\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 \#2 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 24. 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[21.] The numbers $a$ and $b$ are both perfect squares, and are both represented by four-digit decimal numerals. The digits of these two numerals are the same but in the reverse order. If the square root of $a$ divides the square root of $b$, find $a$. \item[5.] The $n$th term, $a_n$, of a sequence of numbers is given by $a_1=1$ and $a_n=a_{n-1}+2n$ for $n>1$. Write an equation expressing $a_n$ as a polynomial in $n$. \item[20.] Two cardboard rectangles each have dimensions 2 and 8. They are placed on a table so that one pair of diagonally opposite corners coincides, but the other pair of diagonally opposite corners does not coincide. Find the area of the region in which the rectangles overlap. \item[15.] Find the sum of the seventeenth powers of the 17 roots of the equation $x^{17} - 3x + 1 = 0$. \item[29.] One hundred pennies are arranged in seven stacks, of which no two stacks contain the same number of pennies. A student counts the number of pennies in each stack and takes 50 pennies in such a way as to disturb the fewest number of stacks. He ends up taking pennies from $N$ stacks. For all such arrangements of pennies, what is the largest possible value of $N$ that will be necessary? \medskip \hrule \medskip \item[] {\bf These are optional. Do any which interest you.} \item[1.] Instead of finding twice the square of a number, a student found twice the square root of that number and got 10 as her answer. What is the correct answer? \item[2.] Diameter $\overline{AB}$ of a circle whose center is $O$ is extended past $A$ to point $P$, and tangent segment $\overline{PT}$ is drawn. On $\overline{AO}$ as diameter, a small semicircle is drawn on the same side of $\overline{AB}$ as point $T$. If $\overline{OT}$ intersects the smaller semicircle at $X$ and $m\angle TPB=40^\circ$, find the degree measure of arc $OX$. \item[3.] In triangle $ABC$, $AC=6$, $BC=8$, and $AB=10$. Squares $ACXY$ and $BCWZ$ are drawn exterior to the triangle (the triangle has no interior point in common with either of the squares). Find the distance between the centers of the two squares. \item[4.] If $a=\log_{10}(\sqrt{13}+\sqrt{3})$, express in terms of $a$ the value of $\log_{10}(\sqrt{13}-\sqrt{3})$. \item[6.] In parallelogram $PQRS$, point $X$ is on $\overline{PQ}$ and $PX:PQ=1:3$. Point $Y$ is on $\overline{PS}$ and $PY:PS=1:4$. The line $XY$ intersects diagonal $\overline{PR}$ in $Z$. Find the numerical value of the ratio $PZ:PR$. \item[7.] The number $N$ has three digits when written in base ten notation. Its cube root is the sum of its three digits. Find $N$. \item[8.] A train leaves the Jerome Avenue subway terminal every 12 minutes. These trains travel down the Jerome branch line, then down the Lexington Avenue trunk line. A train leaves the White Plains Road terminal every 6 minutes. These trains travel down the White Plains branch line, then also join the Lexington Avenue trunk line. All the trains are scheduled so that there are constant intervals between them as they run down Lexington Avenue. How many minutes are there in this interval? \item[9.] Find the largest integer smaller than $\log_4 9 + \log_9 28$. \item[10.] In equilateral triangle $ABC$, $AB=12$. One vertex of a square is at the midpoint of $\overline{AB}$, and the two {\em adjacent\/} vertices are on the other two sides of the triangle. The length of a side of the square may be expressed as $p\sqrt{2} + q\sqrt{6}$ where $p$ and $q$ are rational numbers. Find the ordered pair $(p,q)$. \item[11.] If $0 < x < \frac{\pi}{4}$ and $\cos x + \sin x = \frac{5}{4}$, find the numerical value of $\cos x - \sin x$. \item[12.] Find the least positive remainder when the integer $2^0 + 0^2 + 2^1 + 1^2 + 2^2 + 2^2 + 2^3 + 3^2 + 2^4 + 4^2 + \cdots + 2^{100} + 100^2$ is divided by 8. \item[13.] The number $N$ is represented by a three-digit base ten numeral. $N$ is equal to the cube of its units digit, and is also equal to the square of a two-digit numeral formed by its other two digits. Find $N$. \item[14.] In triangle $ACB$, $AC=7$ and $BC=5$. Squares $ACXY$ and $BCWZ$ are drawn exterior to the triangle (the triangle has no interior point in common with either square). Find the numerical value of $AB^2 + XW^2$. \item[16.] A train leaves Main Street for Times Square every 15 minutes, starting at 8 A.M. A train also leaves Times Square for Main Street on the same schedule. All trains run at a constant speed along the same tracks, and the trip either way takes 90 minutes. Including the trains it meets at each terminal, how many trains will the one leaving Main Street at noon encounter on its trip to Times Square? \item[17.] Altitude $\overline{CD}$ to hypotenuse $\overline{AB}$ of right triangle $ABC$ is a diameter of circle $O$. This circle intersects $\overline{AC}$ in $E$ and $\overline{BC}$ in $F$. If $AC=9$ and $BC=12$, find $EF$. \item[18.] Find the degree-measure of all angles $x$ such that $0^\circ \leq x \leq 180^\circ$ and $$\cos^6 x - \sin^6 x + \frac{(\sin^2 2x)(\cos 2x)}{4}=0.$$ \item[19.] $AB$ and $CA$ are decimal numerals, and $A$, $B$, $C$ are distinct digits. If four times $AB$ equals $CA$, find the ordered triple $(A,B,C)$. \item[22.] In trapezoid $ABCD$, the ratio of the base $AB$ to base $CD$ is $2:3$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect in $X$, and the line through $X$ parallel to $\overline{AB}$ intersects $\overline{AD}$ at $P$. Find the ratio of the area of triangle $PAX$ to the area of triangle $ABD$. \item[23.] From a point $P$ on the hyperbola with equation $x^2-y^2=9-4y$, tangents are drawn to the circle centered at the origin with radius 1. What is the minimal possible length of these tangents? \item[24.] Walking along a bus line, a student found that a bus caught up with her (going in the same direction) every 12 minutes and a bus passed her (going in the opposite direction) every 4 minutes. The student travels at a constant rate, and all buses travel at the same constant rate as one another. The buses leave their terminals at equally spaced intervals of time. How long, in minutes, is each such interval? \item[25.] A boy has as many brothers as he has sisters. His sister has twice as many brothers as she has sisters. How many children are in this family? \item[26.] Radii $\overline{OA}$ and $\overline{OB}$ of circle $O$ meet at an angle of $120^\circ$. A square has one vertex on $\overline{OA}$, another on $\overline{OB}$, and two more on minor arc $AB$. If $OA=13$, the area of the square can be written as $p+q\sqrt{3}$, where $p$ and $q$ are rational. Find the ordered pair $(p,q)$. \item[27.] If $r_1$, $r_2$, \ldots, $r_7$ are the roots of $2x^7 - x^6 - 5x^5 + 17x^4 - 419x^2 - 372x - 10 = 0$, find $(1+r_1)(1+r_2)\cdots(1+r_7)$. \item[28.] Find all ordered pairs $(x,y)$ of {\em positive\/} integers satisfying $$x=\frac{6-x}{y^2-x}.$$ \item[30.] A right angle has its vertex at the centroid of an equilateral triangle of side 1 unit. Find the maximum possible area interior to both the angle and the triangle. \end{enumerate} \end{document}