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|---|---|---|---|---|---|
100 | What is the tens digit of $7^{2011}$? | 4 | 8 | 0 | 8 |
101 | For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | 4 | 4 | 8 |
102 | Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as | -f(-y) | 0 | 8 | 8 |
103 | For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together? | $87.50 | 8 | 0 | 8 |
104 | Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$? | 3 | 7 | 1 | 8 |
105 | If $x \geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}} =$ | $\sqrt[8]{x^7}$ | 0 | 8 | 8 |
106 | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$. | 289 | 8 | 0 | 8 |
107 | There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | 40 | 7 | 1 | 8 |
108 | What is $10 \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}$? | \frac{25}{2} | 8 | 0 | 8 |
109 | Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose? | 14 | 8 | 0 | 8 |
110 | The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: | 2 | 8 | 0 | 8 |
111 | Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
$\textbf{(I)}\ x+y < a+b$
$\textbf{(II)}\ x-y < a-b$
$\textbf{(III)}\ xy < ab$
$\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$ | 1 | 8 | 0 | 8 |
112 | After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was | 1:1 | 8 | 0 | 8 |
113 | A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day? | \frac{4}{9} | 8 | 0 | 8 |
114 | What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$? | 8 | 8 | 0 | 8 |
115 | A jacket and a shirt originally sold for $80$ dollars and $40$ dollars, respectively. During a sale Chris bought the $80$ dollar jacket at a $40\%$ discount and the $40$ dollar shirt at a $55\%$ discount. The total amount saved was what percent of the total of the original prices? | 45\% | 0 | 8 | 8 |
116 | Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? | 64 | 8 | 0 | 8 |
117 | Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? | 1 | 8 | 0 | 8 |
118 | Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? | \frac{\pi}{3} | 0 | 8 | 8 |
119 | Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? | \frac{1}{2} | 6 | 2 | 8 |
120 | An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? | \frac{10a}{r} | 0 | 8 | 8 |
121 | The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is | 18 | 8 | 0 | 8 |
122 | For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$? | 13\sqrt{2} | 0 | 8 | 8 |
123 | $\sqrt{8}+\sqrt{18}=$ | $5\sqrt{2}$ | 8 | 0 | 8 |
124 | What is the hundreds digit of $(20! - 15!)?$ | 0 | 7 | 1 | 8 |
125 | How many positive integer factors of $2020$ have more than $3$ factors? | 7 | 8 | 0 | 8 |
126 | Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda? | 23\% | 0 | 8 | 8 |
127 | Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? | 48 | 8 | 0 | 8 |
128 | Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? | 10 | 8 | 0 | 8 |
129 | Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $4$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$? | \frac{1}{10} | 8 | 0 | 8 |
130 | A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$? | -13.5 | 8 | 0 | 8 |
131 | What is the greatest number of consecutive integers whose sum is $45?$ | 90 | 8 | 0 | 8 |
132 | What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 | 8 | 0 | 8 |
133 | A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | \frac{1}{12} | 8 | 0 | 8 |
134 | The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: | 16 | 0 | 8 | 8 |
135 | The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to: | a^{-2}+b^{-2} | 0 | 8 | 8 |
136 | A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is: | 9 \text{ inches} | 8 | 0 | 8 |
137 | A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?
[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label("$A$",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5)); draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label("$A$",(6.5,0),NW); dot((6.5,0)); [/asy] | 2(w+h)^2 | 0 | 8 | 8 |
138 | Of the following expressions the one equal to $\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is: | \frac{a^2b^2}{b^3 - a^3} | 0 | 8 | 8 |
139 | What is the value of $\frac{11! - 10!}{9!}$? | 100 | 8 | 0 | 8 |
140 | Let $A, M$, and $C$ be digits with $(100A+10M+C)(A+M+C) = 2005$. What is $A$? | 4 | 8 | 0 | 8 |
141 | Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. | -1/9 | 8 | 0 | 8 |
142 | Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? | \frac{25}{32} | 6 | 2 | 8 |
143 | If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is: | 7 | 8 | 0 | 8 |
144 | $\frac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$ | $\frac{2}{3}$ | 8 | 0 | 8 |
145 | The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is | 28 | 8 | 0 | 8 |
146 | How many integers $n \geq 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that
\[|z_1| = |z_2| = \dots = |z_n| = 1 \text{ and } z_1 + z_2 + \dots + z_n = 0,\]
then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex plane? | 2 | 8 | 0 | 8 |
147 | If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? | \frac{x(x+2)(x+5)}{(x+1)(x+3)} | 0 | 8 | 8 |
148 | The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? | 26 | 8 | 0 | 8 |
149 | A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire.
The length of the shortest wire that will go around them is: | 12\sqrt{3}+14\pi | 0 | 8 | 8 |
150 | Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the start do they pass each other? | 47 | 8 | 0 | 8 |
151 | The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be | 130 | 8 | 0 | 8 |
152 | Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25$, and $36$. What is the sum of the areas of the six rectangles? | 182 | 8 | 0 | 8 |
153 | 650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? | \frac{5}{2} | 0 | 8 | 8 |
154 | In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.
Thereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is: | 210 | 8 | 0 | 8 |
155 | The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? | 14 | 0 | 8 | 8 |
156 | Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$? | 2 | 8 | 0 | 8 |
157 | The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 9 | 8 | 0 | 8 |
158 | In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(60)); draw(circle(14*dir(60),1)); draw((1,0)--(6,0)); draw(circle((7,0),1)); draw((8,0)--(13,0)); draw(circle((14,0),1)); draw(circle((10.5,6.0621778264910705273460621952706),1)); draw((13.5,0.86602540378443864676372317075294)--(11,5.1961524227066318805823390245176)); draw((10,6.9282032302755091741097853660235)--(7.5,11.258330249197702407928401219788)); [/asy] | 39 | 8 | 0 | 8 |
159 | Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$. What is the value of $\frac{x+3y}{3x-y}$? | 2 | 8 | 0 | 8 |
160 | The product $8 \times .25 \times 2 \times .125 =$ | $\frac{1}{2}$ | 8 | 0 | 8 |
161 | In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ | 6.4 | 0 | 8 | 8 |
162 | Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy]
unitsize(6mm);
defaultpen(linewidth(.8pt));
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw(rotate(90)*p);
draw(rotate(180)*p);
draw(rotate(270)*p);
[/asy] | 3 | 8 | 0 | 8 |
163 | Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 | 8 | 0 | 8 |
164 | If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is | 18 | 8 | 0 | 8 |
165 | If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is | 4\sqrt{3} | 0 | 8 | 8 |
166 | Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\frac{1}{2}$ and $\frac{2}{3}$. Armed with this information, what number should Carol choose to maximize her chance of winning? | \frac{13}{24} | 8 | 0 | 8 |
167 | The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: | 6\frac{2}{3} | 0 | 8 | 8 |
168 | Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$ | $3\sqrt{35}$ | 8 | 0 | 8 |
169 | A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet? | $48\pi - 36 \sqrt {3}$ | 8 | 0 | 8 |
170 | The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: | \frac{880}{\pi} | 0 | 8 | 8 |
171 | $\frac{(.2)^3}{(.02)^2} =$ | 20 | 8 | 0 | 8 |
172 | Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? | 6 | 8 | 0 | 8 |
173 | A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | \frac{50}{99} | 8 | 0 | 8 |
174 | Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? | \frac{5}{256} | 0 | 8 | 8 |
175 | Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of segment $AB$ is | 70 | 8 | 0 | 8 |
176 | If $f(x)=\frac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\sqrt{-1}$, is equal to | 0 | 8 | 0 | 8 |
177 | Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$. She places the rods with lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? | 17 | 7 | 1 | 8 |
178 | Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its digits is 9.
This information allows Malcolm to determine Isabella's house number. What is its units digit? | 8 | 8 | 0 | 8 |
179 | Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take? | 18 | 8 | 0 | 8 |
180 | Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters? | 34 | 8 | 0 | 8 |
181 | The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. The amount of money in the company fund before any bonuses were paid was | 995 | 8 | 0 | 8 |
182 | $6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = $ | $6^7$ | 8 | 0 | 8 |
183 | Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes? | 45 | 8 | 0 | 8 |
184 | Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school? | 8 | 8 | 0 | 8 |
185 | What is the value of $\frac{(2112-2021)^2}{169}$? | 49 | 8 | 0 | 8 |
186 | A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game? | 37 | 7 | 1 | 8 |
187 | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$? | \sqrt{30} | 0 | 8 | 8 |
188 | Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$? | \frac{125}{12} | 8 | 0 | 8 |
189 | Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? | 9 | 8 | 0 | 8 |
190 | The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? | 24000 | 8 | 0 | 8 |
191 | In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$? | 12 | 8 | 0 | 8 |
192 | If $f(x)=3x+2$ for all real $x$, then the statement:
"$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$"
is true when | $b \le a/3$ | 6 | 2 | 8 |
193 | If $a, b, c$ are positive integers less than $10$, then $(10a + b)(10a + c) = 100a(a + 1) + bc$ if: | $b+c=10$ | 8 | 0 | 8 |
194 | Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$? | 3 | 8 | 0 | 8 |
195 | A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? | 15 | 8 | 0 | 8 |
196 | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$
Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth? | $(6,2,1)$ | 0 | 8 | 8 |
197 | In a collection of red, blue, and green marbles, there are $25\%$ more red marbles than blue marbles, and there are $60\%$ more green marbles than red marbles. Suppose that there are $r$ red marbles. What is the total number of marbles in the collection? | 3.4r | 0 | 8 | 8 |
198 | If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by | 11-k | 0 | 8 | 8 |
199 | If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\cdots$ and $F(1)=2$, then $F(101)$ equals: | 52 | 8 | 0 | 8 |
Subsets and Splits