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Difference between revisions of "Mock AIME 5 2005-2006 Problems"

 
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\begin{center}
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{\huge 2006 Mock AIME 5} \\
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{\large Prepared by Jeffrey Wang (paladin8)}
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\end{center}
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<math>1</math>. 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>.
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<math>2</math>. 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>.
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<math>3</math>. A {\em hailstone} 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 {\em hailstone} numbers and <math>b</math> be the number of three-digit {\em hailstone} numbers. Find <math>a+b</math>.
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<math>4</math>. 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>).
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<math>5</math>. Find the largest prime divisor of <math>25^2+72^2</math>.
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<math>6</math>. <math>P_1</math>, <math>P_2</math>, and <math>P_3</math> are polynomials defined by:
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\begin{center}
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<math>P_1(x) = 1+x+x^3+x^4+\cdots+x^{96}+x^{97}+x^{99}+x^{100}</math> \\
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<math>P_2(x) = 1-x+x^2-\cdots-x^{99}+x^{100}</math> \\
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<math>P_3(x) = 1+x+x^2+\cdots+x^{66}+x^{67}</math>.
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\end{center}
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Find the number of distinct complex roots of <math>P_1 \cdot P_2 \cdot P_3</math>.
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<math>7</math>. 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>.
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<math>8</math>. 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?
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<math>9</math>. <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?
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<math>10</math>. Find the smallest positive integer <math>n</math> such that <math>\displaystyle \binom{2n}{n}</math> is divisible by all the primes between <math>10</math> and <math>30</math>.
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<math>11</math>. 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>)
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<math>12</math>. 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>.
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<math>13</math>. 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>.
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<math>14</math>. Let <math>ABC</math> be a triangle such that <math>AB = 68</math>, <math>BC = 100</math>, and <math>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>.
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<math>15</math>. <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.
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The problems can be found [http://wangsblog.com/jeffrey/MockAIME5.pdf here].
 
The problems can be found [http://wangsblog.com/jeffrey/MockAIME5.pdf here].

Revision as of 03:15, 25 February 2007

\begin{center} {\huge 2006 Mock AIME 5} \\ {\large Prepared by Jeffrey Wang (paladin8)} \end{center}

$1$. Suppose $n$ is a positive integer. Let $f(n)$ be the sum of the distinct positive prime divisors of $n$ less than $50$ (e.g. $f(12) = 2+3 = 5$ and $f(101) = 0$). Evaluate the remainder when $f(1)+f(2)+\cdots+f(99)$ is divided by $1000$.

\vskip .6in

$2$. A circle $\omega_1$ of radius $6\sqrt{2}$ is internally tangent to a larger circle $\omega_2$ of radius $12\sqrt{2}$ such that the center of $\omega_2$ lies on $\omega_1$. A diameter $AB$ of $\omega_2$ is drawn tangent to $\omega_1$. A second line $l$ is drawn from $B$ tangent to $\omega_1$. Let the line tangent to $\omega_2$ at $A$ intersect $l$ at $C$. Find the area of $\triangle ABC$.

\vskip .6in

$3$. A {\em hailstone} number $n = d_1d_2 \cdots d_k$, where $d_i$ denotes the $i$th digit in the base-$10$ representation of $n$ for $i = 1,2, \ldots,k$, is a positive integer with distinct nonzero digits such that $d_m < d_{m-1}$ if $m$ is even and $d_m > d_{m-1}$ if $m$ is odd for $m = 1,2,\ldots,k$ (and $d_0 = 0$). Let $a$ be the number of four-digit {\em hailstone} numbers and $b$ be the number of three-digit {\em hailstone} numbers. Find $a+b$.

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$4$. Let $m$ and $n$ be integers such that $1 < m \le 10$ and $m < n \le 100$. Given that $x = \log_m{n}$ and $y = \log_n{m}$, find the number of ordered pairs $(m,n)$ such that $\displaystyle \lfloor x \rfloor = \lceil y \rceil$. ($\lfloor a \rfloor$ is the greatest integer less than or equal to $a$ and $\lceil a \rceil$ is the least integer greater than or equal to $a$).

\vskip .6in

$5$. Find the largest prime divisor of $25^2+72^2$.

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$6$. $P_1$, $P_2$, and $P_3$ are polynomials defined by:

\begin{center} $P_1(x) = 1+x+x^3+x^4+\cdots+x^{96}+x^{97}+x^{99}+x^{100}$ \\ $P_2(x) = 1-x+x^2-\cdots-x^{99}+x^{100}$ \\ $P_3(x) = 1+x+x^2+\cdots+x^{66}+x^{67}$. \end{center}

Find the number of distinct complex roots of $P_1 \cdot P_2 \cdot P_3$.

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$7$. A coin of radius $1$ is flipped onto an $500 \times 500$ square grid divided into $2500$ equal squares. Circles are inscribed in $n$ of these $2500$ squares. Let $P_n$ 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 $P$ be the probability that, when flipped onto the grid, the coin lands completely within one of the smaller squares. Let $n_0$ smallest value of $n$ such that $P_n > P$. Find the value of $\displaystyle \left\lfloor \frac{n_0}{3} \right\rfloor$.

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$8$. Let $P$ be a polyhedron with $37$ 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 $P$ can have?

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$9$. $13$ nondistinguishable residents are moving into $7$ 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 $4$ residents?

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$10$. Find the smallest positive integer $n$ such that $\displaystyle \binom{2n}{n}$ is divisible by all the primes between $10$ and $30$.

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$11$. Let $A$ be a subset of consecutive elements of $S = \{n, n+1, \ldots, n+999\}$ where $n$ is a positive integer. Define $\displaystyle \mu(A) = \sum_{k \in A} \tau(k)$, where $\tau(k) = 1$ if $k$ has an odd number of divisors and $\tau(k) = 0$ if $k$ has an even number of divisors. For how many $n \le 1000$ does there exist an $A$ such that $|A| = 620$ and $\mu(A) = 11$? ($|X|$ denotes the cardinality of the set $X$, or the number of elements in $X$)

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$12$. Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$ and $E$ be the point on $BC$ between $D$ and $C$ such that $BD = CE$. Extend $AE$ to meet the circumcircle of $ABC$ at $F$. If the area of triangle $FAC$ is $\displaystyle \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

\vskip .6in

$13$. Let $S$ be the set of positive integers with only odd digits satisfying the following condition: any $x \in S$ with $n$ digits must be divisible by $5^n$. Let $A$ be the sum of the $20$ smallest elements of $S$. Find the remainder upon dividing $A$ by $1000$.

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$14$. Let $ABC$ be a triangle such that $AB = 68$, $BC = 100$, and $CA = 112$. Let $H$ be the orthocenter of $\triangle ABC$ (intersection of the altitudes). Let $D$ be the midpoint of $BC$, $E$ be the midpoint of $CA$, and $F$ be the midpoint of $AB$. Points $X$, $Y$, and $Z$ are constructed on $HD$, $HE$, and $HF$, respectively, such that $D$ is the midpoint of $XH$, $E$ is the midpoint of $YH$, and $F$ is the midpoint of $ZH$. Find $AX+BY+CZ$.

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$15$. $2006$ colored beads are placed on a necklace (circular ring) such that each bead is adjacent to two others. The beads are labeled $a_0$, $a_1$, $\ldots$, $a_{2005}$ around the circle in order. Two beads $a_i$ and $a_j$, where $i$ and $j$ are non-negative integers, satisfy $a_i = a_j$ if and only if the color of $a_i$ is the same as the color of $a_j$. Given that there exists no non-negative integer $m < 2006$ and positive integer $n < 1003$ such that $a_m = a_{m-n} = a_{m+n}$, where all subscripts are taken $\pmod{2006}$, find the minimum number of different colors of beads on the necklace.

The problems can be found here.

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