quad
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− | \text{(P) } 15 \quad
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− | \text{(Q) } 16 \quad
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− | \text{(R) } 17 \quad
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− | \text{(S) } 18 \quad
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− | \text{(T) } 223 \quad
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− | \text{(U) } 678 \quad
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− | \text{(V) } 2007 \quad </math>
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− |
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− |
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− | [[2007 iTest Problems/Problem 22|Solution]]
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− |
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− | ===Problem 23===
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− | Find the product of the non-real roots of the equation
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− | <cmath>(x^2-3x)^2+5(x^2-3x)+6=0</cmath>
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− |
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− | <math>\text{(A) } 0\quad
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− | \text{(B) } 1\quad
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− | \text{(C) } -1\quad
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− | \text{(D) } 2\quad
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− | \text{(E) } -2\quad
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− | \text{(F) } 3\quad
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− | \text{(G) } -3\quad
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− | \text{(H) } 4\quad
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− | \text{(I) } -4\quad</math>
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− |
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− | <math>\text{(J) } 5\quad
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− | \text{(K) } -5\quad
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− | \text{(L) } 6\quad
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− | \text{(M) } -6\quad
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− | \text{(N) } 3+2i\quad
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− | \text{(O) } 3-2i\quad</math>
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− |
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− | <math>\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad
| |
− | \text{(Q) } 8\quad
| |
− | \text{(R) } -8\qquad
| |
− | \text{(S) } 12\quad
| |
− | \text{(T) } -12\quad
| |
− | \text{(U) } 42\quad</math>
| |
− |
| |
− | <math>\text{(V) Ying} \quad
| |
− | \text{(W) } 207</math>
| |
− |
| |
− |
| |
− | [[2007 iTest Problems/Problem 23|Solution]]
| |
− |
| |
− | ===Problem 24===
| |
− | Let <math>N</math> be the smallest positive integer such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube.
| |
− | Find the remainder when <math>N</math> is divided by <math>25</math>.
| |
− |
| |
− | <math>\text{(A) }0 \quad
| |
− | \text{(B) }1 \quad
| |
− | \text{(C) }2 \quad
| |
− | \text{(D) }3 \quad
| |
− | \text{(E) }4 \quad
| |
− | \text{(F) }5 \quad
| |
− | \text{(G) }6 \quad
| |
− | \text{(H) }7 \quad
| |
− | \text{(I) } 8\quad</math>
| |
− |
| |
− | <math>\text{(J) }9 \quad
| |
− | \text{(K) }10 \quad
| |
− | \text{(L) }11 \quad
| |
− | \text{(M) }12 \quad
| |
− | \text{(N) }13 \quad
| |
− | \text{(O) }14 \quad
| |
− | \text{(P) }15 \quad
| |
− | \text{(Q) }16 \quad</math>
| |
− |
| |
− | <math>\text{(R) }17 \quad
| |
− | \text{(S) }18 \quad
| |
− | \text{(T) }19 \quad
| |
− | \text{(U) }20 \quad
| |
− | \text{(V) }21 \quad
| |
− | \text{(W) }22 \quad
| |
− | \text{(X) }23 </math>
| |
− |
| |
− |
| |
− | [[2007 iTest Problems/Problem 24|Solution]]
| |
− |
| |
− | ===Problem 25===
| |
− |
| |
− | Ted's favorite number is equal to
| |
− | <cmath>1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}</cmath>
| |
− |
| |
− | Find the remainder when Ted's favorite number is divided by 25.
| |
− |
| |
− | <math>\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</math>
| |
− |
| |
− | <math>\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</math>
| |
− |
| |
− | <math>\text{(R) } 17\qquad
| |
− | \text{(S) } 18\qquad
| |
− | \text{(T) } 19\qquad
| |
− | \text{(U) } 20\qquad
| |
− | \text{(V) } 21\qquad
| |
− | \text{(W) } 22\qquad
| |
− | \text{(X) } 23\qquad
| |
− | \text{(Y) } 24</math>
| |
− |
| |
− |
| |
− | [[2007 iTest Problems/Problem 25|Solution]]
| |
− |
| |
| ==Short Answer Section== | | ==Short Answer Section== |
| | | |
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?
Solution
Problem 27
The face diagonal of a cube has length
. Find the value of n given that
is the
of the cube.
Solution
Problem 28
The space diagonal (interior diagonal) of a cube has length 6. Find the
of the cube.
Solution
Problem 29
Let
be equal to the sum
. Find the remainder when
is divided by
.
Solution
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
.
Solution
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.
Solution
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?
Solution
Problem 33
How many
four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?
Solution
Problem 34
Let
be the probability that a randomly selected divisor of
is a multiple of
. If
and
are relatively prime positive integers, find
.
Solution
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.
Solution
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
.
Solution
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.
Solution
Problem 38
Find the largest positive integer that is equal to the cube of the sum of its digits.
Solution
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
.
Solution
Problem 40
Let
be the sum of all
such that
and
. Compute
.
Solution
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
.
Solution
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
.
Solution
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.
Solution
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
.
Solution
Problem 45
Find the sum of all positive integers
such that
, where
represent distinct base
digits,
.
Solution
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
.
Solution
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
.
Solution
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
.
Solution
Problem 49
How many
-element subsets of
are there, the sum of whose elements is divisible by
?
Solution
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
).
Solution
Ultimate Question
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
Solution
Problem 52
Let
. Let
be the region consisting of points
of the Cartesian plane satisfying both
and
. Find the area of region
.
Solution
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
.
Solution
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
.
Solution
Problem 55
Let
, and let
. Let
be the smallest real solution of
. Find the value of
.
Solution
Problem 56
Let
. In the binary expansion of
, how many of the first
digits to the right of the radix point are
's?
Solution
Problem 57
Let
. How many positive integers are within
of exactly
perfect squares? (Note:
is considered a perfect square.)
Solution
Problem 58
Let
. For natural numbers
, we define
Compute the value of
.
Solution
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
.
Solution
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
.
Solution
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.
Solution
Problem TB2
Factor completely over integer coefficients the polynomial
. Demonstrate that your factorization is complete.
Solution
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.
Solution
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.
Solution
See Also