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Difference between revisions of "Mock AIME II 2012 Problems"

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==Problem 3==
 
==Problem 3==
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The <math>\textit{digital root}</math> of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the <math>\textit{digital root}</math> of <math>237</math> is <math>3</math> (<math>2+3+7=12, 1+2=3</math>). Find the <math>\textit{digital root}</math> of <math>2012^{2012^{2012}}</math>.
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The <math>\textit{digital root}</math> of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the <math>\textit{digital root}</math> of <math>237</math> is <math>3</math> <math>(2+3+7=12</math>, <math>1+2=3)</math>. Find the <math>\textit{digital root}</math> of <math>2012^{2012^{2012}}</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 3| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 3| Solution]]
  
 
==Problem 4==
 
==Problem 4==
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Let <math>\triangle ABC</math> be a triangle, and let <math>I_A</math>, <math>I_B</math>, and <math>I_C</math> be the points where the angle bisectors of <math>A</math>, <math>B</math>, and <math>C</math>, respectfully, intersect the sides opposite them.  Given that <math>AI_B=5</math>, <math>CI_B=4</math>, and <math>CI_A=3</math>, then the ratio <math>AI_C:BI_C</math> can be written in the form <math>m/n</math> where <math>m</math> and <math>n</math> are positive relatively prime integers.  Find <math>m+n</math>.
+
Let <math>\triangle ABC</math> be a triangle, and let <math>I_A</math>, <math>I_B</math>, and <math>I_C</math> be the points where the angle bisectors of <math>A</math>, <math>B</math>, and <math>C</math>, respectfully, intersect the sides opposite them.  Given that <math>AI_B=5</math>, <math>CI_B=4</math>, and <math>CI_A=3</math>, then the ratio <math>AI_C:BI_C</math> can be written in the form <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive relatively prime integers.  Find <math>m+n</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 4| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 4| Solution]]
  
 
==Problem 5==
 
==Problem 5==
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A fair die with <math>12</math> sides numbered <math>1</math> through <math>12</math> inclusive is rolled <math>n</math> times. The probability that the sum of the rolls is <math>2012</math> is nonzero and is equivalent to the probability that a sum of <math>k</math> is rolled. Find the minimum value of k.
+
A fair die with <math>12</math> sides numbered <math>1</math> through <math>12</math> inclusive is rolled <math>n</math> times. The probability that the sum of the rolls is <math>2012</math> is nonzero and is equivalent to the probability that a sum of <math>k</math> is rolled. Find the minimum value of <math>k</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 5| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 5| Solution]]
−
 
  
 
==Problem 6==
 
==Problem 6==
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A circle with radius <math>5</math> and center in the first quadrant is placed so that it is tangent to the <math>y</math>-axis.  If the line passing through the origin that is tangent to the circle has slope <math>\dfrac{1}{2}</math>, then the <math>y</math>-coordinate of the center of the circle can be written in the form <math>\dfrac{m+\sqrt{n}}{p}</math> where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math> \text{gcd}(m,p)=1 </math>.  Find <math>m+n+p</math>.
+
A circle with radius <math>5</math> and center in the first quadrant is placed so that it is tangent to the <math>y</math>-axis.  If the line passing through the origin that is tangent to the circle has slope <math>\dfrac{1}{2}</math>, then the <math>y</math>-coordinate of the center of the circle can be written in the form <math>\dfrac{m+\sqrt{n}}{p}</math> where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>m</math> and <math>p</math> are relatively prime.  Find <math>m+n+p</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 6| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 6| Solution]]
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==Problem 8==
 
==Problem 8==
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Let <math>A</math> be a point outside circle <math>\Omega</math> with center <math>O</math> and radius <math>9</math> such that the tangents from <math>A</math> to <math>\Omega</math>, <math>AB</math> and <math>AC</math>, form <math>\angle BAO=15^{\circ}</math>. Let <math>AO</math> first intersect the circle at <math>D</math>, and extend the parallel to <math>AB</math> from <math>D</math> to meet the circle at <math>E</math>. The length <math>EC^2=m+k\sqrt{n}</math>, where <math>m</math>,<math>n</math>, and <math>k</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>.
+
Let <math>A</math> be a point outside circle <math>\Omega</math> with center <math>O</math> and radius <math>9</math> such that the tangents from <math>A</math> to <math>\Omega</math>, <math>AB</math> and <math>AC</math>, form <math>\angle BAO=15^{\circ}</math>. Let <math>AO</math> first intersect the circle at <math>D</math>, and extend the parallel to <math>AB</math> from <math>D</math> to meet the circle at <math>E</math>. The length <math>EC^2=m+k\sqrt{n}</math>, where <math>m</math>, <math>n</math>, and <math>k</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 8| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 8| Solution]]
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==Problem 10==
 
==Problem 10==
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Call a set of positive integers <math>\mathcal{S}</math> <math>\textit{lucky}</math> if it can be split into two nonempty disjoint subsets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> with <math>A\cap B=S</math> such that the product of the elements in <math>\mathcal{A}</math> and the product of the elements in <math>\mathcal{B}</math> sum up to the cardinality of <math>\mathcal{S}</math>. Find the number of <math>\textit{lucky}</math> sets such that the largest element is less than <math>15</math>. (Disjoint subsets have no elements in common, and the cardinality of a set is the number of elements in the set.)
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Call a set of positive integers <math>\mathcal{S}</math> <math>\textit{lucky}</math> if it can be split into two nonempty disjoint subsets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> with <math>A\cup B=S</math> such that the product of the elements in <math>\mathcal{A}</math> and the product of the elements in <math>\mathcal{B}</math> sum up to the cardinality of <math>\mathcal{S}</math>. Find the number of <math>\textit{lucky}</math> sets such that the largest element is less than <math>15</math>. (Disjoint subsets have no elements in common, and the cardinality of a set is the number of elements in the set.)
  
 
[[Mock AIME II 2012 Problems/Problem 10| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 10| Solution]]
  
 
==Problem 11==
 
==Problem 11==
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There exist real values of <math>a</math> and <math>b</math> such that <math>a+b=n</math>, <math>a^2+b^2=2n</math>, and <math>a^3+b^3=3n</math> for some value of <math>n</math>.  Let <math>S</math> be the sum of all possible values of <math>a^4+b^4</math>.  Find <math>S</math>.
 +
 
[[Mock AIME II 2012 Problems/Problem 11| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 11| Solution]]
 +
 +
==Problem 12==
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Let <math>\log_{a}b=5\log_{b}ac^4=3\log_{c}a^2b</math>. Assume the value of <math>\log_ab</math> has three real solutions <math>x,y,z</math>. If <math>\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 12| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 12| Solution]]
 +
 +
==Problem 13==
 +
Regular octahedron <math>ABCDEF</math> (such that points <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are coplanar and form the vertices of a square) is divided along plane <math> \mathcal{P} </math>, parallel to line <math> BC </math>, into two polyhedra of equal volume. The cosine of the acute angle plane <math> \mathcal{P} </math> makes with plane <math> BCDE </math> is <math> \frac{1}{3} </math>. Given that <math> AB=30 </math>, find the area of the cross section made by plane <math> \mathcal{P}  </math> with octahedron <math> ABCDEF </math>.
  
 
[[Mock AIME II 2012 Problems/Problem 13| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 13| Solution]]
 +
 +
==Problem 14==
 +
Call a number a <math>\textit{near Carmichael number}</math> if for all prime divisors <math>p</math> of the <math>\textit{near Carmichael number}</math>, <math>n</math>, <math>(p-1)</math> divides <math>(n-1)</math> and <math>n</math> is not prime. Find the sum of all two digit <math>\textit{near Carmichael numbers}</math>.
  
 
[[Mock AIME II 2012 Problems/Problem 14| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 14| Solution]]
 +
 +
==Problem 15==
 +
Define <math>a_n=\sum_{i=0}^{n}f(i)</math> for <math>n\ge 0</math> and <math>a_n=0</math>.  Given that <math>f(x)</math> is a polynomial, and <math>a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125, \cdots</math> is an arithmetic sequence, find the smallest positive integer value of <math>x</math> such that <math>f(x)<-2012</math>. 
  
 
[[Mock AIME II 2012 Problems/Problem 15| Solution]]
 
[[Mock AIME II 2012 Problems/Problem 15| Solution]]

Latest revision as of 20:16, 8 March 2021

Problem 1

Given that \[\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},\] where $m$ and $n$ are positive relatively prime integers, find the remainder when $m+n$ is divided by $1000$.

Solution

Problem 2

Let $\{a_n\}$ be a recursion defined such that $a_1=1, a_2=20$, and $a_n=\sqrt{\left| a_{n-1}^2-a_{n-2}^2 \right|}$ where $n\ge 3$, and $n$ is an integer. If $a_m=k$ for $k$ being a positive integer greater than $1$ and $m$ being a positive integer greater than 2, find the smallest possible value of $m+k$.

Solution

Problem 3

The $\textit{digital root}$ of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the $\textit{digital root}$ of $237$ is $3$ $(2+3+7=12$, $1+2=3)$. Find the $\textit{digital root}$ of $2012^{2012^{2012}}$.

Solution

Problem 4

Let $\triangle ABC$ be a triangle, and let $I_A$, $I_B$, and $I_C$ be the points where the angle bisectors of $A$, $B$, and $C$, respectfully, intersect the sides opposite them. Given that $AI_B=5$, $CI_B=4$, and $CI_A=3$, then the ratio $AI_C:BI_C$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.

Solution

Problem 5

A fair die with $12$ sides numbered $1$ through $12$ inclusive is rolled $n$ times. The probability that the sum of the rolls is $2012$ is nonzero and is equivalent to the probability that a sum of $k$ is rolled. Find the minimum value of $k$.

Solution

Problem 6

A circle with radius $5$ and center in the first quadrant is placed so that it is tangent to the $y$-axis. If the line passing through the origin that is tangent to the circle has slope $\dfrac{1}{2}$, then the $y$-coordinate of the center of the circle can be written in the form $\dfrac{m+\sqrt{n}}{p}$ where $m$, $n$, and $p$ are positive integers, and $m$ and $p$ are relatively prime. Find $m+n+p$.

Solution

Problem 7

Given $x, y$ are positive real numbers that satisfy $3x+4y+1=3\sqrt{x}+2\sqrt{y}$, then the value $xy$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Solution

Problem 8

Let $A$ be a point outside circle $\Omega$ with center $O$ and radius $9$ such that the tangents from $A$ to $\Omega$, $AB$ and $AC$, form $\angle BAO=15^{\circ}$. Let $AO$ first intersect the circle at $D$, and extend the parallel to $AB$ from $D$ to meet the circle at $E$. The length $EC^2=m+k\sqrt{n}$, where $m$, $n$, and $k$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+k$.

Solution

Problem 9

In $\triangle ABC$, $AB=12$, $AC=20$, and $\angle ABC=120^\circ$. $D, E,$ and $F$ lie on $\overline{AC}, \overline{AB}$, and $\overline{BC}$, respectively. If $AE=\frac{1}{4}AB, BF=\frac{1}{4}BC$, and $AD=\frac{1}{4}AC$, the area of $\triangle DEF$ can be expressed in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$ where $a, b, c, d, e$ are all positive integers, and $b$ and $d$ do not have any perfect squares greater than $1$ as divisors. Find $a+b+c+d+e$.

Solution

Problem 10

Call a set of positive integers $\mathcal{S}$ $\textit{lucky}$ if it can be split into two nonempty disjoint subsets $\mathcal{A}$ and $\mathcal{B}$ with $A\cup B=S$ such that the product of the elements in $\mathcal{A}$ and the product of the elements in $\mathcal{B}$ sum up to the cardinality of $\mathcal{S}$. Find the number of $\textit{lucky}$ sets such that the largest element is less than $15$. (Disjoint subsets have no elements in common, and the cardinality of a set is the number of elements in the set.)

Solution

Problem 11

There exist real values of $a$ and $b$ such that $a+b=n$, $a^2+b^2=2n$, and $a^3+b^3=3n$ for some value of $n$. Let $S$ be the sum of all possible values of $a^4+b^4$. Find $S$.

Solution

Problem 12

Let $\log_{a}b=5\log_{b}ac^4=3\log_{c}a^2b$. Assume the value of $\log_ab$ has three real solutions $x,y,z$. If $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

Problem 13

Regular octahedron $ABCDEF$ (such that points $B$, $C$, $D$, and $E$ are coplanar and form the vertices of a square) is divided along plane $\mathcal{P}$, parallel to line $BC$, into two polyhedra of equal volume. The cosine of the acute angle plane $\mathcal{P}$ makes with plane $BCDE$ is $\frac{1}{3}$. Given that $AB=30$, find the area of the cross section made by plane $\mathcal{P}$ with octahedron $ABCDEF$.

Solution

Problem 14

Call a number a $\textit{near Carmichael number}$ if for all prime divisors $p$ of the $\textit{near Carmichael number}$, $n$, $(p-1)$ divides $(n-1)$ and $n$ is not prime. Find the sum of all two digit $\textit{near Carmichael numbers}$.

Solution

Problem 15

Define $a_n=\sum_{i=0}^{n}f(i)$ for $n\ge 0$ and $a_n=0$. Given that $f(x)$ is a polynomial, and $a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125, \cdots$ is an arithmetic sequence, find the smallest positive integer value of $x$ such that $f(x)<-2012$.

Solution

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