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Difference between revisions of "Mock AIME 1 2006-2007/Problems"
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1. <math>\triangle ABC</math> has positive integer side lengths of <math>x</math>,<math>y</math>, and <math>17</math>.  The angle bisector of <math>\angle BAC</math> hits <math>BC</math> at <math>D</math>. If <math>\angle C=90^\circ</math>, and the maximum value of <math>\frac{[ABD]}{[ACD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note that <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
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==Problem 1==
 +
<math>\triangle ABC</math> has positive integer side lengths of <math>x</math>,<math>y</math>, and <math>17</math>.  The angle bisector of <math>\angle BAC</math> hits <math>BC</math> at <math>D</math>. If <math>\angle C=90^\circ</math>, and the maximum value of <math>\frac{[ABD]}{[ACD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note that <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
  
 +
[[Mock AIME 1 2006-2007/Problem 1|Solution]]
  
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2. Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of positive integers such that for all elements <math>n</math> in <math>\mathcal{S}</math>, we have that <math>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>.
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==Problem 2==
 +
Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of positive integers such that for all elements <math>n</math> in <math>\mathcal{S}</math>, we have that <math>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)</math>.
  
 +
[[Mock AIME 1 2006-2007/Problem 2|Solution]]
  
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3. Let <math>\triangle ABC</math> have <math>BC=\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder when <math>n^{2007}</math> is divided by <math>1000</math>.
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==Problem 3==
 +
Let <math>\triangle ABC</math> have <math>BC=\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder when <math>n^{2007}</math> is divided by <math>1000</math>.
  
 +
[[Mock AIME 1 2006-2007/Problem 3|Solution]]
  
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4. <math>\triangle ABC</math> has all of it's verticies on the parabola <math>y=x^{2}</math>. The slopes of <math>AB</math> and <math>BC</math> are <math>10</math> and <math>-9</math>, respectively. If the x-coordinate of the triangle's centroid is <math>1</math>, find the area of <math>\triangle ABC</math>.
+
==Problem 4==
 +
<math>\triangle ABC</math> has all of it's verticies on the parabola <math>y=x^{2}</math>. The slopes of <math>AB</math> and <math>BC</math> are <math>10</math> and <math>-9</math>, respectively. If the x-coordinate of the triangle's centroid is <math>1</math>, find the area of <math>\triangle ABC</math>.
  
 +
[[Mock AIME 1 2006-2007/Problem 4|Solution]]
  
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5. Let <math>p</math> be a prime and <math>f(n)</math> satisfy <math>0\le f(n) <p</math> for all integers <math>n</math>. <math>\lfloor x\rfloor</math> is the greatest integer less than or equal to <math>x</math>. If for fixed <math>n</math>, there exists an integer <math>0\le y < p</math> such that:
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==Problem 5==
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Let <math>p</math> be a prime and <math>f(n)</math> satisfy <math>0\le f(n) <p</math> for all integers <math>n</math>. <math>\lfloor x\rfloor</math> is the greatest integer less than or equal to <math>x</math>. If for fixed <math>n</math>, there exists an integer <math>0\le y < p</math> such that:
  
 
<math>ny-p\left\lfloor \frac{ny}{p}\right\rfloor=1</math>
 
<math>ny-p\left\lfloor \frac{ny}{p}\right\rfloor=1</math>
−
 
  
 
then <math>f(n)=y</math>. If there is no such <math>y</math>, then <math>f(n)=0</math>. If <math>p=11</math>, find the sum: <math>f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})</math>.
 
then <math>f(n)=y</math>. If there is no such <math>y</math>, then <math>f(n)=0</math>. If <math>p=11</math>, find the sum: <math>f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})</math>.
  
 +
[[Mock AIME 1 2006-2007/Problem 5|Solution]]
  
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6. Let <math>P_{1}: y=x^{2}+\frac{101}{100}</math> and <math>P_{2}: x=y^{2}+\frac{45}{4}</math> be two parabolas in the cartesian plane. Let <math>\mathcal{L}</math> be the common tangent of <math>P_{1}</math> and <math>P_{2}</math> that has a rational slope. If <math>\mathcal{L}</math> is written in the form <math>ax+by=c</math> for positive integers <math>a,b,c</math> where <math>\gcd(a,b,c)=1</math>. Find <math>a+b+c</math>.
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==Problem 6==
 +
Let <math>P_{1}: y=x^{2}+\frac{101}{100}</math> and <math>P_{2}: x=y^{2}+\frac{45}{4}</math> be two parabolas in the cartesian plane. Let <math>\mathcal{L}</math> be the common tangent of <math>P_{1}</math> and <math>P_{2}</math> that has a rational slope. If <math>\mathcal{L}</math> is written in the form <math>ax+by=c</math> for positive integers <math>a,b,c</math> where <math>\gcd(a,b,c)=1</math>. Find <math>a+b+c</math>.
  
 +
[[Mock AIME 1 2006-2007/Problem 6|Solution]]
  
 +
==Problem 7==
 +
Let <math>\triangle ABC</math> have <math>AC=6</math> and <math>BC=3</math>. Point <math>E</math> is such that <math>CE=1</math> and <math>AE=5</math>. Construct point <math>F</math> on segment <math>BC</math> such that <math>\angle AEB=\angle AFB</math>. <math>EF</math> and <math>AB</math> are extended to meet at <math>D</math>. If <math>\frac{[AEF]}{[CFD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math> (note: <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
  
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7. Let <math>\triangle ABC</math> have <math>AC=6</math> and <math>BC=3</math>. Point <math>E</math> is such that <math>CE=1</math> and <math>AE=5</math>. Construct point <math>F</math> on segment <math>BC</math> such that <math>\angle AEB=\angle AFB</math>. <math>EF</math> and <math>AB</math> are extended to meet at <math>D</math>. If <math>\frac{[AEF]}{[CFD]}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math> (note: <math>[ABC]</math> denotes the area of <math>\triangle ABC</math>).
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[[Mock AIME 1 2006-2007/Problem 7|Solution]]
  
 +
==Problem 8==
 +
Let <math>ABCDE</math> be a convex pentagon with <math>AB\sqrt{2}=BC=CD=DE</math>, <math>\angle ABC=150^\circ</math>, <math>\angle BCD=75^\circ</math>, and <math>\angle CDE=165^\circ</math>. If <math>\angle ABE=\frac{m}{n}^\circ</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
  
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8. Let <math>ABCDE</math> be a convex pentagon with <math>AB\sqrt{2}=BC=CD=DE</math>, <math>\angle ABC=150^\circ</math>, <math>\angle BCD=75^\circ</math>, and <math>\angle CDE=165^\circ</math>. If <math>\angle ABE=\frac{m}{n}^\circ</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
+
[[Mock AIME 1 2006-2007/Problem 8|Solution]]
  
 +
==Problem 9==
 +
Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</math> and <math>a_{10}=1</math>. Let <math>S</math> denote the infinite sum: <math>a_{10}+a_{11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then compute the sum of the positive prime factors of <math>n</math>.
  
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9. Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</math> and <math>a_{10}=1</math>. Let <math>S</math> denote the infinite sum: <math>a_{10}+a_{11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then compute the sum of the positive prime factors of <math>n</math>.
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[[Mock AIME 1 2006-2007/Problem 9|Solution]]
  
 +
==Problem 10==
 +
In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
  
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10. In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
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[[Mock AIME 1 2006-2007/Problem 10|Solution]]
  
 +
==Problem 11==
 +
Let <math>\mathcal{S}_{n}</math> be the set of strings with only 0's or 1's with length <math>n</math> such that any 3 adjacent place numbers sum to at least 1. For example, <math>00100</math> works, but <math>10001</math> does not. Find the number of elements in <math>\mathcal{S}_{11}</math>.
  
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11. Let <math>\mathcal{S}_{n}</math> be the set of strings with only 0's or 1's with length <math>n</math> such that any 3 adjacent place numbers sum to at least 1. For example, <math>00100</math> works, but <math>10001</math> does not. Find the number of elements in <math>\mathcal{S}_{11}</math>.
+
[[Mock AIME 1 2006-2007/Problem 11|Solution]]
  
 +
==Problem 12==
 +
Let <math>k</math> be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, <math>m</math>, that satisfies <math>14m+1=k</math>. Find the number of possible values of <math>m</math> between <math>0</math> and <math>10^{2007}</math>.
  
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12. Let <math>k</math> be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, <math>m</math>, that satisfies <math>14m+1=k</math>. Find the number of possible values of <math>m</math> between <math>0</math> and <math>10^{2007}</math>.
+
[[Mock AIME 1 2006-2007/Problem 12|Solution]]
  
 +
==Problem 13==
 +
Let <math>a_{n}</math>, <math>b_{n}</math>, and <math>c_{n}</math> be geometric sequences with different common ratios and let <math>a_{n}+b_{n}+c_{n}=d_{n}</math> for all integers <math>n</math>. If <math>d_{1}=1</math>, <math>d_{2}=2</math>, <math>d_{3}=3</math>, <math>d_{4}=-7</math>, <math>d_{5}=13</math>, and <math>d_{6}=-16</math>, find <math>d_{7}</math>.
  
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13. Let <math>a_{n}</math>, <math>b_{n}</math>, and <math>c_{n}</math> be geometric sequences with different common ratios and let <math>a_{n}+b_{n}+c_{n}=d_{n}</math> for all integers <math>n</math>. If <math>d_{1}=1</math>, <math>d_{2}=2</math>, <math>d_{3}=3</math>, <math>d_{4}=-7</math>, <math>d_{5}=13</math>, and <math>d_{6}=-16</math>, find <math>d_{7}</math>.
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[[Mock AIME 1 2006-2007/Problem 13|Solution]]
  
 +
==Problem 14==
 +
Three points <math>A</math>, <math>B</math>, and <math>T</math> are fixed such that <math>T</math> lies on segment <math>AB</math>, closer to point <math>A</math>. Let <math>AT=m</math> and <math>BT=n</math> where <math>m</math> and <math>n</math> are positive integers. Construct circle <math>O</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>.  Let <math>P</math> be the point such that circle <math>O</math> is the incircle of <math>\triangle APB</math>. Construct <math>M</math> as the midpoint of <math>AB</math>. Let <math>f(m,n)</math> denote the maximum value <math>\tan^{2}\angle AMP</math> for fixed <math>m</math> and <math>n</math> where <math>n>m</math>. If <math>f(m,49)</math> is an integer, find the sum of all possible values of <math>m</math>.
  
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14. Three points <math>A</math>, <math>B</math>, and <math>T</math> are fixed such that <math>T</math> lies on segment <math>AB</math>, closer to point <math>A</math>. Let <math>AT=m</math> and <math>BT=n</math> where <math>m</math> and <math>n</math> are positive integers. Construct circle <math>O</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>.  Let <math>P</math> be the point such that circle <math>O</math> is the incircle of <math>\triangle APB</math>. Construct <math>M</math> as the midpoint of <math>AB</math>. Let <math>f(m,n)</math> denote the maximum value <math>\tan^{2}\angle AMP</math> for fixed <math>m</math> and <math>n</math> where <math>n>m</math>. If <math>f(m,49)</math> is an integer, find the sum of all possible values of <math>m</math>.
+
[[Mock AIME 1 2006-2007/Problem 14|Solution]]
  
 +
==Problem 15==
 +
Let <math>S</math> be the set of integers <math>0,1,2,...,10^{11}-1</math>. An element <math>x\in S</math> (in) is chosen at random. Let <math>\star (x)</math> denote the sum of the digits of <math>x</math>.  The probability that <math>\star (x)</math> is divisible by 11 is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the last 3 digits of <math>m+n</math>
  
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15. Let <math>S</math> be the set of integers <math>0,1,2,...,10^{11}-1</math>. An element <math>x\in S</math> (in) is chosen at random. Let <math>\star (x)</math> denote the sum of the digits of <math>x</math>.  The probability that <math>\star (x)</math> is divisible by 11 is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the last 3 digits of <math>m+n</math>
+
[[Mock AIME 1 2006-2007/Problem 15|Solution]]
  
  
 
[[Mock AIME 1 2006-2007]]
 
[[Mock AIME 1 2006-2007]]

Revision as of 14:27, 25 August 2006

Problem 1

$\triangle ABC$ has positive integer side lengths of $x$,$y$, and $17$. The angle bisector of $\angle BAC$ hits $BC$ at $D$. If $\angle C=90^\circ$, and the maximum value of $\frac{[ABD]}{[ACD]}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive intgers, find $m+n$. (Note that $[ABC]$ denotes the area of $\triangle ABC$).

Solution

Problem 2

Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$, we have that $\star (n)=12$ and $0\le n< 10^{7}$. If $m$ is the number of elements in $\mathcal{S}$, compute $\star(m)$.

Solution

Problem 3

Let $\triangle ABC$ have $BC=\sqrt{7}$, $CA=1$, and $AB=3$. If $\angle A=\frac{\pi}{n}$ where $n$ is an integer, find the remainder when $n^{2007}$ is divided by $1000$.

Solution

Problem 4

$\triangle ABC$ has all of it's verticies on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the x-coordinate of the triangle's centroid is $1$, find the area of $\triangle ABC$.

Solution

Problem 5

Let $p$ be a prime and $f(n)$ satisfy $0\le f(n) <p$ for all integers $n$. $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$. If for fixed $n$, there exists an integer $0\le y < p$ such that:

$ny-p\left\lfloor \frac{ny}{p}\right\rfloor=1$

then $f(n)=y$. If there is no such $y$, then $f(n)=0$. If $p=11$, find the sum: $f(1)+f(2)+...+f(p^{2}-1)+f(p^{2})$.

Solution

Problem 6

Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the cartesian plane. Let $\mathcal{L}$ be the common tangent of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$. Find $a+b+c$.

Solution

Problem 7

Let $\triangle ABC$ have $AC=6$ and $BC=3$. Point $E$ is such that $CE=1$ and $AE=5$. Construct point $F$ on segment $BC$ such that $\angle AEB=\angle AFB$. $EF$ and $AB$ are extended to meet at $D$. If $\frac{[AEF]}{[CFD]}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$ (note: $[ABC]$ denotes the area of $\triangle ABC$).

Solution

Problem 8

Let $ABCDE$ be a convex pentagon with $AB\sqrt{2}=BC=CD=DE$, $\angle ABC=150^\circ$, $\angle BCD=75^\circ$, and $\angle CDE=165^\circ$. If $\angle ABE=\frac{m}{n}^\circ$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

Problem 9

Let $a_{n}$ be a geometric sequence for $n\in\mathbb{Z}$ with $a_{0}=1024$ and $a_{10}=1$. Let $S$ denote the infinite sum: $a_{10}+a_{11}+a_{12}+...$. If the sum of all distinct values of $S$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then compute the sum of the positive prime factors of $n$.

Solution

Problem 10

In $\triangle ABC$, $AB$, $BC$, and $CA$ have lengths $3$, $4$, and $5$, respectively. Let the incircle, circle $I$, of $\triangle ABC$ touch $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Construct three circles, $A''$, $B''$, and $C''$, externally tangent to the other two and circles $A''$, $B''$, and $C''$ are internally tangent to the circle $I$ at $A'$, $B'$, and $C'$, respectively. Let circles $A''$, $B''$, $C''$, and $I$ have radii $a$, $b$, $c$, and $r$, respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$.

Solution

Problem 11

Let $\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$.

Solution

Problem 12

Let $k$ be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, $m$, that satisfies $14m+1=k$. Find the number of possible values of $m$ between $0$ and $10^{2007}$.

Solution

Problem 13

Let $a_{n}$, $b_{n}$, and $c_{n}$ be geometric sequences with different common ratios and let $a_{n}+b_{n}+c_{n}=d_{n}$ for all integers $n$. If $d_{1}=1$, $d_{2}=2$, $d_{3}=3$, $d_{4}=-7$, $d_{5}=13$, and $d_{6}=-16$, find $d_{7}$.

Solution

Problem 14

Three points $A$, $B$, and $T$ are fixed such that $T$ lies on segment $AB$, closer to point $A$. Let $AT=m$ and $BT=n$ where $m$ and $n$ are positive integers. Construct circle $O$ with a variable radius that is tangent to $AB$ at $T$. Let $P$ be the point such that circle $O$ is the incircle of $\triangle APB$. Construct $M$ as the midpoint of $AB$. Let $f(m,n)$ denote the maximum value $\tan^{2}\angle AMP$ for fixed $m$ and $n$ where $n>m$. If $f(m,49)$ is an integer, find the sum of all possible values of $m$.

Solution

Problem 15

Let $S$ be the set of integers $0,1,2,...,10^{11}-1$. An element $x\in S$ (in) is chosen at random. Let $\star (x)$ denote the sum of the digits of $x$. The probability that $\star (x)$ is divisible by 11 is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute the last 3 digits of $m+n$

Solution


Mock AIME 1 2006-2007

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