Difference between revisions of "Mock AIME 1 2006-2007/Problems"
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− | 1 | + | ==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]] | ||
− | 2 | + | ==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]] | ||
− | 3 | + | ==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]] | ||
− | 4 | + | ==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]] | ||
− | 5 | + | ==Problem 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: | |
<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]] | ||
− | 6 | + | ==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>). | ||
− | + | [[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>. | ||
− | 8 | + | [[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>. | ||
− | + | [[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>. | ||
− | 10 | + | [[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>. | ||
− | + | [[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>. | ||
− | + | [[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>. | ||
− | + | [[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>. | ||
− | 14 | + | [[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> | ||
− | + | [[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
Contents
Problem 1
has positive integer side lengths of ,, and . The angle bisector of hits at . If , and the maximum value of where and are relatively prime positive intgers, find . (Note that denotes the area of ).
Problem 2
Let be the sum of the digits of a positive integer . is the set of positive integers such that for all elements in , we have that and . If is the number of elements in , compute .
Problem 3
Let have , , and . If where is an integer, find the remainder when is divided by .
Problem 4
has all of it's verticies on the parabola . The slopes of and are and , respectively. If the x-coordinate of the triangle's centroid is , find the area of .
Problem 5
Let be a prime and satisfy for all integers . is the greatest integer less than or equal to . If for fixed , there exists an integer such that:
then . If there is no such , then . If , find the sum: .
Problem 6
Let and be two parabolas in the cartesian plane. Let be the common tangent of and that has a rational slope. If is written in the form for positive integers where . Find .
Problem 7
Let have and . Point is such that and . Construct point on segment such that . and are extended to meet at . If where and are positive integers, find (note: denotes the area of ).
Problem 8
Let be a convex pentagon with , , , and . If where and are relatively prime positive integers, find .
Problem 9
Let be a geometric sequence for with and . Let denote the infinite sum: . If the sum of all distinct values of is where and are relatively prime positive integers, then compute the sum of the positive prime factors of .
Problem 10
In , , , and have lengths , , and , respectively. Let the incircle, circle , of touch , , and at , , and , respectively. Construct three circles, , , and , externally tangent to the other two and circles , , and are internally tangent to the circle at , , and , respectively. Let circles , , , and have radii , , , and , respectively. If where and are positive integers, find .
Problem 11
Let be the set of strings with only 0's or 1's with length such that any 3 adjacent place numbers sum to at least 1. For example, works, but does not. Find the number of elements in .
Problem 12
Let be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, , that satisfies . Find the number of possible values of between and .
Problem 13
Let , , and be geometric sequences with different common ratios and let for all integers . If , , , , , and , find .
Problem 14
Three points , , and are fixed such that lies on segment , closer to point . Let and where and are positive integers. Construct circle with a variable radius that is tangent to at . Let be the point such that circle is the incircle of . Construct as the midpoint of . Let denote the maximum value for fixed and where . If is an integer, find the sum of all possible values of .
Problem 15
Let be the set of integers . An element (in) is chosen at random. Let denote the sum of the digits of . The probability that is divisible by 11 is where and are relatively prime positive integers. Compute the last 3 digits of