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Difference between revisions of "Mock AIME 6 2006-2007 Problems"

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==Problem 1==
 
==Problem 1==
  
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Let <math>T</math> be the sum of all positive integers of the form <math>2^r\cdot3^s</math>, where <math>r</math> and <math>s</math> are nonnegative integers that do not exceed <math>4</math>.  Find the remainder when <math>T</math> is divided by 1000.
+
Let <math>T</math> be the sum of all positive integers of the form <math>2^r\cdot3^s</math>, where <math>r</math> and <math>s</math> are nonnegative integers that do not exceed <math>4</math>.  Find the remainder when <math>T</math> is divided by <math>1000</math>.
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 1|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 1|Solution]]
  
 
==Problem 2==  
 
==Problem 2==  
 +
Draw in the diagonals of a regular octagon.  What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 2|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
 +
Alvin, Simon, and Theodore are running around a <math>1000</math>-meter circular track starting at different positions.  Alvin is running in the opposite direction of Simon and Theodore.  He is also the fastest, running twice as fast as Simon and three times as fast as Theodore.  If Alvin meets Simon for the first time after running <math>312</math> meters, and Simon meets Theodore for the first time after running <math>2526</math> meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 3|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
 +
Let <math>R</math> be a set of <math>13</math> points in the plane, no three of which lie on the same line.  At most how many ordered triples of points <math>(A,B,C)</math> in <math>R</math> exist such that <math>\angle ABC</math> is obtuse?
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 4|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
 +
Let <math>S(n)</math> be the sum of the squares of the digits of <math>n</math>.  How many positive integers <math>n>2007</math> satisfy the inequality <math>n-S(n)\le 2007</math>?
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 5|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
 +
<math>C_1</math> is a circle with radius <math>164</math> and <math>C_2</math> is a circle internally tangent to <math>C_1</math> that passes through the center of <math>C_1</math>.  <math>\overline{AB}</math> is a chord in <math>C_1</math> of length <math>320</math> tangent to <math>C_2</math> at <math>D</math> where <math>AD>BD</math>.  Given that <math>BD=a-b\sqrt{c}</math> where <math>a,b,c</math> are positive integers and <math>c</math> is not divisible by the square of any prime, what is <math>a+b+c</math>?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 6|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
 +
Let <math>P_n(x)=1+x+x^2+\cdots+x^n</math> and <math>Q_n(x)=P_1\cdot P_2\cdots P_n</math> for all integers <math>n\ge 1</math>.  How many more distinct complex roots does <math>Q_{1004}</math> have than <math>Q_{1003}</math>?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 7|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
 +
A sequence of positive reals defined by <math>a_0=x</math>, <math>a_1=y</math>, and <math>a_n\cdot a_{n+2}=a_{n+1}</math> for all integers <math>n\ge 0</math>.  Given that <math>a_{2007}+a_{2008}=3</math> and <math>a_{2007}\cdot a_{2008}=\frac 13</math>, find <math>x^3+y^3</math>.
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 8|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
<math>ABC</math> is a triangle with integer side lengths.  Extend <math>\overline{AC}</math> beyond <math>C</math> to point <math>D</math> such that <math>CD=120</math>.  Similarly, extend <math>\overline{CB}</math> beyond <math>B</math> to point <math>E</math> such that <math>BE=112</math> and <math>\overline{BA}</math> beyond <math>A</math> to point <math>F</math> such that <math>AF=104</math>.  If triangles <math>CBD</math>, <math>BAE</math>, and <math>ACF</math> all have the same area, what is the minimum possible area of triangle <math>ABC</math>?
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 9|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
Given a point <math>P</math> in the coordinate plane, let <math>T_k(P)</math> be the <math>90^\circ</math> rotation of <math>P</math> around the point <math>(2000-k,k)</math>.  Let <math>P_0</math> be the point <math>(2007,0)</math> and <math>P_{n+1}=T_n(P_n)</math> for all integers <math>n\ge 0</math>.  If <math>P_m</math> has a <math>y</math>-coordinate of <math>433</math>, what is <math>m</math>?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 10|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
Each face of an octahedron is randomly colored blue or red.  A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges.  The probability that it can do so without traveling along an edge that is shared by two faces of the same color is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 11|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
Let <math>x_k</math> be the largest positive rational solution <math>x</math> to the equation <math>(2007-x)(x+2007^{-k})^k=1</math> for all integers <math>k\ge 2</math>.  For each <math>k</math>, let <math>x_k=\frac{a_k}{b_k}</math>, where <math>a_k</math> and <math>b_k</math> are relatively prime positive integers.  If <cmath>S=\sum_{k=2}^{2007} (2007b_k-a_k),</cmath> what is the remainder when <math>S</math> is divided by <math>1000</math>?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 12|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
Consider two circles of different sizes that do not intersect.  The smaller circle has center <math>O</math>.  Label the intersection of their common external tangents <math>P</math>.  A common internal tangent interesects the common external tangents at points <math>A</math> and <math>B</math>.  Given that the radius of the larger circle is <math>11</math>, <math>PO=3</math>, and <math>AB=20\sqrt{2}</math>, what is the square of the area of triangle <math>PBA</math>?
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 13|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
A rational <math>\frac{1}{k}</math>, where <math>k</math> is a positive integer, is said to be <math>\textit{n-unsound}</math> if its base <math>N</math> representation terminates.  Let <math>S_n</math> be the set of all <math>\textit{n-unsound}</math> rationals.  The sum of all the elements in the union set <math>S_2\cup S_3\cup\cdots\cup S_{14}</math> is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m+n</math>.
 +
 
[[Mock AIME 6 2006-2007 Problems/Problem 14|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
For any finite sequence of positive integers <math>A=(a_1,a_2,\cdots,a_n)</math>, let <math>f(A)</math> be the sequence of the differences between consecutive terms of <math>A</math>. i.e. <math>f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})</math>.  Let <math>F^k(A)</math> denote <math>F</math> applied <math>k</math> times to <math>A</math>.  If all of the sequences <math>A, f(A), f^2(A),\cdots, f^{n-2}(A)</math> are strictly increasing and the only term of <math>f^{n01}(A)</math> is <math>1</math>, we call the sequence <math>A</math> <math>\textit{superpositive}</math>.  How many sequences <math>A</math> with at least two terms and no terms exceeding <math>18</math> are <math>\textit{superpositive}</math>?
  
 
[[Mock AIME 6 2006-2007 Problems/Problem 15|Solution]]
 
[[Mock AIME 6 2006-2007 Problems/Problem 15|Solution]]

Latest revision as of 20:04, 7 December 2018

Problem 1

Let $T$ be the sum of all positive integers of the form $2^r\cdot3^s$, where $r$ and $s$ are nonnegative integers that do not exceed $4$. Find the remainder when $T$ is divided by $1000$.

Solution

Problem 2

Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?

Solution

Problem 3

Alvin, Simon, and Theodore are running around a $1000$-meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running $312$ meters, and Simon meets Theodore for the first time after running $2526$ meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?

Solution

Problem 4

Let $R$ be a set of $13$ points in the plane, no three of which lie on the same line. At most how many ordered triples of points $(A,B,C)$ in $R$ exist such that $\angle ABC$ is obtuse?

Solution

Problem 5

Let $S(n)$ be the sum of the squares of the digits of $n$. How many positive integers $n>2007$ satisfy the inequality $n-S(n)\le 2007$?

Solution

Problem 6

$C_1$ is a circle with radius $164$ and $C_2$ is a circle internally tangent to $C_1$ that passes through the center of $C_1$. $\overline{AB}$ is a chord in $C_1$ of length $320$ tangent to $C_2$ at $D$ where $AD>BD$. Given that $BD=a-b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime, what is $a+b+c$?

Solution

Problem 7

Let $P_n(x)=1+x+x^2+\cdots+x^n$ and $Q_n(x)=P_1\cdot P_2\cdots P_n$ for all integers $n\ge 1$. How many more distinct complex roots does $Q_{1004}$ have than $Q_{1003}$?

Solution

Problem 8

A sequence of positive reals defined by $a_0=x$, $a_1=y$, and $a_n\cdot a_{n+2}=a_{n+1}$ for all integers $n\ge 0$. Given that $a_{2007}+a_{2008}=3$ and $a_{2007}\cdot a_{2008}=\frac 13$, find $x^3+y^3$.

Solution

Problem 9

$ABC$ is a triangle with integer side lengths. Extend $\overline{AC}$ beyond $C$ to point $D$ such that $CD=120$. Similarly, extend $\overline{CB}$ beyond $B$ to point $E$ such that $BE=112$ and $\overline{BA}$ beyond $A$ to point $F$ such that $AF=104$. If triangles $CBD$, $BAE$, and $ACF$ all have the same area, what is the minimum possible area of triangle $ABC$?

Solution

Problem 10

Given a point $P$ in the coordinate plane, let $T_k(P)$ be the $90^\circ$ rotation of $P$ around the point $(2000-k,k)$. Let $P_0$ be the point $(2007,0)$ and $P_{n+1}=T_n(P_n)$ for all integers $n\ge 0$. If $P_m$ has a $y$-coordinate of $433$, what is $m$?

Solution

Problem 11

Each face of an octahedron is randomly colored blue or red. A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges. The probability that it can do so without traveling along an edge that is shared by two faces of the same color is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 12

Let $x_k$ be the largest positive rational solution $x$ to the equation $(2007-x)(x+2007^{-k})^k=1$ for all integers $k\ge 2$. For each $k$, let $x_k=\frac{a_k}{b_k}$, where $a_k$ and $b_k$ are relatively prime positive integers. If \[S=\sum_{k=2}^{2007} (2007b_k-a_k),\] what is the remainder when $S$ is divided by $1000$?

Solution

Problem 13

Consider two circles of different sizes that do not intersect. The smaller circle has center $O$. Label the intersection of their common external tangents $P$. A common internal tangent interesects the common external tangents at points $A$ and $B$. Given that the radius of the larger circle is $11$, $PO=3$, and $AB=20\sqrt{2}$, what is the square of the area of triangle $PBA$?

Solution

Problem 14

A rational $\frac{1}{k}$, where $k$ is a positive integer, is said to be $\textit{n-unsound}$ if its base $N$ representation terminates. Let $S_n$ be the set of all $\textit{n-unsound}$ rationals. The sum of all the elements in the union set $S_2\cup S_3\cup\cdots\cup S_{14}$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 15

For any finite sequence of positive integers $A=(a_1,a_2,\cdots,a_n)$, let $f(A)$ be the sequence of the differences between consecutive terms of $A$. i.e. $f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})$. Let $F^k(A)$ denote $F$ applied $k$ times to $A$. If all of the sequences $A, f(A), f^2(A),\cdots, f^{n-2}(A)$ are strictly increasing and the only term of $f^{n01}(A)$ is $1$, we call the sequence $A$ $\textit{superpositive}$. How many sequences $A$ with at least two terms and no terms exceeding $18$ are $\textit{superpositive}$?

Solution

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