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Difference between revisions of "Mock AIME 1 2013 Problems"

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== Problem 13 ==
 
== Problem 13 ==
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In acute <math>\triangle ABC</math>, <math>H</math> is the orthocenter, <math>G</math> is the centroid, and <math>M</math> is the midpoint of BC. It is obvious that <math>AM \ge GM</math>, but <math>GM \ge HM</math> does not always hold. If <math>[ABC] = 162</math>, <math>BC=18</math>, then the value of <math>GM</math> which produces the smallest value of <math>AB</math> such that <math>GM \ge HM</math> can be expressed in the form <math>a+b\sqrt{c}</math>, for <math>b</math> squarefree. Compute <math>a+b+c</math>.
+
In acute <math>\triangle ABC</math>, <math>H</math> is the orthocenter, <math>G</math> is the centroid, and <math>M</math> is the midpoint of <math>BC</math>. It is obvious that <math>AM \ge GM</math>, but <math>GM \ge HM</math> does not always hold. If <math>[ABC] = 162</math>, <math>BC=18</math>, then the value of <math>GM</math> which produces the smallest value of <math>AB</math> such that <math>GM \ge HM</math> can be expressed in the form <math>a+b\sqrt{c}</math>, for <math>b</math> squarefree. Compute <math>a+b+c</math>.
  
 
[[2013 Mock AIME I Problems/Problem 13|Solution]]
 
[[2013 Mock AIME I Problems/Problem 13|Solution]]

Revision as of 17:16, 13 June 2016

Problem 1

Two circles $C_1$ and $C_2$, each of unit radius, have centers $A_1$ and $A_2$ such that $A_1A_2=6$. Let $P$ be the midpoint of $A_1A_2$ and let $C_3$ be a circle externally tangent to both $C_1$ and $C_2$. $C_1$ and $C_3$ have a common tangent that passes through $P$. If this tangent is also a common tangent to $C_2$ and $C_1$, find the radius of circle $C_3$.

Solution

Problem 2

Find the number of ordered positive integer pairs $(a,b,c)$ such that $a$ evenly divides $b$, $b+1$ evenly divides $c$, and $c-a=10$.

Solution

Problem 3

Let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and let $\{x\}=x-\lfloor x\rfloor$. If $x=(7+4\sqrt{3})^{2^{2013}}$, compute $x\left(1-\{x\}\right)$.

Solution


Problem 4

Compute the number of ways to fill in the following magic square such that:

1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),

2. all entries are nonnegative integers less than or equal to ten, and

3. entries CAN repeat in a column, row, or diagonal.

[asy] size(100); defaultpen(linewidth(0.7)); int i; for(i=0; i<4; i=i+1) { draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6)); } label("$1$", (1,5)); label("$9$", (3,5)); label("$3$", (1,1)); [/asy]

Solution

Problem 5

In quadrilateral $ABCD$, $AC\cap BD=M$. Also, $MA=6, MB=8, MC=4, MD=3$, and $BC=2CD$. The perimeter of $ABCD$ can be expressed in the form $\frac{p\sqrt{q}}{r}$ where $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.

Solution

Problem 6

Find the number of integer values $k$ can have such that the equation \[7\cos x+5\sin x=2k+1\] has a solution.

Solution

Problem 7

Let $S$ be the set of all $7$th primitive roots of unity with imaginary part greater than $0$. Let $T$ be the set of all $9$th primitive roots of unity with imaginary part greater than $0$. (A primitive $n$th root of unity is a $n$th root of unity that is not a $k$th root of unity for any $1 \le k < n$.)Let $C=\sum_{s\in S}\sum_{t\in T}(s+t)$. The absolute value of the real part of $C$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime numbers. Find $m+n$.

Solution

Problem 8

Let $\textbf{u}=4\textbf{i}+3\textbf{j}$ and $\textbf{v}$ be two perpendicular vectors in the $x-y$ plane. If there are $n$ vectors $\textbf{r}_i$ for $i=1, 2, \ldots, n$ in the same plane having projections of $1$ and $2$ along $\textbf{u}$ and $\textbf{v}$ respectively, then find \[\sum_{i=1}^{n}\|\textbf{r}_i\|^2.\] (Note: $\textbf{i}$ and $\textbf{j}$ are unit vectors such that $\textbf{i}=(1,0)$ and $\textbf{j}=(0,1)$, and the projection of a vector $\textbf{a}$ onto $\textbf{b}$ is the length of the vector that is formed by the origin and the foot of the perpendicular of $\textbf{a}$ onto $\textbf{b}$.)

Solution

Problem 9

In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If $E$ is the expected number of turns it takes to short circuit all of the lights, find $100E$.

Solution

Problem 10

Let $T_n$ denote the $n$th triangular number, i.e. $T_n=1+2+3+\cdots+n$. Let $m$ and $n$ be relatively prime positive integers so that \[\sum_{i=3}^\infty \sum_{k=1}^\infty \left(\dfrac{3}{T_i}\right)^k=\dfrac{m}{n}.\] Find $m+n$.

Solution

Problem 11

Let $a,b,$ and $c$ be the roots of the equation $x^3+2x-1=0$, and let $X$ and $Y$ be the two possible values of $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.$ Find $(X+1)(Y+1)$.

Solution

Problem 12

In acute triangle $ABC$, the orthocenter $H$ lies on the line connecting the midpoint of segment $AB$ to the midpoint of segment $BC$. If $AC=24$, and the altitude from $B$ has length $14$, find $AB\cdot BC$.

Solution

Problem 13

In acute $\triangle ABC$, $H$ is the orthocenter, $G$ is the centroid, and $M$ is the midpoint of $BC$. It is obvious that $AM \ge GM$, but $GM \ge HM$ does not always hold. If $[ABC] = 162$, $BC=18$, then the value of $GM$ which produces the smallest value of $AB$ such that $GM \ge HM$ can be expressed in the form $a+b\sqrt{c}$, for $b$ squarefree. Compute $a+b+c$.

Solution

Problem 14

Let \begin{align*}P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum_{j=1}^{2014}j^2x^{2014-j}.\end{align*} If $a_1, a_2, \cdots a_{2013}$ are its roots, then compute the remainder when $a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}$ is divided by 997.

Solution

Problem 15

Let $S$ be the set of integers $n$ such that $n | (a^{n+1}-a)$ for all integers $a$. Compute the remainder when the sum of the elements in $S$ is divided by $1000$.

Solution

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