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Mock AIME I 2012 Problems

Problem 1

A circle of maximal area is inscribed in the region bounded by the graph of $y=-x^2 -7x + 12$ and the $x$-axis. The radius of this circle is $\dfrac{\sqrt{p} + q}{r}$, where $p$, $q$, and $r$ are integers and $q$ and $r$ are relatively prime. What is $p+q+r$?

Solution

Problem 2

A permutation of the numbers $\{2$, $3$, $4$, $\cdots$, $9\}$ is called golden if every two consecutive numbers in the permutation are relatively prime. How many golden permutations are there?

Solution

Problem 3

Triangle $MNO$ has $MN=11$, $NO=21$, $MO=23$. The trisection points of $MO$ are $E$ and $F$, with $ME<MF$. Segments $NE$ and $NF$ are extended to points $E'$ and $F'$ such that $ME' || NF$ and $OF' || NE$. The area of pentagon $MNOF'E'$ is $p\sqrt{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 4

Consider the polynomial $p(x)=3x^2+2x+1$. Let $p^n (x) = p(p^{n-1}(x))$ and $p^1 (x) = p(x)$. The product of the roots of $p^5 (x)$ can be expressed in the form $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m$ is divided by $1000$.

Solution

Problem 5

Define a triple inequality to be an inequality of the form $r_i > r_j > r_k$. Bradley has a list of 50 real numbers $(r_1, r_2, \cdots, r_{50})$, but he will only give Batra triple inequalities consisting of 3 numbers in the list. For example, if Bradley gives Batra only the triple inequality $r_7 > r_2 > r_{49}$, Batra knows that the 7th number is greater than the 2nd number which is greater than the 49th number, but nothing else. Let $N$ be the maximum number of (distinct) triple inequalities that Bradley could give Batra such that Batra cannot determine the order of the full list of 50 numbers. Find the remainder when $N$ is divided by 1000.

Solution

Problem 6

Eli has 5 strings. He begins by taking one end of one of the strings, and tying it to another end of a string (possibly the same string) chosen at random. He continues on by tying two randomly selected untied ends together until there are no more untied ends, and only loops remain. The expected value of the number of loops that he will have in the end can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Solution

Problem 7

Let two circles $O_1, O_2$ with radii $3, 5$ in the plane be centered at points $M, N$, respectively. Consider a point $P$ in the plane such that $PM = \sqrt{33}, PN = 7$. Denote the intersections of the line $PM$ with $O_1$ as $A, B$, and the intersections of the line $PN$ with $O_2$ as $C, D$. Let $O_1$ and $O_2$ intersect at points $X, Y$ such that $XY=2$. If $Z$ equals the area of $\triangle{PMN}$, $24Z$ can be written in the form $a\sqrt{b}+c\sqrt{d}+e$ where $b, d$ are distinct squarefree integers. Find $a+b+c+d+e$.

Solution

Problem 8

Suppose that the complex number $z$ satisfies $\left|z\right| = \left|z^2+1\right|$. If $K$ is the maximum possible value of $\left|z\right|$, $K^4$ can be expressed in the form $\dfrac{r+\sqrt{s}}{t}$. Find $r+s+t$.

Solution

Problem 9

A class of seven students is doing a Secret Santa, for which all seven students have contributed a gift each. However, one of the seven students has not arrived yet. The teacher decides to randomly assign each of the six students present to a gift that is not their own. If $P$ is the probability that the seventh student is left with his own gift once he arrives, find $\lfloor1000P\rfloor$.

Solution

Problem 10

Consider the function $f(n,x) = \dfrac{\sin{x} + \sin{2x} + \sin{3x} + \cdots + \sin{(n-1)x} + \sin{nx}}{\cos{x} + \cos{2x} + \cos{3x} + \cdots + \cos{(n-1)x} + \cos{nx}}$. Find the sum of all $x$ for which $f(23,x)=f(33,x)$, where $x$ is measured in degrees and $100<x<200$.

Solution

Problem 11

Triangle $ABC$ has sides of length 13, 14, and 15. Set $S$ is the set of all points $O$ for which the circle of radius $1$ centered at $O$ intersects $ABC$ exactly $3$ times. The points in $S$ enclose a finite region whose area can be expressed in the form $\dfrac{p-q\pi}{r}$, where $p$, $q$, and $r$ are positive integers such that $p$ and $r$ are relatively prime. Find $p+q+r$.

Solution

Problem 12

Let $P(x)$ be a polynomial of degree 10 satisfying $P(x^2) = P(x)P(x-1)$. Find the maximum possible sum of the coefficients of $P(x)$.

Solution

Problem 13

Eduardo and Silvie play the Factor Game. Eduardo initially picks a number $N$; Silvie then subtracts a factor of $N$ from $N$ to get $N'$. Eduardo then subtracts a factor of $N'$ from $N'$ to get $N''$ and so on. Neither player is allowed to subtract 1 at any time. The loser is the person who must subtract a number from itself and get zero. Let $S$ be the set of all positive integers less than 1000 which Eduardo can choose as the initial number $N$ to ensure that he wins (assuming optimal play). Find the remainder when the sum of the elements of $S$ is divided by 1000.

Solution

Problem 14

Let $S$ be the set of complex numbers of the form $c+di$ such that $c+di = (a+bi)^{12}$ for some integers $a$ and $b$. Find the largest integer that must divide $d$ for all numbers in $S$.

Solution

Problem 15

Paula the Painter initially paints every complex number black. When Paula toggles a complex number, she paints it white if it was previously black, and black if it was previously white. For each $k=1,2,\dots,20$, Paula progressively toggles the roots of $x^{2k}+x^k+1$. Let $N$ be the number of complex numbers are white at the end of this process. Find $N$.

Solution

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