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|---|---|---|---|
1,900 | For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 | 91.40625 |
1,901 | Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? | 1 : 14 | 0 |
1,902 | A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black? | \frac{49}{512} | 42.1875 |
1,903 | When the repeating decimal $0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: | 15 | 93.75 |
1,904 | The fraction $\frac{\sqrt{a^2+x^2}-\frac{x^2-a^2}{\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to: | \frac{2a^2}{(a^2+x^2)^{\frac{3}{2}}} | 2.34375 |
1,905 | Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $\frac{1}{6}$, independent of the outcome of any other toss.) | \frac{36}{91} | 2.34375 |
1,906 | Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Randy's trip? | \frac{300}{7} | 70.3125 |
1,907 | A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?
[asy]
size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));
[/asy]
Note: 1 mile = 5280 feet | \frac{\pi}{10} | 36.71875 |
1,908 | How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles? | 5 | 98.4375 |
1,909 | Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. | 20 | 0 |
1,910 | Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | \frac{3}{4} | 9.375 |
1,911 | The area in square units of the region enclosed by parallelogram $ABCD$ is | 8 | 1.5625 |
1,912 | Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
\item $\frac{a}{2} \le x \le 2a$
\item $\frac{a}{2} \le y \le 2a$
\item $x+y \ge a$
\item $x+a \ge y$
\item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with | 6 | 28.125 |
1,913 | The sum of three numbers is $98$. The ratio of the first to the second is $\frac {2}{3}$,
and the ratio of the second to the third is $\frac {5}{8}$. The second number is: | 30 | 98.4375 |
1,914 | A man has $10,000 to invest. He invests $4000 at 5% and $3500 at 4%. In order to have a yearly income of $500, he must invest the remainder at: | 6.4\% | 87.5 |
1,915 | Which pair of numbers does NOT have a product equal to $36$? | {\frac{1}{2},-72} | 0 |
1,916 | How many ordered triples (x,y,z) of integers satisfy the system of equations below?
\begin{array}{l} x^2-3xy+2y^2-z^2=31 \ -x^2+6yz+2z^2=44 \ x^2+xy+8z^2=100\ \end{array} | 0 | 75.78125 |
1,917 | If the length of a diagonal of a square is $a + b$, then the area of the square is: | \frac{1}{2}(a+b)^2 | 0 |
1,918 | Three $\Delta$'s and a $\diamondsuit$ will balance nine $\bullet$'s. One $\Delta$ will balance a $\diamondsuit$ and a $\bullet$.
How many $\bullet$'s will balance the two $\diamondsuit$'s in this balance? | 3 | 95.3125 |
1,919 | If $A=20^{\circ}$ and $B=25^{\circ}$, then the value of $(1+\tan A)(1+\tan B)$ is | 2 | 98.4375 |
1,920 | What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \times 5?$ | \frac{1}{5} | 53.90625 |
1,921 | A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$? | 2 | 53.125 |
1,922 | The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | \frac{1}{19} | 46.09375 |
1,923 | On a $10000 order a merchant has a choice between three successive discounts of 20%, 20%, and 10% and three successive discounts of 40%, 5%, and 5%. By choosing the better offer, he can save: | $345 | 0 |
1,924 | The number of positive integers $k$ for which the equation
\[kx-12=3k\]has an integer solution for $x$ is | 6 | 100 |
1,925 | One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? | 3.5 | 2.34375 |
1,926 | Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$, rounded to the one decimal place, is | 4.2 | 6.25 |
1,927 | A telephone number has the form \text{ABC-DEF-GHIJ}, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,
$D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd
digits; and $A + B + C = 9$. Find $A$. | 8 | 61.71875 |
1,928 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines? | 6 | 40.625 |
1,929 | Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? | \frac{5}{16} | 45.3125 |
1,930 | Triangle $\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$,
are on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$
have equal areas, then that area is | 6 | 0.78125 |
1,931 | A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? | 26 | 67.96875 |
1,932 | The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | \frac{1}{19} | 53.125 |
1,933 | A calculator has a squaring key $\boxed{x^2}$ which replaces the current number displayed with its square. For example, if the display is $\boxed{000003}$ and the $\boxed{x^2}$ key is depressed, then the display becomes $\boxed{000009}$. If the display reads $\boxed{000002}$, how many times must you depress the $\boxed{x^2}$ key to produce a displayed number greater than $500$? | 4 | 86.71875 |
1,934 | For how many positive integer values of $N$ is the expression $\dfrac{36}{N+2}$ an integer? | 7 | 88.28125 |
1,935 | There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that
\[A \log_{200} 5 + B \log_{200} 2 = C.\]What is $A + B + C$? | 6 | 95.3125 |
1,936 | A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | 2\sqrt{5} | 27.34375 |
1,937 | The values of $a$ in the equation: $\log_{10}(a^2 - 15a) = 2$ are: | 20, -5 | 80.46875 |
1,938 | Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? | 10 | 62.5 |
1,939 | Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 | 33.59375 |
1,940 | Given that $\text{1 mile} = \text{8 furlongs}$ and $\text{1 furlong} = \text{40 rods}$, the number of rods in one mile is | 320 | 100 |
1,941 | For every $n$ the sum of $n$ terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: | 6r - 1 | 97.65625 |
1,942 | Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? | 137 | 64.0625 |
1,943 | Figure $ABCD$ is a trapezoid with $AB \parallel DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$, and $\angle CDA=60^\circ$. The length of $DC$ is | 8 + \sqrt{3} | 33.59375 |
1,944 | A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is: | $5s + 80n$ | 0 |
1,945 | Tom, Dick and Harry started out on a $100$-mile journey. Tom and Harry went by automobile at the rate of $25$ mph, while Dick walked at the rate of $5$ mph. After a certain distance, Harry got off and walked on at $5$ mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was: | 8 | 26.5625 |
1,946 | A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is | 50000000 | 0.78125 |
1,947 | A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? | 20\sqrt{2} | 55.46875 |
1,948 | The ten-letter code $\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\text{CLUE}$? | 8671 | 90.625 |
1,949 | The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\frac{3}{2}$ and center $(0,\frac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? | \frac{9\pi}{8} | 1.5625 |
1,950 | If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? | \frac{1}{2} | 74.21875 |
1,951 | Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? | 31 | 96.09375 |
1,952 | Let $R$ be a rectangle. How many circles in the plane of $R$ have a diameter both of whose endpoints are vertices of $R$? | 5 | 4.6875 |
1,953 | Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals: | 5 | 99.21875 |
1,954 | The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? | 2 | 7.8125 |
1,955 | A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$? | 4 | 38.28125 |
1,956 | 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} | 32.8125 |
1,957 | Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes. If she doesn’t use it any more but leaves the phone on, how many more hours will the battery last? | 8 | 71.875 |
1,958 | A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? | 60 | 73.4375 |
1,959 | The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose? | 27 | 79.6875 |
1,960 | A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the second column $14,15,\ldots,26$ and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system). | 555 | 68.75 |
1,961 | A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$? | 2 | 94.53125 |
1,962 | The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$? | 761 | 85.15625 |
1,963 | All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class? | 13 | 89.84375 |
1,964 | Jeff rotates spinners $P$, $Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number? | 1/3 | 0.78125 |
1,965 | In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 65 | 0 |
1,966 | In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? | \frac{4}{3} | 4.6875 |
1,967 | Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? | 20 | 82.8125 |
1,968 | The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a, y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is: | 0 | 33.59375 |
1,969 | In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: | 24n | 2.34375 |
1,970 | If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$? | 127 | 51.5625 |
1,971 | Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block. | 114 | 0 |
1,972 | If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount: | $\pm 2ax + a^2$ | 0 |
1,973 | On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. | 2 | 75.78125 |
1,974 | Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\prime} = 10$, $BB^{\prime}= 8$, $CC^\prime = 18$, and $DD^\prime = 22$ and $M$ and $N$ are the midpoints of $A^{\prime} C^{\prime}$ and $B^{\prime}D^{\prime}$, respectively, then $MN =$ | 1 | 67.1875 |
1,975 | A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ | 77 | 20.3125 |
1,976 | Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. What is the degree measure of $\angle AMD$? | 75 | 2.34375 |
1,977 | If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation | r^2+r+1=0 | 15.625 |
1,978 | When a student multiplied the number $66$ by the repeating decimal, \(1.\overline{ab}\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$? | 75 | 77.34375 |
1,979 | The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? | 7 | 98.4375 |
1,980 | Bricklayer Brenda takes $9$ hours to build a chimney alone, and bricklayer Brandon takes $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney? | 900 | 87.5 |
1,981 | An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is | 2\pi+4 | 0.78125 |
1,982 | In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD=BC=5$, $AB=4$, and $DC=10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in right triangle $DEF$. Then $CF=$ | 4.0 | 0 |
1,983 | What number is directly above $142$ in this array of numbers?
\[\begin{array}{cccccc}& & & 1 & &\\ & & 2 & 3 & 4 &\\ & 5 & 6 & 7 & 8 & 9\\ 10 & 11 & 12 &\cdots & &\\ \end{array}\] | 120 | 7.03125 |
1,984 | For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides
$\prod_{n=2}^{5300}\text{pow}(n)$? | 77 | 0 |
1,985 | If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for
$\begin{array}{ccc} X & X & X \ & Y & X \ + & & X \ \hline \end{array}$
has the form | $YYZ$ | 0 |
1,986 | With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time? | 60 | 69.53125 |
1,987 | A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? | \frac{3 + \sqrt{5}}{2} | 3.90625 |
1,988 | Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | \frac{223}{286} | 0 |
1,989 | Two equal circles in the same plane cannot have the following number of common tangents. | 1 | 13.28125 |
1,990 | The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \(\begin{tabular}[t]{lllllllll} 89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\ 63, & 84, & 78, & 71, & 80, & 90. & & & \\ \end{tabular}\)
In Mr. Freeman's class, what percent of the students received a grade of C? | 33\frac{1}{3}\% | 0 |
1,991 | A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$? | 4.5 | 19.53125 |
1,992 | A square of side length $1$ and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square? | \frac{2\pi}{9} - \frac{\sqrt{3}}{3} | 0 |
1,993 | (6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by | \div | 53.90625 |
1,994 | Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]
Assume that the planes cutting the prism do not intersect anywhere in or on the prism. | 36 | 74.21875 |
1,995 | For all positive integers $n$ less than $2002$, let
\begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if } n \text{ is divisible by } 13 \text{ and } 14;\\ 13, & \text{if } n \text{ is divisible by } 14 \text{ and } 11;\\ 14, & \text{if } n \text{ is divisible by } 11 \text{ and } 13;\\ 0, & \text{otherwise}. \end{array} \right. \end{eqnarray*}
Calculate $\sum_{n=1}^{2001} a_n$. | 448 | 59.375 |
1,996 | Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese? | 40 | 71.875 |
1,997 | The product $\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{9^{2}}\right)\left(1-\frac{1}{10^{2}}\right)$ equals | \frac{11}{20} | 96.875 |
1,998 | How many positive factors does $23,232$ have? | 42 | 96.09375 |
1,999 | Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$? | $a+4$ | 0 |
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