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Difference between revisions of "2011 AIME II Problems"

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== Problem 9 ==
 
== Problem 9 ==
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Let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_6</math> be nonnegative real numbers such that <math>x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1</math>, and <math>x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}</math>. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac{p}{q}</math> is the maximum possible value of <math>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2</math>. Find <math>p + q</math>.
+
Let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_6</math> be nonnegative real numbers such that <math>x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1</math>, and <math>x_1x_3x_5 + x_2x_4x_6 \ge {\scriptstyle\frac{1}{540}}</math>. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac{p}{q}</math> is the maximum possible value of <math>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2</math>. Find <math>p + q</math>.
  
 
[[2011 AIME II Problems/Problem 9|Solution]]
 
[[2011 AIME II Problems/Problem 9|Solution]]

Revision as of 16:46, 1 April 2011

2011 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Gary purchased a large bevarage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drank twice as much, he would have wasted only 2/9 as much bevarage. Find m+n.

Solution

Problem 2

On square ABCD, point E lies on side AD and point F lies on side BC, so that BE=EF=FD=30. Find the area of the square ABCD.

Solution

Problem 3

The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

Solution

Problem 4

In triangle ABC, AB=(20/11)AC. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of intersection of AC and the line BM. The ratio of CP to PA can be expresses in the form m/n, where m and n are relatively prime positive integers. Find m+n.

Solution

Problem 5

The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms.

Solution

Problem 6

Define an ordered quadruple (a, b, c, d) as interesting if $1 \le a<b<c<d \le 10$, and a+d>b+c. How many interesting ordered quadruples are there?

Solution

Problem 7

Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.

Solution

Problem 8

Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Then the maximum possible value of the real part of $\sum_{j = 1}^{12} w_j$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Solution

Problem 9

Let $x_1$, $x_2$, $\dots$, $x_6$ be nonnegative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$, and $x_1x_3x_5 + x_2x_4x_6 \ge {\scriptstyle\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p + q$.

Solution

Problem 10

A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by 1000.

Solution

Problem 11

Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n$, $m_{i,i} = 10$; for $1 \le i \le n - 1$, $m_{i+1,i} = m_{i,i+1} = 3$; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$. Then $\sum_{n=1}^{\infty} \frac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. Note: The determinant of the $1 \times 1$ matrix $[a]$ is $a$, and the determinant of the $2 \times 2$ matrix $\left[ {\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right] = ad - bc$; for $n \ge 2$, the determinant of an $n \times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\dots$ $a_n$ is equal to $a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1}a_nC_n$, where $C_i$ is the determinant of the $(n - 1) \times (n - 1)$ matrix formed by eliminating the row and column containing $a_i$.

Solution

Problem 12

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 13

Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP \gt CP$ (Error compiling LaTeX. Unknown error_msg). Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$, respectively. Given that $AB = 12$ and $\angle O_1PO_2 = 120\textdegree$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$.

Solution

Problem 14

There are $N$ permutations $(a_1, a_2, \dots, a_{30})$ of $1, 2, \dots, 30$ such that for $m \in \{2,3,5\}$, $m$ divides $a_{n+m} - a_n$ for all integers $n$ with $1 \le n < n+m \le 30$. Find the remainder when $N$ is divided by 1000.

Solution

Problem 15

Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers, and none of $a$, $b$, or $c$ is divisible by the square of a prime. Find $a + b + c + d + e$.

Solution

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