Practice (TheColoringMethod)

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843
Points $A,B,C,D$ and $E$ are located in 3-dimensional space with $AB=BC=CD=DE=EA=2$ and $\angle ABC=\angle CDE=\angle DEA=90^o$. The plane of $\triangle ABC$ is parallel to $\overline{DE}$. What is the area of $\triangle BDE$?

849
The $8\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$?


850
Mary is $20\%$ older than Sally, and Sally is $40\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?

852
Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?

855

A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?


856
The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?


858
Suppose $\cos x=0$ and $\cos (x+z)=\frac{1}{2}$. What is the smallest possible positive value of $z$?

859
Circles with centers $A$ and $B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $C$ and $D$, respectively. Lines $AB$ and $CD$ intersect at $E$, and $AE=5$. What is $CD$?


860
Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. What is $\frac{r}{s}$?


861
The function $\displaystyle f$ has the property that for each real number $\displaystyle x$ in its domain, $\displaystyle 1\/x$ is also in its domain and $f(x)+f\left(\frac{1}{x}\right)=x$ What is the largest set of real numbers that can be in the domain of $f$?

862
Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$?


864
Let $S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$ and $S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$. What is the ratio of the area of $S_2$ to the area of $S_1$?

865
A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $\frac{1}{2}$. What is $r$?

866
Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$ of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\ldots ,x^{100})$. If $A^{100}(S)=(1\/2^{50})$, then what is $x$?

867

The expression $(x+y+z)^{2006}+(x-y-z)^{2006}$ can be simplified by expanding it and combining like terms. How many terms are there in the simplified expression?


868
How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties? $(1)$ No two consecutive integers belong to $S$. $(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

869
What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$?

870
For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$?

872
Mary is about to pay for five items at the grocery store. The prices of the items are $7.99$, $4.99$, $2.99$, $1.99$, and $0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $20.00$ that she will receive in change?

873
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?

874
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?

875
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?

876
The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?

877
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

879
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?