Difference between revisions of "Mock AIME 5 2005-2006 Problems"
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− | + | == Problem 1 == | |
+ | Suppose <math>n</math> is a positive integer. Let <math>f(n)</math> be the sum of the distinct positive prime divisors of <math>n</math> less than <math>50</math> (e.g. <math>f(12) = 2+3 = 5</math> and <math>f(101) = 0</math>). Evaluate the remainder when <math>f(1)+f(2)+\cdots+f(99)</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 1|Solution]] | ||
+ | |||
+ | == Problem 2 == | ||
+ | A circle <math>\omega_1</math> of radius <math>6\sqrt{2}</math> is internally tangent to a larger circle <math>\omega_2</math> of radius <math>12\sqrt{2}</math> such that the center of <math>\omega_2</math> lies on <math>\omega_1</math>. A diameter <math>AB</math> of <math>\omega_2</math> is drawn tangent to <math>\omega_1</math>. A second line <math>l</math> is drawn from <math>B</math> tangent to <math>\omega_1</math>. Let the line tangent to <math>\omega_2</math> at <math>A</math> intersect <math>l</math> at <math>C</math>. Find the area of <math>\triangle ABC</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 2|Solution]] | ||
+ | |||
+ | == Problem 3 == | ||
+ | A <math>\emph hailstone</math> number <math>n = d_1d_2 \cdots d_k</math>, where <math>d_i</math> denotes the <math>i</math>th digit in the base-<math>10</math> representation of <math>n</math> for <math>i = 1,2, \ldots,k</math>, is a positive integer with distinct nonzero digits such that <math>d_m < d_{m-1}</math> if <math>m</math> is even and <math>d_m > d_{m-1}</math> if <math>m</math> is odd for <math>m = 1,2,\ldots,k</math> (and <math>d_0 = 0</math>). Let <math>a</math> be the number of four-digit hailstone numbers and <math>b</math> be the number of three-digit hailstone numbers. Find <math>a+b</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 3|Solution]] | ||
+ | |||
+ | == Problem 4 == | ||
+ | Let <math>m</math> and <math>n</math> be integers such that <math>1 < m \le 10</math> and <math>m < n \le 100</math>. Given that <math>x = \log_m{n}</math> and <math>y = \log_n{m}</math>, find the number of ordered pairs <math>(m,n)</math> such that <math> \displaystyle \lfloor x \rfloor = \lceil y \rceil</math>. (<math>\lfloor a \rfloor</math> is the greatest integer less than or equal to <math>a</math> and <math>\lceil a \rceil</math> is the least integer greater than or equal to <math>a</math>). | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 4|Solution]] | ||
+ | |||
+ | == Problem 5 == | ||
+ | Find the largest prime divisor of <math>25^2+72^2</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 5|Solution]] | ||
+ | |||
+ | == Problem 6 == | ||
+ | <math>P_1</math>, <math>P_2</math>, and <math>P_3</math> are polynomials defined by: | ||
+ | |||
+ | : <math>P_1(x) = 1+x+x^3+x^4+\cdots+x^{96}+x^{97}+x^{99}+x^{100}</math> | ||
+ | : <math>P_2(x) = 1-x+x^2-\cdots-x^{99}+x^{100}</math> | ||
+ | : <math>P_3(x) = 1+x+x^2+\cdots+x^{66}+x^{67}</math> | ||
+ | |||
+ | Find the number of distinct complex roots of <math>P_1 \cdot P_2 \cdot P_3</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 6|Solution]] | ||
+ | |||
+ | == Problem 7 == | ||
+ | A coin of radius <math>1</math> is flipped onto an <math>500 \times 500</math> square grid divided into <math>2500</math> equal squares. Circles are inscribed in <math>n</math> of these <math>2500</math> squares. Let <math>P_n</math> be the probability that, given that the coin lands completely within one of the smaller squares, it also lands completely within one of the circles. Let <math>P</math> be the probability that, when flipped onto the grid, the coin lands completely within one of the smaller squares. Let <math>n_0</math> smallest value of <math>n</math> such that <math>P_n > P</math>. Find the value of <math>\displaystyle \left\lfloor \frac{n_0}{3} \right\rfloor</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 7|Solution]] | ||
+ | |||
+ | == Problem 8 == | ||
+ | Let <math>P</math> be a polyhedron with <math>37</math> faces, all of which are equilateral triangles, squares, or regular pentagons with equal side length. Given there is at least one of each type of face and there are twice as many pentagons as triangles, what is the sum of all the possible number of vertices <math>P</math> can have? | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 8|Solution]] | ||
+ | |||
+ | == Problem 9 == | ||
+ | <math>13</math> nondistinguishable residents are moving into <math>7</math> distinct houses in Conformistville, with at least one resident per house. In how many ways can the residents be assigned to these houses such that there is at least one house with <math>4</math> residents? | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 9|Solution]] | ||
+ | |||
+ | == Problem 10 == | ||
+ | Find the smallest positive integer <math>n</math> such that <math>\displaystyle {2n \choose n}</math> is divisible by all the primes between <math>10</math> and <math>30</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 10|Solution]] | ||
+ | |||
+ | == Problem 11 == | ||
+ | Let <math>A</math> be a subset of consecutive elements of <math>S = \{n, n+1, \ldots, n+999\}</math> where <math>n</math> is a positive integer. Define <math>\displaystyle \mu(A) = \sum_{k \in A} \tau(k)</math>, where <math>\tau(k) = 1</math> if <math>k</math> has an odd number of divisors and <math>\tau(k) = 0</math> if <math>k</math> has an even number of divisors. For how many <math>n \le 1000</math> does there exist an <math>A</math> such that <math>|A| = 620</math> and <math>\mu(A) = 11</math>? (<math>|X|</math> denotes the cardinality of the set <math>X</math>, or the number of elements in <math>X</math>) | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 11|Solution]] | ||
+ | |||
+ | == Problem 12 == | ||
+ | Let <math>ABC</math> be a triangle with <math>AB = 13</math>, <math>BC = 14</math>, and <math>AC = 15</math>. Let <math>D</math> be the foot of the altitude from <math>A</math> to <math>BC</math> and <math>E</math> be the point on <math>BC</math> between <math>D</math> and <math>C</math> such that <math>BD = CE</math>. Extend <math>AE</math> to meet the circumcircle of <math>ABC</math> at <math>F</math>. If the area of triangle <math>FAC</math> is <math>\displaystyle \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 12|Solution]] | ||
+ | |||
+ | == Problem 13 == | ||
+ | Let <math>S</math> be the set of positive integers with only odd digits satisfying the following condition: any <math>x \in S</math> with <math>n</math> digits must be divisible by <math>5^n</math>. Let <math>A</math> be the sum of the <math>20</math> smallest elements of <math>S</math>. Find the remainder upon dividing <math>A</math> by <math>1000</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 13|Solution]] | ||
+ | |||
+ | == Problem 14 == | ||
+ | Let <math>ABC</math> be a triangle such that <math>AB = 68</math>, <math>BC = 100</math>, and <math>\displaystyle CA = 112</math>. Let <math>H</math> be the orthocenter of <math>\triangle ABC</math> (intersection of the altitudes). Let <math>D</math> be the midpoint of <math>BC</math>, <math>E</math> be the midpoint of <math>CA</math>, and <math>F</math> be the midpoint of <math>AB</math>. Points <math>X</math>, <math>Y</math>, and <math>Z</math> are constructed on <math>HD</math>, <math>HE</math>, and <math>HF</math>, respectively, such that <math>D</math> is the midpoint of <math>XH</math>, <math>E</math> is the midpoint of <math>YH</math>, and <math>F</math> is the midpoint of <math>ZH</math>. Find <math>AX+BY+CZ</math>. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 14|Solution]] | ||
+ | |||
+ | == Problem 15 == | ||
+ | <math>2006</math> colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled <math>a_0</math>, <math>a_1</math>, <math>\ldots</math>, <math>a_{2005}</math> around the circle in order. Two beads <math>a_i</math> and <math>a_j</math>, where <math>i</math> and <math>j</math> are non-negative integers, satisfy <math>a_i = a_j</math> if and only if the color of <math>a_i</math> is the same as the color of <math>a_j</math>. Given that there exists no non-negative integer <math>m < 2006</math> and positive integer <math>n < 1003</math> such that <math>a_m = a_{m-n} = a_{m+n}</math>, where all subscripts are taken <math>\pmod{2006}</math>, find the minimum number of different colors of beads on the necklace. | ||
+ | |||
+ | [[Mock AIME 5 2005-2006 Problems/Problem 15|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | * [[Mock AIME]] | ||
+ | * A [http://wangsblog.com/jeffrey/MockAIME5.pdf .pdf] version of the problems | ||
+ | |||
+ | {{Mock AIME box|year=2005-2006|n=5|source=76847|before=[[Mock AIME 4 2005-2006]]|after=[[Mock AIME 1 2006-2007]]}} |
Latest revision as of 20:52, 27 February 2007
Contents
Problem 1
Suppose is a positive integer. Let be the sum of the distinct positive prime divisors of less than (e.g. and ). Evaluate the remainder when is divided by .
Problem 2
A circle of radius is internally tangent to a larger circle of radius such that the center of lies on . A diameter of is drawn tangent to . A second line is drawn from tangent to . Let the line tangent to at intersect at . Find the area of .
Problem 3
A number , where denotes the th digit in the base- representation of for , is a positive integer with distinct nonzero digits such that if is even and if is odd for (and ). Let be the number of four-digit hailstone numbers and be the number of three-digit hailstone numbers. Find .
Problem 4
Let and be integers such that and . Given that and , find the number of ordered pairs such that . ( is the greatest integer less than or equal to and is the least integer greater than or equal to ).
Problem 5
Find the largest prime divisor of .
Problem 6
, , and are polynomials defined by:
Find the number of distinct complex roots of .
Problem 7
A coin of radius is flipped onto an square grid divided into equal squares. Circles are inscribed in of these squares. Let be the probability that, given that the coin lands completely within one of the smaller squares, it also lands completely within one of the circles. Let be the probability that, when flipped onto the grid, the coin lands completely within one of the smaller squares. Let smallest value of such that . Find the value of .
Problem 8
Let be a polyhedron with faces, all of which are equilateral triangles, squares, or regular pentagons with equal side length. Given there is at least one of each type of face and there are twice as many pentagons as triangles, what is the sum of all the possible number of vertices can have?
Problem 9
nondistinguishable residents are moving into distinct houses in Conformistville, with at least one resident per house. In how many ways can the residents be assigned to these houses such that there is at least one house with residents?
Problem 10
Find the smallest positive integer such that is divisible by all the primes between and .
Problem 11
Let be a subset of consecutive elements of where is a positive integer. Define , where if has an odd number of divisors and if has an even number of divisors. For how many does there exist an such that and ? ( denotes the cardinality of the set , or the number of elements in )
Problem 12
Let be a triangle with , , and . Let be the foot of the altitude from to and be the point on between and such that . Extend to meet the circumcircle of at . If the area of triangle is , where and are relatively prime positive integers, find .
Problem 13
Let be the set of positive integers with only odd digits satisfying the following condition: any with digits must be divisible by . Let be the sum of the smallest elements of . Find the remainder upon dividing by .
Problem 14
Let be a triangle such that , , and . Let be the orthocenter of (intersection of the altitudes). Let be the midpoint of , be the midpoint of , and be the midpoint of . Points , , and are constructed on , , and , respectively, such that is the midpoint of , is the midpoint of , and is the midpoint of . Find .
Problem 15
colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled , , , around the circle in order. Two beads and , where and are non-negative integers, satisfy if and only if the color of is the same as the color of . Given that there exists no non-negative integer and positive integer such that , where all subscripts are taken , find the minimum number of different colors of beads on the necklace.
See also
Mock AIME 5 2005-2006 (Problems, Source) | ||
Preceded by Mock AIME 4 2005-2006 |
Followed by Mock AIME 1 2006-2007 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |