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Difference between revisions of "Mock AIME I 2015 Problems"

(→‎Problem 11)
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[[Mock AIME I 2015 Problems/Problem 11|Solution]]
 
[[Mock AIME I 2015 Problems/Problem 11|Solution]]
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For convenience, let's use <math>a, b, c</math> instead of <math>\alpha, \beta, \gamma</math>. Define a polynomial <math>P(x)</math> such that <math>P(x) = (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc</math>. Let <math>j = ab + ac + bc</math> and <math>k = -abc</math>. Then, our polynomial becomes <math>P(x) = x^3 - (a+b+c)x^2 + jx + k</math>.
 
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Note that we want to compute <math>-\frac{j}{k}</math>.
 
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From the given information, we know that the coefficient of the <math>x^2</math> term is <math>6</math>, and we also know that <math>P(-1) = -33</math>, or in other words, <math>-j + k = -26</math>. By Newton's Sums (since we are given <math>a^3 + b^3 + c^3</math>), we also find that <math>6j + k = 43</math>. Solving this system, we find that <math>(j, k) \in (\frac{69}{7}, -\frac{113}{7})</math>. Thus, <math>\frac{j}{-k} = \frac{69}{113}</math>, so our final answer is <math>69 + 113 = \boxed{182}</math>.
 
  
 
== Problem 12 ==
 
== Problem 12 ==

Latest revision as of 13:48, 12 February 2017

Problem 1

David, Justin, Richard, and Palmer are demonstrating a "math magic" concept in front of an audience. There are four boxes, labeled A, B, C, and D, and each one contains a different number. First, David pulls out the numbers in boxes A and B and reports that their product is $14$. Justin then claims that the product of the numbers in boxes B and C is $16$, and Richard states the product of the numbers in boxes C and D to be $18$. Finally, Palmer announces the product of the numbers in boxes D and A. If $k$ is the number that Palmer says, what is $20k$?

Solution

Problem 2

Suppose that $x$ and $y$ are real numbers such that $\log_x 3y = \tfrac{20}{13}$ and $\log_{3x}y=\tfrac23$. The value of $\log_{3x}3y$ can be expressed in the form $\tfrac ab$ where $a$ and $b$ are positive relatively prime integers. Find $a+b$.

Solution

Problem 3

Let $A,B,C$ be points in the plane such that $AB=25$, $AC=29$, and $\angle BAC<90^\circ$. Semicircles with diameters $\overline{AB}$ and $\overline{AC}$ intersect at a point $P$ with $AP=20$. Find the length of line segment $\overline{BC}$.

Solution

Problem 4

At the AoPS Carnival, there is a "Weighted Dice" game show. This game features two identical looking weighted 6 sided dice. For each integer $1\leq i\leq 6$, Die A has $\tfrac{i}{21}$ probability of rolling the number $i$, while Die B has a $\tfrac{7-i}{21}$ probability of rolling $i$. During one session, the host randomly chooses a die, rolls it twice, and announces that the sum of the numbers on the two rolls is $10$. Let $P$ be the probability that the die chosen was Die A. When $P$ is written as a fraction in lowest terms, find the sum of the numerator and denominator.

Solution

Problem 5

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the probability that the two marbles are of opposite color is $\tfrac12$. Let $k_1<k_2<\cdots<k_{100}$ be the $100$ smallest possible values for the total number of marbles in the urn. Compute the remainder when \[k_1+k_2+k_3+\cdots+k_{100}\] is divided by $1000$.

Solution

Problem 6

Find the number of $5$ digit numbers using only the digits $1,2,3,4,5,6,7,8$ such that every pair of adjacent digits is no more than $1$ apart. For instance, $12345$ and $33234$ are acceptable numbers, while $13333$ and $56789$ are not.

Solution

Problem 7

For all points $P$ in the coordinate plane, let $P'$ denote the reflection of $P$ across the line $y=x$. For example, if $P=(3,1)$, then $P'=(1,3)$. Define a function $f$ such that for all points $P$, $f(P)$ denotes the area of the triangle with vertices $(0,0)$, $P$, and $P'$. Determine the number of lattice points $Q$ in the first quadrant such that $f(Q)=8!$.

Solution

Problem 8

Let $a,b,c$ be consecutive terms (in that order) in an arithmetic sequence with common difference $d$. Suppose $\cos b$ and $\cos d$ are roots of a monic quadratic $p(x)$ with $p(-\tfrac{1}{2})=\tfrac1{2014}$. Then \[|\cos a+\cos b+\cos c+\cos d\,|=\frac pq\] for positive relatively prime integers $p$ and $q$. Find the remainder when $p+q$ is divided by $1000$.

Solution

Problem 9

Compute the number of positive integer triplets $(a,b,c)$ with $1\le a,b,c\le 500$ that satisfy the following properties:

(a) $abc$ is a perfect square,

(b) $(a+7b)c$ is a power of $2$,

(c) $a$ is a multiple of $b$.

Solution

Problem 10

Let $f$ be a function defined along the rational numbers such that $f(\tfrac mn)=\tfrac1n$ for all relatively prime positive integers $m$ and $n$. The product of all rational numbers $0<x<1$ such that \[f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}\] can be written in the form $\tfrac pq$ for positive relatively prime integers $p$ and $q$. Find $p+q$.

Solution

Problem 11

Suppose $\alpha$, $\beta$, and $\gamma$ are complex numbers that satisfy the system of equations \begin{align*}\alpha+\beta+\gamma&=6,\\\alpha^3+\beta^3+\gamma^3&=87,\\(\alpha+1)(\beta+1)(\gamma+1)&=33.\end{align*} If $\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mn$ for positive relatively prime integers $m$ and $n$, find $m+n$.

Solution

Problem 12

Alpha and Beta play a game on the number line below.

[asy] import olympiad; size(140); dot(origin); real t = .2; draw((-3,0)--(3,0), Arrows(6)); for(int i=-2; i<3; i=i+1) { draw((i,t)--(i,-t)); label(string(i), (i,-t), dir(270), fontsize(8));} [/asy] Both players start at $0$. Each turn, Alpha has an equal chance of moving $1$ unit in either the positive or negative directions while Beta has a $\tfrac{2}{3}$ chance of moving $1$ unit in the positive direction and a $\tfrac{1}{3}$ chance of moving $1$ unit in the negative direction. The two alternate turns with Alpha going first. If a player reaches $2$ at any point in the game, he wins; however, if a player reaches $-2$, he loses and the other player wins. If $\tfrac pq$ is the probability that Alpha beats Beta, where $p$ and $q$ are relatively prime positive integers, find $p+q$.

Solution

Problem 13

Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed inside a circle of radius $r$. Furthermore, for each positive integer $1\leq i\leq 6$ let $M_i$ be the midpoint of the segment $\overline{A_iA_{i+1}}$, where $A_7\equiv A_1$. Suppose that hexagon $M_1M_2M_3M_4M_5M_6$ can also be inscribed inside a circle. If $A_1A_2=A_3A_4=5$ and $A_5A_6=23$, then $r^2$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.

Solution

Problem 14

Consider a set of $\tfrac{n(n+1)}{2}$ pennies laid out in the formation of an equilateral triangle with "side length" $n$. You wish to move some of the pennies so that the triangle is flipped upside down. For example, with $n=2$, you could take the top penny and move it to the bottom to accomplish this task, as shown:

[asy] size(180); defaultpen(linewidth(0.8)); filldraw(circle((-2,0),1)^^circle((2,0),1)^^circle((0,2*sqrt(3)),1)^^circle((10,0),1)^^circle((14,0),1)^^circle((12,-2*sqrt(3)),1),grey); draw((-1.25,2*sqrt(3))..(-4.5,0)..(-1.25,-2*sqrt(3)),linetype("4 4"),EndArrow); draw(circle((0,-2*sqrt(3)),1),linetype("4 4")); [/asy]

Let $S_n$ be the minimum number of pennies for which this can be done in terms of $n$. Find $S_{50}$.

Solution

Problem 15

Let $\triangle ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $O$ denote its circumcenter and $H$ its orthocenter. The circumcircle of $\triangle AOH$ intersects $AB$ and $AC$ at $D$ and $E$ respectively. Suppose $\tfrac{AD}{AE}=\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m-n$.

Solution

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