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Western Mass.\ ARML Summer Homework \#4

NO CALCULATORS
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Send all of your answers (including your answers to any optional questions) in a single message to {\tt rjyanco1@aol.com} by Wednesday, August 28.

These are intended to be five-minute questions (two questions in ten minutes), but take as much time as you need.

\begin{enumerate}

\item[] {\bf Please do these five:}

\item[4.] How many ways can a set of 6 different elements be divided into 3 subsets of 2 each?

\item[6.] For {\em how many\/} natural numbers $N$ less than or equal to 12 is $4^N + 5^N + 6^N + 7^N$ a multiple of 11?

\item[7.] Express in simplest form the real number $\left(\sqrt[3]{\sqrt{75}-\sqrt{12}}\right)^{-2}$

\item[25.] Find the largest natural number $A$ such that $x$ and $x^3$ leave the same remainder when divided by $A$, for any integer $x>5$.

\item[30.] A chord of a triangle is a line segment whose endpoints lie on the sides of the triangle (but not at the vertices).  For a triangle whose sides are 4, 5, 6, find the length of the shortest chord that divides the triangle into two regions of equal area.

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\item[] {\bf These are optional.  Do any which interest you.}

\item[1.] The integer 1234321 is a perfect square.  Find its positive square root.

\item[2.] Points $P$, $Q$, $R$, and $S$ are chosen on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively, of square $ABCD$ so that $AP:PB = BQ:QC = CR:RD = DS:SA = 3:1$.  Find the ratio of the area of a square $PQRS$ to that of $ABCD$.

\item[3.] The minute hand of a clock travels $k\pi$ radians between 1 P.M.\ and 2:35 P.M.  Find $k$.

\item[5.] The roots of the equation $x^2-qx+p=0$ are the squares of the roots of the equation $x^2-px+q=0$.  Find the ordered pair of nonzero real numbers $(p,q)$.

\item[8.] Line $\ell$ is drawn through the centroid (intersection of the medians) of triangle $ABC$.  Points $B$ and $C$ are on the opposite side of line $\ell$ from point $A$.  The (perpendicular) distances from $A$ and $B$ to $\ell$ are 10 and 6, respectively.  Find the distance from point $C$ to $\ell$.

\item[9.] If $$A=1+\frac{1}{1+\frac{1}{1+\cdots}} \qquad {\rm and} \qquad B=2+\frac{1}{2+\frac{1}{2+\cdots}}$$ (where both fractions are assumed to converge), find the numerical value of $$A+B-\left(\frac{1}{A} + \frac{1}{B}\right).$$

\item[10.] If $A$ is an acute angle and $(\sin A)(\cos A)=\frac{60}{169}$, find the numerical value of $\sin A + \cos A$.

\item[11.] $A$ and $B$ both represent {\em nonzero\/} digits.  If the base ten numeral $AB$ divides (without remainder) the base ten numeral A0B (whose middle digit is zero), find all possible values for the integer $AB$.

\item[12.] In triangle $ABC$, $AB=20$, $BC=30$, and $\overline{BD}$ is an angle bisector (point $D$ is on $\overline{AC}$).  Point $E$ is chosen on $\overline{BC}$ so that $\overline{DE} \parallel \overline{AB}$, and point $K$ is chosen on $\overline{DC}$ so that $\overline{EX} \parallel \overline{BD}$.  If $AD-KC=1$, find the length of $AC$.

\item[13.] By using a ``yardstick'' that was too long, a dealer made a profit of 20\% on some cloth he sold, instead of 30\%.  The cloth was sold by the (linear) yard.  How many inches were there in his ``yard'' stick?

\item[14.] Points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively, of square $ABCD$ to form square $WXYZ$.  If the area of square $WXYZ$ is $\frac{5}{8}$ that of $ABCD$ and $AW<WB$, find the numerical value of the ratio $AW:WB$.

\item[15.] How many fractions $\overline{a}{b}$ are there such that $0<a<b$, $\frac{a}{b}$ is in lowest terms, and $b$ divides 24 (note that $b$ could {\em equal\/} 24)?

\item[16.] If $x$ and $y$ are real numbers such that $xy=7$, find the smallest possible value of $20x^2 + 5y^2$.

\item[17.] If all numbers are written in base ten notation, how many digits are in the numeral representing $61224^2$?

\item[18.] Define an interior diagonal of a three-dimensional polyhedron $P$ to be a line segment whose endpoints are distinct vertices of $P$ and which (except for its endpoints) lies entirely in the interior of $P$.  Find the maximum possible number of interior diagonals if $P$ has 10 vertices.

\item[19.] A fleeble factory was supposed to fill a certain order for fleebles in 12 days.  The factory produced 25\% more fleebles each day that it was supposed to, and as a result it filled the order in 10 days, producing 42 extra fleebles besides.  If the same number of fleebles were produced each day, what was that number?

\item[20.] Find the value of $\arcsin \frac{5}{13} + \arccos \frac{5}{13}$, where ``arc'' denotes principal value.

\item[21.] If $x$ and $y$ are nonnegative real numbers such that $x^2 + 4y^2 = 50$, find the largest possible value of $x+2y$.

\item[22.] In triangle $ABC$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{AC}$.  The area of triangle $ADE$ is half that of triangle $ABC$, and $AD:DB = 4:3$.  Find the ratio $AE:EC$.

\item[23.] Some children bought sticks of gum for 1{\rm\rlap/c} each, rolls for 10{\rm\rlap/c} each, and candy bars for 50{\rm\rlap/c} each.  They spent \$5 and got 100 items.  How many sticks of gum did they buy?

\item[24.] If $r_1$, $r_2$, \ldots, $r_5$ are the distinct roots of the equation $3x^5 + 8x^4 + 3x^3 + x^2 - 4x + 1 = 0$, find the numerical value of $(1+r_1)(1+r_2)\cdots(1+r_5)$.

\item[26.] Find the area of a triangle whose sides have lengths $a$, $b$, and $c$ if $4a^2b^2 - (a^2+b^2-c^2)^2=16$.

\item[27.] Find all ordered triples of integers $(a,b,c)$ such that $0<a<b<c$ and $a^2+b^2+c^2=90$.

\item[28.] If $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$, find the numerical value of $\cos(x-y) + \cos(y-z) + \cos(z-x)$.

\item[29.] A cube is to be colored, with six colors, so that each face is a single different color.  Two colorings are considered the same if two cubes so colored can be placed next to each other so that the colorings of {\em corresponding\/} faces are identical.  Colorings that are mirror images of each other are considered distinct.  How many such distinct colorings are there?

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