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|---|---|---|---|
1,300 | If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately | 20^{\circ} | 0 |
1,301 | After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members? | 11 | 46.875 |
1,302 | A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | \frac{1}{3} | 92.96875 |
1,303 | The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil? | 4 | 92.96875 |
1,304 | What is equal to $\frac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$? | \frac{1}{2} | 80.46875 |
1,305 | Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling? | 4 | 99.21875 |
1,306 | What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$? | -280 | 57.03125 |
1,307 | If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: | 100t+10u+1 | 83.59375 |
1,308 | For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals: | \frac{2}{7} | 44.53125 |
1,309 | Find the area of the shaded region. | 6\dfrac{1}{2} | 0 |
1,310 | The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216 \pi$. What is the length $AB$? | 20 | 95.3125 |
1,311 | At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$? | 2 | 88.28125 |
1,312 | Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the nth set. Then $S_{21}$ equals: | 4641 | 90.625 |
1,313 | A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches? | 88 | 89.84375 |
1,314 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | 95.3125 |
1,315 | A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? | \frac{1}{5} | 6.25 |
1,316 | One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? | 125 | 98.4375 |
1,317 | A shopper buys a $100$ dollar coat on sale for $20\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\%$ is paid on the final selling price. The total amount the shopper pays for the coat is | 81.00 | 71.875 |
1,318 | Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game? | \frac{1}{4} | 4.6875 |
1,319 | Through a point $P$ inside the $\triangle ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas.
If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: | \frac{1}{2} | 25.78125 |
1,320 | ABCD is a square with side of unit length. Points E and F are taken respectively on sides AB and AD so that AE = AF and the quadrilateral CDFE has maximum area. In square units this maximum area is: | \frac{5}{8} | 33.59375 |
1,321 | Let N = $69^5 + 5 \cdot 69^4 + 10 \cdot 69^3 + 10 \cdot 69^2 + 5 \cdot 69 + 1$. How many positive integers are factors of $N$? | 216 | 100 |
1,322 | Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | 9 | 0.78125 |
1,323 | Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city? | 23 | 35.9375 |
1,324 | The number halfway between $\frac{1}{8}$ and $\frac{1}{10}$ is | \frac{1}{9} | 0 |
1,325 | If the ratio of $2x-y$ to $x+y$ is $\frac{2}{3}$, what is the ratio of $x$ to $y$? | \frac{5}{4} | 96.09375 |
1,326 | At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | 32 | 73.4375 |
1,327 | Let a binary operation $\star$ on ordered pairs of integers be defined by $(a,b)\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\star (0,0)$ and $(x,y)\star (3,2)$ represent identical pairs, $x$ equals: | $6$ | 0 |
1,328 | 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} | 0.78125 |
1,329 | Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? | 55 | 98.4375 |
1,330 | If $10^{\log_{10}9} = 8x + 5$ then $x$ equals: | \frac{1}{2} | 94.53125 |
1,331 | In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$. If the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is | 12 | 67.1875 |
1,332 | When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to: | x^{ - 1}y^{ - 1} | 67.1875 |
1,333 | A poll shows that $70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? | 0.189 | 93.75 |
1,334 | Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her? | 5 | 68.75 |
1,335 | Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way? | 1 | 54.6875 |
1,336 | Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? | 100 | 97.65625 |
1,337 | Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: | n_1n_2 | 0 |
1,338 | What is the area of the shaded figure shown below? | 6 | 3.125 |
1,339 | The sum of the base-10 logarithms of the divisors of $10^n$ is $792$. What is $n$? | 11 | 71.875 |
1,340 | Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? | 15 | 96.09375 |
1,341 | Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: | 1 | 99.21875 |
1,342 | The perimeter of the polygon shown is | 28 | 2.34375 |
1,343 | In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T''$ as a point of tangency. If $TP=4$ and $T'Q=9$ then $r$ is
[asy]
unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label("O",O,W); label("T",T,N); label("T'",T1,S); label("T''",T2,NE); label("P",P,NE); label("Q",Q,S); draw(O--T2); label("$r$",midpoint(O--T2),NW); draw(T--P); label("4",midpoint(T--P),N); draw(T1--Q); label("9",midpoint(T1--Q),S); draw(P--Q);
[/asy] | 6 | 6.25 |
1,344 | $\frac{2}{1-\frac{2}{3}}=$ | 6 | 85.9375 |
1,345 | What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8? | 4 | 92.96875 |
1,346 | In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$ | 80 | 0 |
1,347 | In $\triangle ABC$ with integer side lengths, $\cos A = \frac{11}{16}$, $\cos B = \frac{7}{8}$, and $\cos C = -\frac{1}{4}$. What is the least possible perimeter for $\triangle ABC$? | 9 | 28.125 |
1,348 | For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? | 6 | 92.96875 |
1,349 | 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 | 85.9375 |
1,350 | What is the smallest prime number dividing the sum $3^{11}+5^{13}$? | 2 | 54.6875 |
1,351 | Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is: | \text{(1) and (4)} | 0 |
1,352 | On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches. | .34 | 7.8125 |
1,353 | The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$? | 40 | 0.78125 |
1,354 | A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle? | 2.37 | 27.34375 |
1,355 | Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $4$ per gallon. How many miles can Margie drive on $20$ worth of gas? | 160 | 78.125 |
1,356 | Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? | -10 | 97.65625 |
1,357 | Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? | 5 | 7.8125 |
1,358 | What is $100(100-3)-(100 \cdot 100-3)$? | -297 | 92.1875 |
1,359 | Two lines with slopes $\frac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$ ? | 6 | 90.625 |
1,360 | A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 40 | 7.8125 |
1,361 | Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? | \frac{1}{6} | 71.875 |
1,362 | Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then the total area of the shaded triangles is nearest | 6 | 71.09375 |
1,363 | On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was $6:57\textsc{am}$, and the sunset as $8:15\textsc{pm}$. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun really set? | $5:21\textsc{pm}$ | 0 |
1,364 | At a store, when a length or a width is reported as $x$ inches that means it is at least $x - 0.5$ inches and at most $x + 0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle? | 3.75 | 100 |
1,365 | In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? | \frac{4}{7} | 96.09375 |
1,366 | What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? | 3127 | 92.96875 |
1,367 | The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? | 4 | 82.03125 |
1,368 | A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (asumming that the letter $A$ is one place to the right of the letter $Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $1+2$ places to the right, the third time it is replaced by the letter that is $1+2+3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $s$ in the message "Lee's sis is a Mississippi miss, Chriss!?" | s | 10.9375 |
1,369 | Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 400 | 29.6875 |
1,370 | Six pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni? | \frac{2}{3} | 96.875 |
1,371 | If $S = i^n + i^{-n}$, where $i = \sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: | 3 | 94.53125 |
1,372 | All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? | 70 | 15.625 |
1,373 | Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$? | 0 | 95.3125 |
1,374 | Find the units digit of the decimal expansion of $\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}$. | 9 | 0.78125 |
1,375 | If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m \ne 0$, $n \ne 0$, then the sum of the roots is: | -1 | 94.53125 |
1,376 | For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? | all | 0 |
1,377 | Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$? | 7 | 63.28125 |
1,378 | The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$? | 929 | 0 |
1,379 | The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to | $1.05 | 0 |
1,380 | Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$? | 2\sqrt{3} | 4.6875 |
1,381 | An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$. How many small triangles are required? | 100 | 92.96875 |
1,382 | A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.) | 36 | 0 |
1,383 | 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 |
1,384 | The square of $5-\sqrt{y^2-25}$ is: | y^2-10\sqrt{y^2-25} | 88.28125 |
1,385 | Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: | 1 - \frac{1}{n^2} | 93.75 |
1,386 | There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$ | \frac{1}{2001} | 42.96875 |
1,387 | A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation? | \sqrt{13} | 35.9375 |
1,388 | Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$ | 9 | 92.1875 |
1,389 | A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 | 63.28125 |
1,390 | Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. The maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \dots, L_{100}\}$ is | 4351 | 4.6875 |
1,391 | A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $x$-coordinates of $P$ and $Q$) is: | 4 | 17.1875 |
1,392 | A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 | 21.875 |
1,393 | How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$? | 2 | 89.84375 |
1,394 | What is the value of \(2^{1+2+3}-(2^1+2^2+2^3)?\) | 50 | 55.46875 |
1,395 | Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work for the final week to earn the tickets? | 12 | 82.8125 |
1,396 | A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is | 30\pi - 9\sqrt3 | 0.78125 |
1,397 | Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? | 128 | 0 |
1,398 | After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? | 24 | 21.875 |
1,399 | Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? | 60 | 13.28125 |
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