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Western Mass.\ ARML Summer Homework \#3

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 21.

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[6.] Find the smallest natural number $N$ such that $\frac{N}{2}$ is a perfect square and $\frac{N}{3}$ is a perfect cube.

\item[7.] A circular path is 330 meters in circumference.  A man makes a mark on the path, then walks around it several times, making a mark every 75 meters.  He stops when the mark he is about to make coincides with his very first mark.  When he is done, what is the shortest (positive) distance (measured along the circular path) between two of the
marks?

\item[3.] How many five-digit numbers (in base ten notation, with the leftmost digit not equal to zero) are there such that each digit is strictly greater than the sum of the digits to its right (in particular, the tens digit is larger than the units digit)?

\item[5.] Line $XY$ is tangent to circle $O$ (with center $O$) at $X$ and to circle $P$ (with center $P$) at $Y$.  The radii of the circles are 5 and 8, respectively, and points $O$ and $P$ are both on the same side of line $XY$.  If $XY=\sqrt{7}$, find the length of $OP$.

\item[11.] In triangle $ABC$, $AB=3$, $BC=4$, $AC=6$.  If $\overline{BC}$ is extended through $C$ to $D$ and $BC=CD$, find $AD$.

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

\item[1.] The sequence $\{a_n\}$ is defined by $a_1 = a_4 = 1$, $a_2=a_3=a_5=-1$, $a_6=a_1a_2$, $a_7=a_2a_3$, and in general for $k>5$, $a_k = a_{k-5}a_{k-4}$.  Find $k_{1983}$.

\item[2.] The numbers 1, 2, 3, \ldots, 1000 are written in a row.  Sam started at 1 and circled every 24th number in red.  Janet started at 1 and circled every 15th number in blue.  What is the smallest possible (positive) difference between a red number and a blue number?

\item[4.] If $0<A<\pi$, $0<B<\pi$, and $\sin A + \sin B = \cos A + \cos B$, find the numerical value of $A+B$.

\item[8.] In a plane, points $A$ and $B$ are on the same side of line $L$.  They are each 3 cm from line $L$ and they are 4 cm from each other.  Find the radius of the circle through points $A$ and $B$ that is tangent to line $L$.

\item[9.] In right triangle $ABC$, leg $AC=4$ and leg $BC=8$.  A square is drawn exterior to the triangle with $\overline{AB}$ as one side.  Find the distance from $C$ to the intersection of the diagonals of the square.

\item[10.] The sequence $\{a_i\}$ is defined as follows:  $a_1 = 3^{1983}$, and for $i>1$, $a_i$ is the sum of the digits in the decimal representation of $a_{i-1}$.  Find the numerical value of $a_{10}$.

\item[12.] How many ordered pairs $(x,y)$ of positive integers are there such that both $x$ and $y$ are less than 100 and the expression $\log_{10}x + \log_{10}y$ has an {\em integral\/} value?

\item[13.] In triangle $ABC$, $\sin^2 A + \sin^2 B = 1$.  Find the degree-measure of angle $C$.

\item[14.] The number $N$ is represented by the base $q$ numeral 1441.  When divided by eleven, $N$ leaves a remainder of 1.  If $1<q<10$, find $q$.

\item[15.] A certain plane geometric figure may be made to coincide with itself if it is rotated in a plane about a fixed point $P$ through an angle of $48^\circ$.  What is the smallest positive angle through which you can be {\em sure\/} the figure may be rotated (about point $P$) and still coincide with itself?

\item[16.] In triangle $ABC$, $AC=6$, $BC=8$, and $AB=10$.  A line dividing the triangle into two regions of equal area is perpendicular to $\overline{AB}$ at $X$.  Find $BX$.

\item[17.] If $m,n > 1$ and for all $x>0$, $\log_n x = 3 \log_m x$, write an equation expressing $m$ explicitly in terms of $n$.

\item[18.] In trapezoid $ABCD$, $AB=BC=CD=6$ and $AD=8$.  Points $P$ and $R$ are chosen on line $AD$ (on opposite sides of point $A$), so that there exists a point $Q$ on the plane for which $B$ is the midpoint of $\overline{PQ}$, $C$ is the midpoint of $\overline{RQ}$, and $BQ=QC$.  Find $PQ$.

\item[19.] The sequence $\{a_i\}$ is defined by setting $a_1=7$ and, for $i>1$, taking $a_i$ to be the sum of the digits in the decimal representation of $(a_{i-1})^2$.  Find $a_{1983}$.

\item[20.] Let $n$ be the maximum number of points that can be arranged in a plane so that of any four of them, there are three that determine an equilateral triangle.  Find $n$.

\item[21.] Find all values of $a$ such that the three equations

$$\left\{\matrix{ax & + & y & = & 1 \cr
                          x & + & y & = & 2 \cr
                          x & - & y & = & a}\right.$$
are satisfied simultaneously by some ordered pair $(a,b)$.

\item[22.] Points $M$ and $N$ are on side $\overline{AC}$ of triangle $ABC$, and points $P$ and $Q$ are on side $\overline{AB}$.  The lines $MP$, $NQ$, and $BC$ are parallel, and they divide the triangle into three regions of equal area.  If $NQ=4$, find $BC$.

\item[23.] In triangle $ABC$, $m\angle ACB=90$ and $m\angle ABC=45$.  Points $X$, $Y$, $Z$ are on sides $\overline{AC}$, $\overline{CB}$, $\overline{BA}$, respectively, so that $AX:XC=CY:YB=BZ:ZA=2:1$.  If $\overline{CZ}$ intersects $\overline{XY}$ in $P$, find the degree-measure of angle $CPY$.

\item[24.] The symbol $[x]$ represents the largest integer not exceeding $x$.  Find all positive integral values of $n$ for which the expression $\left[\frac{n^2}{3}\right]$ represents a prime number (1 is not prime).

\item[25.] A sequence of digits has the property that each pair of successive digits, taken in the order written, forms a decimal numeral representing a multiple of either 17 or 23.  If the first digit is 9 and the sequence is finite, what is its last (possible) digit?

\item[26.] In parallelogram $ABCD$, $AB=4$, $AD=9$, and $m\angle BAD=30$.  The circle through points $A$, $B$, and $D$ intersects $\overline{BC}$ at $B$ and $X$.  Find $XC$.

\item[27.] A right circular cone has a base with radius 12.  A plane parallel to the plane of the base cuts the cone into two equal volumes.  Find, in radical form, the radius of the circle of intersection of this plane with the cone.

\item[28.] If $[x]$ denotes the greatest integer not exceeding $x$ and $\{x\} = x-[x]$, find all ordered triples $(x,y,z)$ of real numbers such that $x+[y]+\{z\}=1.1$, $y+[z]+\{x\}=2.2$, $z+[x]+\{y\}=3.3$.

\item[29.] If $0 < x < \pi$ and $2^{\tan x} = 8^{\sin x}$, find the numerical value of $\cos x$.

\item[30.] Acute triangle $ABC$ is inscribed in a circle.  Altitudes $\overline{AM}$ and $\overline{CN}$ are extended to meet the circle again at $P$ and $Q$, respectively.  If $PQ:AC=7:2$, find the numerical value of $\sin B$.

\end{enumerate}

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