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| − | {{AIME Problems|year=2010|n=I}}
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| − | == Problem 1 ==
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| − | Maya lists all the positive divisors of <math>2010^2</math>. She then randomly selects two distinct divisors from this list. Let <math>p</math> be the probability that exactly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
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| − | [[2010 AIME I Problems/Problem 1|Solution]]
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| − | == Problem 2 ==
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| − | Find the remainder when <math>9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}</math> is divided by <math>1000</math>.
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| − | [[2010 AIME I Problems/Problem 2|Solution]]
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| − | == Problem 3 ==
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| − | Suppose that <math>y = \frac34x</math> and <math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>.
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| − | [[2010 AIME I Problems/Problem 3|Solution]]
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| − | == Problem 4 ==
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| − | Jackie and Phil have two fair coins and a third coin that comes up heads with probability <math>\frac47</math>. Jackie flips the three coins, and then Phil flips the three coins. Let <math>\frac {m}{n}</math> be the probability that Jackie gets the same number of heads as Phil, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
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| − | [[2010 AIME I Problems/Problem 4|Solution]]
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| − | == Problem 5 ==
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| − | Positive integers <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> satisfy <math>a > b > c > d</math>, <math>a + b + c + d = 2010</math>, and <math>a^2 - b^2 + c^2 - d^2 = 2010</math>. Find the number of possible values of <math>a</math>.
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| − | [[2010 AIME I Problems/Problem 5|Solution]]
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| − | == Problem 6 ==
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| − | How many positive integers <math>N</math> less than <math>1000</math> are there such that the equation <math>x^{\lfloor x\rfloor} = N</math> has a solution for <math>x</math>? (The notation <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.)
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| − | [[2010 AIME I Problems/Problem 6|Solution]]
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| − | == Problem 7 ==
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| − | Define an ordered triple <math>(A, B, C)</math> of sets to be minimally intersecting if <math>|A \cap B| = |B \cap C| = |C \cap A| = 1</math> and <math>A \cap B \cap C = \emptyset</math>. For example, <math>(\{1,2\},\{2,3\},\{1,3,4\})</math> is a minimally intersecting triple. Let <math>N</math> be the number of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>.
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| − | '''Note''': <math>|S|</math> represents the number of elements in the set <math>S</math>.
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| − | [[2010 AIME I Problems/Problem 7|Solution]]
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| − | == Problem 8 ==
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| − | For a real number <math>a</math>, let <math>\lfloor a \rfloor</math> denominate the greatest integer less than or equal to <math>a</math>. Let <math>\mathcal{R}</math> denote the region in the coordinate plane consisting of points <math>(x,y)</math> such that <math>\lfloor x \rfloor ^2 + \lfoor y \rfloor ^2 = 25</math>. The region <math>\mathcal{R}</math> is completely contained in a disk of radius <math>r</math> (a disk is the union of a circle and its interior). The minimum value of <math>r</math> can be written as <math>\frac {\sqrt {m}}{n}</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by the square of any prime. Find <math>m + n</math>.
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| − | [[2010 AIME I Problems/Problem 8|Solution]]
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| − | == Problem 9 ==
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| − | Let <math>(a,b,c)</math> be the real solution of the system of equations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
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| − | [[2010 AIME I Problems/Problem 9|Solution]]
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| − | == Problem 10 ==
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| − | Let <math>N</math> be the number of ways to write <math>2010</math> in the form <math>2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0</math>, where the <math>a_i</math>'s are integers, and <math>0 \le a_i \le 99</math>. An example of such a representation is <math>1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0</math>. Find <math>N</math>.
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| − | [[2010 AIME I Problems/Problem 10|Solution]]
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| − | == Problem 11 ==
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| − | Let <math>\mathcal{R}</math> be the region consisting of the set of points in the coordinate plane that satisfy both <math>|8 - x| + y \le 10</math> and <math>3y - x \ge 15</math>. When <math>\mathcal{R}</math> is revolved around the line whose equation is <math>3y - x = 15</math>, the volume of the resulting solid is <math>\frac {m\pi}{n\sqrt {p}}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
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| − | [[2010 AIME I Problems/Problem 11|Solution]]
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| − | == Problem 12 ==
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| − | Let <math>M \ge 3</math> be an integer and let <math>S = \{3,4,5,\ldots,m\}</math>. Find the smallest value of <math>m</math> such that for every partition of <math>S</math> into two subsets, at least one of the subsets contains integers <math>a</math>, <math>b</math>, and <math>c</math> (not necessarily distinct) such that <math>ab = c</math>.
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| − | '''Note''': a partition of <math>S</math> is a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>.
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| − | [[2010 AIME I Problems/Problem 12|Solution]]
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| − | == Problem 13 ==
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| − | Rectangle <math>ABCD</math> and a semicircle with diameter <math>AB</math> are coplanar and have nonoverlapping interiors. Let <math>\mathcal{R}</math> denote the region enclosed by the semicircle and the rectangle. Line <math>\ell</math> meets the semicircle, segment <math>AB</math>, and segment <math>CD</math> at distinct points <math>N</math>, <math>U</math>, and <math>T</math>, respectively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose that <math>AU = 84</math>, <math>AN = 126</math>, and <math>UB = 168</math>. Then <math>DA</math> can be represented as <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>.
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| − | [[2010 AIME I Problems/Problem 13|Solution]]
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| − | == Problem 14 ==
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| − | For each positive integer n, let <math>f(n) = \sum_{k = 1}^{\infty} \lfloor log_{10} (kn) \rfloor</math>. Find the largest value of n for which <math>f(n) \le 300</math>.
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| − | '''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.
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| − | [[2010 AIME I Problems/Problem 14|Solution]]
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| − | == Problem 15 ==
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| − | In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac {AM}{CM} = \frac {p}{q}</math>. Find <math>p + q</math>.
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| − | [[2010 AIME I Problems/Problem 15|Solution]]
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| − | == See also ==
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| − | * [[American Invitational Mathematics Examination]]
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| − | * [[AIME Problems and Solutions]]
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| − | * [[Mathematics competition resources]]
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