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1,000 | The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? | 6 | 31.25 |
1,001 | The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | 28.90625 |
1,002 | The area of the rectangular region is | .088 m^2 | 0 |
1,003 | Let $a$, $b$, $c$, and $d$ be positive integers with $a < 2b$, $b < 3c$, and $c<4d$. If $d<100$, the largest possible value for $a$ is | 2367 | 85.15625 |
1,004 | Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? | \frac{32}{5} | 0 |
1,005 | Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$? | 10 | 86.71875 |
1,006 | When simplified, the third term in the expansion of $(\frac{a}{\sqrt{x}} - \frac{\sqrt{x}}{a^2})^6$ is: | \frac{15}{x} | 85.9375 |
1,007 | Let $f(n) = \frac{x_1 + x_2 + \cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \cdots, n$, the set of possible values of $f(n)$ is: | $\{0, -\frac{1}{n}\}$ | 0 |
1,008 | Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8? | 13 | 81.25 |
1,009 | Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | 12 | 17.1875 |
1,010 | Three primes $p, q$, and $r$ satisfy $p + q = r$ and $1 < p < q$. Then $p$ equals | 2 | 87.5 |
1,011 | Let $n$ be the number of ways $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals: | 19 | 72.65625 |
1,012 | For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | 10 | 92.1875 |
1,013 | If the sequence $\{a_n\}$ is defined by
$a_1=2$
$a_{n+1}=a_n+2n$
where $n\geq1$.
Then $a_{100}$ equals | 9902 | 99.21875 |
1,014 | Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? | 10 | 63.28125 |
1,015 | The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? | \frac{12}{25} | 90.625 |
1,016 | The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
[Note: Abscissas means x-coordinate.] | $y = x$, $y = x-2$ | 0 |
1,017 | A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3$). If $\measuredangle CDA=8^\circ$, what is the largest value $n$ can have? | 10 | 0.78125 |
1,018 | If $a \otimes b = \dfrac{a + b}{a - b}$, then $(6\otimes 4)\otimes 3 =$ | 4 | 100 |
1,019 | Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? | 18 | 22.65625 |
1,020 | If $y=x^2+px+q$, then if the least possible value of $y$ is zero $q$ is equal to: | \frac{p^2}{4} | 88.28125 |
1,021 | If $g(x)=1-x^2$ and $f(g(x))=\frac{1-x^2}{x^2}$ when $x\not=0$, then $f(1/2)$ equals | 1 | 87.5 |
1,022 | How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? | 36 | 95.3125 |
1,023 | George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? | 4\frac{1}{2} | 0 |
1,024 | A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals: | 14\frac{14}{23} | 0.78125 |
1,025 | Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | 37 | 89.0625 |
1,026 | Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? | 91 | 78.90625 |
1,027 | The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square? | 9 | 99.21875 |
1,028 | The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$? | 16:3 | 67.1875 |
1,029 | The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is: | 3 | 91.40625 |
1,030 | If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to | \frac{8}{9} | 89.0625 |
1,031 | Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus? | 60 | 59.375 |
1,032 | Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? | 50 | 96.875 |
1,033 | The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots$ consists of $1$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is | 2419 | 85.15625 |
1,034 | Call a number prime-looking if it is composite but not divisible by $2, 3,$ or $5.$ The three smallest prime-looking numbers are $49, 77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$? | 100 | 70.3125 |
1,035 | How many whole numbers between 1 and 1000 do not contain the digit 1? | 728 | 92.96875 |
1,036 | A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$? | \frac{5}{16} | 1.5625 |
1,037 | A circle with center $O$ has area $156\pi$. Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$, and point $O$ is outside $\triangle ABC$. What is the side length of $\triangle ABC$? | $6$ | 0 |
1,038 | The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? | 2S + 12 | 89.84375 |
1,039 | When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is | 826 | 71.875 |
1,040 | The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click.
The speed of the train in miles per hour is approximately the number of clicks heard in: | 20 seconds | 0 |
1,041 | When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is | 2x-2 | 85.9375 |
1,042 | \frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \frac{33}{32} + \frac{65}{64} - 7 = | -\frac{1}{64} | 35.15625 |
1,043 | Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday? | 2017 | 87.5 |
1,044 | If $x, y$ and $2x + \frac{y}{2}$ are not zero, then
$\left( 2x + \frac{y}{2} \right)^{-1} \left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]$ equals | \frac{1}{xy} | 97.65625 |
1,045 | There are $52$ people in a room. what is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true? | 5 | 17.1875 |
1,046 | Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? | \frac{1}{9} | 96.875 |
1,047 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$? | 20 | 0 |
1,048 | The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? | 34 | 90.625 |
1,049 | Find the degree measure of an angle whose complement is 25% of its supplement. | 60 | 100 |
1,050 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | 500 | 21.875 |
1,051 | A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);[/asy] | \frac{\sqrt{2} - 1}{2} | 4.6875 |
1,052 | Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$\begin{array}{ccccc} {}&{}&{}&{}&55\\ 55&-&7^2&=&6\\ 6&-&2^2&=&2\\ 2&-&1^2&=&1\\ 1&-&1^2&=&0\\ \end{array}$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$? | 3 | 33.59375 |
1,053 | What is the value of $1234 + 2341 + 3412 + 4123$ | 11110 | 53.125 |
1,054 | If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is: | -16 | 97.65625 |
1,055 | Let $x=-2016$. What is the value of $|| |x|-x|-|x||-x$ ? | 4032 | 21.875 |
1,056 | If $9^{x + 2} = 240 + 9^x$, then the value of $x$ is: | 0.5 | 100 |
1,057 | A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | \frac{1}{5} | 7.03125 |
1,058 | Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is
[asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle); [/asy] | 24 | 88.28125 |
1,059 | What is the value of \(\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}\)? | 2 | 80.46875 |
1,060 | A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is | 72 | 10.15625 |
1,061 | If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-n representation of $b$ is | 80 | 89.84375 |
1,062 | How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ | 100 | 97.65625 |
1,063 | The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is: | 33 | 92.96875 |
1,064 | Let $c = \frac{2\pi}{11}.$ What is the value of
\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\] | 1 | 77.34375 |
1,065 | An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) | 1 | 39.84375 |
1,066 | If $2x+1=8$, then $4x+1=$ | 15 | 100 |
1,067 | A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replaced by m-8, the median of the new list would be m-4. What is m? | 20 | 8.59375 |
1,068 | A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $a/b$ that it will take more than four draws until the third shiny penny appears and $a/b$ is in lowest terms, then $a+b=$ | 66 | 35.9375 |
1,069 | The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$? | 13 | 75 |
1,070 | In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY / YZ$, in this rectangle is | 5 | 0 |
1,071 | In rectangle $ABCD$, $\overline{AB}=20$ and $\overline{BC}=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $\overline{AE}$? | 20 | 18.75 |
1,072 | What is the value of $\frac{(2112-2021)^2}{169}$? | 49 | 40.625 |
1,073 | The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time? | 6:00 | 85.9375 |
1,074 | Mientka Publishing Company prices its bestseller Where's Walter? as follows:
$C(n) = \begin{cases} 12n, & \text{if } 1 \le n \le 24 \\ 11n, & \text{if } 25 \le n \le 48 \\ 10n, & \text{if } 49 \le n \end{cases}$
where $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ books cost less than $24$ books. For how many values of $n$ is it cheaper to buy more than $n$ books than to buy exactly $n$ books? | 6 | 7.03125 |
1,075 | The value of $\frac{1}{16}a^0+\left (\frac{1}{16a} \right )^0- \left (64^{-\frac{1}{2}} \right )- (-32)^{-\frac{4}{5}}$ is: | 1 | 0.78125 |
1,076 | For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie? | [201,400] | 0 |
1,077 | In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? | 8 | 87.5 |
1,078 | The number $25^{64} \cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is | 14 | 78.90625 |
1,079 | If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals | 0 | 93.75 |
1,080 | In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is | 2\sqrt{3}-3 | 0.78125 |
1,081 | A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player? | 400,000 | 0 |
1,082 | A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? | 200 | 64.84375 |
1,083 | For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$? | 197 | 32.03125 |
1,084 | Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? | \frac{2+\sqrt{3}}{3} | 25 |
1,085 | At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? | 51\% | 75 |
1,086 | Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$ | 100 | 28.90625 |
1,087 | When the fraction $\frac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be | 19 | 99.21875 |
1,088 | Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$? | 30 | 0 |
1,089 | A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$. | 4(36 + \sqrt{113}) | 0 |
1,090 | Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? | 18 | 0 |
1,091 | For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\frac {\sqrt {m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$. | 69 | 0 |
1,092 | The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins? | 22 | 80.46875 |
1,093 | Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there? | 1524 | 1.5625 |
1,094 | Suppose that \(\begin{array}{c} a \\ b \\ c \end{array}\) means $a+b-c$.
For example, \(\begin{array}{c} 5 \\ 4 \\ 6 \end{array}\) is $5+4-6 = 3$.
Then the sum \(\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\) + \(\begin{array}{c} 4 \\ 1 \\ 6 \end{array}\) is | 1 | 19.53125 |
1,095 | Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?
[asy]
size((50));
dot((5,0));
dot((5,5));
dot((0,5));
dot((-5,5));
dot((-5,0));
dot((-5,-5));
dot((0,-5));
dot((5,-5));
[/asy] | \frac{2}{7} | 47.65625 |
1,096 | There are several sets of three different numbers whose sum is $15$ which can be chosen from $\{ 1,2,3,4,5,6,7,8,9 \}$. How many of these sets contain a $5$? | 4 | 86.71875 |
1,097 | How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}?$ | 9 | 97.65625 |
1,098 | A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$? | 20 | 75 |
1,099 | Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$? | \frac{1}{16} | 3.125 |
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