\documentclass[12pt]{article} \addtolength{\voffset}{-.5in} \addtolength{\textheight}{1in} \addtolength{\hoffset}{-.5in} \addtolength{\textwidth}{1in} \begin{document} \pagestyle{empty} \begin{center} Western Mass.\ ARML Summer Homework \#1 NO CALCULATORS \end{center} Send all of your answers (including your answers to any optional questions) in a single message to {\tt rjyanco1@aol.com} by Wednesday, July 17. 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[20.] Find all ordered pairs $(x,y)$ of real numbers for which $x^2 + xy + x = 14$ and $y^2 + xy + y = 28$. \item[2.] The cube of a certain integer has a decimal representation consisting of ten digits, of which the two leftmost, as well as the rightmost, are the digit 7. Find the integer whose cube has this form. \item[4.] Find the numerical value of $\cos 15^\circ(\sin 75^\circ + \cos 45^\circ)+\sin 15^\circ(\cos 75^\circ - \sin 45^\circ)$. \item[16.] If $x$ and $y$ are real numbers, with $x>y$ and $xy=1$, find the minimum possible value for $\frac{x^2+y^2}{x-y}$. \item[30.] A regular 11-gon is inscribed in a circle. How many triangles are there whose three vertices are all vertices of the 11-gon and whose interiors contain the center of the circle? \medskip \hrule \medskip \item[] {\bf These are optional. Do any which interest you.} \item[1.] In square $ABCD$, points $P$, $Q$, $R$, $S$ are chosen on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DA}$, respectively, so that $AP:PB = BQ:QC = CR:RD = DS:SA = 1:3$. Find the ratio of the area of $PQRS$ to that of $ABCD$. \item[3.] Find all ordered pairs $(x,y)$ of real numbers such that $3^{x^2-2xy}=1$ and $2\log_3 x = \log_3(y+3)$. \item[5.] A student guesses at random at three true-false questions. What is the probability that she gets at least two correct answers? \item[6.] In rectangle $ABCD$, $AB=6$ and diagonal $BD=10$. Circle $O$ (with center $O$) is inscribed in triangle $CBD$, and circle $P$ (with center $P$) is inscribed in triangle $BAD$. Find $OP$. \item[7.] For all real nonzero numbers, $f(x)=1-\frac{1}{x}$ and $g(x)=1-x$. If $h(x)=f[g(x)]$ for what value of $x$ does $h(x)=8$? \item[8.] Find the numerical value of $$\frac{\cos 15^\circ + \sin 15^\circ}{\cos 15^\circ - \sin 15^\circ}.$$ \item[9.] Side $\overline{BC}$ of triangle $ABC$ is extended through $C$ to $X$ so that $BC=CX$. Similarly, side $\overline{CA}$ is extended through $A$ to $Y$ so that $CA=AY$, and side $\overline{AB}$ is extended through $B$ to $Z$ so that $AB=BZ$. Find the ratio of the area of triangle $XYZ$ to that of triangle $ABC$. \item[10.] Find the value of $c$ for which the roots of $x^3 - 6x^2 - 24x + c = 0$ form an arithmetic progression. \item[11.] Three ferryboats start at a terminal at noon and go to different destinations. Ferryboat $A$ reaches its destination after 20 minutes, boat $B$ after 15 minutes, and boat $C$ after 32 minutes. Upon reaching their destinations, the boats return to the terminal, then make another trip, and so on. The trip back to the terminal in each case is the same length, and takes the same time, as the trip out. What is the least number of hours after which the three ferries will again dock at the terminal simultaneously? \item[12.] In right triangle $ABC$, leg $AC=\sin \theta$ and leg $BC=\cos\theta$. Find the length of the longer leg if the length of the median to the hypotenuse $\overline{AB}$ is $\tan\theta$. \item[13.] Points $M$, $N$, and $P$ are the respective midpoints of sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of triangle $ABC$. A point $X$ is chosen outside of the plane of triangle $ABC$. Points $D$, $E$, $F$ are chosen such that $M$, $N$, and $P$ are respective midpoints of $\overline{XD}$, $\overline{XE}$, and $\overline{XF}$. Find the ratio of the area of triangle $DEF$ to that of triangle $ABC$. \item[14.] For all real numbers $x$, the function $f(x)$ satisfies $2f(x) + f(1-x) = x^2$. Find $f(5)$. \item[15.] In triangle $ABC$, $AB=5$ and $AC=8$. Point $P$ is on $\overline{BC}$ and $BP:PC=3:5$. Find the ratio of the radius of the circle through $A$, $B$, and $P$ to theradius of the circle through $A$, $C$, and $P$. \item[17.] If $[x]$ denotes the ``greatest integer'' function, find the largest prime number $p$ such that $\left[\frac{n^2}{3}\right]=p$ for some integer $n$. \item[18.] Square $ABCD$ has area 1 square unit. Point $P$ is 5 units from its center. Set $S$ is the set of points that can be obtained by rotating point $P$ $90^\circ$ counterclockwise about some point on or inside the square. Find the area of set $S$. \item[19.] Circle $O$ passes through vertex $D$ of square $ABCD$, and is tangent to sides $\overline{AB}$ and $\overline{BC}$. If $AB=1$, the radius of circle $O$ can be expressed as $p+q\sqrt{2}$. Find the ordered pair of rational numbers $(p,q)$. \item[21.] In a rectangular coordinate system, a tangent from the point $(24,7)$ to the circle whose equation is $x^2 + y^2 = 400$ has point of tangency $(a,b)$ where $b>0$. Find $a$. \item[22.] Angle $ABC$ is a right angle and $CB=1$. $D$ is a point on ray $BC$ such that $DB=3$ and $E$ is the point on ray $BA$ such that $m\angle DEC$ is maximum. Find the distance $BE$. \item[23.] An ordinary pack of playing cards is shuffled, and two cards are dealt face up. Find the probability that at least one of these is a spade. \item[24.] In convex quadrilateral $PQRS$, diagonals $\overline{PR}$ and $\overline{QS}$ intersect at $T$, with $PT:TR=5:4$ and $QT:TS=2:5$. Point $X$ is chosen between $T$ and $S$ so that $QT=TX$, and $\overline{RX}$ is extended its own length to $Y$. If point $Y$ is {\em outside\/} the quadrilateral, find the ratio of the area of triangle $PSY$ so that of triangle $QRT$. \item[25.] Point $P$ is chosen along leg $\overline{BC}$ of right triangle $ABC$ so that $BP=PA$. If leg $BC=10$ and leg $AC=4$, find $BP$. \item[26.] Five identical black socks and five identical brown socks are in a drawer. Two socks are picked at random. Find the probability that the two socks picked will match. \item[27.] The roots of $f(x)=0$ are 2, 3, 7, 5, and 9. The roots of $g(x)=0$ are 3, 5, 7, 8, and $-1$. Find all solutions of the equation $\frac{f(x)}{g(x)}=0$. \item[28.] A set of distinct, nonzero real numbers is placed along the circumference of a circle. Each of the numbers is equal to the product of the two numbers adjacent to it. What is the least possible number of numbers in the set? \item[29.] In equilateral triangle $ABC$ of edge length 1, $D$ is on $\overline{BC}$ so that $m\angle DAC=45^\circ$. Find the area of triangle $DAC$. \end{enumerate} \vfill \end{document}