Inequality

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75   
Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

263   
Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge {\scriptstyle\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2$. Find $p+q$.

287   
Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

352   
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?

407   
What is the greatest possible perimeter of an isosceles triangle with sides of length $5x + 20$, $3x + 76$ and $x + 196$?

439   
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

463   
Real numbers $a$ and $b$ are chosen with $1 < a < b$ such that no triangle with positive area has side lengths $1$, $a$, and $b$ or $\frac{1}{b}$, $\frac{1}{a}$, and $1$. What is the smallest possible value of $b$?

492   
$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?

500   
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?

611   
Suppose that $\left|x+y\right|+\left|x-y\right|=2$. What is the maximum possible value of $x^2-6x+y^2$?

648   
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimum value for $n$?

682   
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$?

692   
The set of real numbers $x$ for which \[\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1\] is the union of intervals of the form $a< x \le b$. What is the sum of the lengths of these intervals?

1009   
Let $a, b$ and $c$ be numbers with $0 < a < b < c$. Which of the following is impossible?

1261   
For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius 10 centered at $(5, 5)$?

1263   
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?

1286   
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true? $\textbf{(I)}\ x+y < a+b\qquad$ $\textbf{(II)}\ x-y < a-b\qquad$ $\textbf{(III)}\ xy < ab\qquad$ $\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$

1324   
For how many integers $x$ is the number $x^4-51x^2+50$ negative?

1495   
The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$. What is $b - a$?

1504   
A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?

1546   
In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?


1720   
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

1803   
If $0 \le a_1 \le a_2 \le a_3 \le \cdots \le a_n \le 1$, find the maximum value of $$\sum_{1 \le i < j \le n}(a_j-a_i+1)^2+4 \sum_{i=i}^n a_i^2$$

1884   
What percent of the interval with endpoints \u22125 and 5 consists of real numbers $x$ satisfying the inequality $x + 1 > \frac{8}{x -1}$?

1899   
Shaina has one stick of length $a$ cm and another of length $b$ cm, where $a \ne b$. She needs a third stick with length strictly between 8 cm and 26 cm to make the third side of a triangle. What is the product $ab$?

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