Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
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stringlengths 1
1.22k
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float64 0
100
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|---|---|---|---|
1,700 | Chloe chooses a real number uniformly at random from the interval $[ 0,2017 ]$. Independently, Laurent chooses a real number uniformly at random from the interval $[ 0 , 4034 ]$. What is the probability that Laurent's number is greater than Chloe's number? | \frac{3}{4} | 25.78125 |
1,701 | Let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\lfloor x \rfloor +51=0$ is | 4 | 32.03125 |
1,702 | Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? | 20\% | 93.75 |
1,703 | Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$? | 10 | 96.09375 |
1,704 | Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$ | 7 | 74.21875 |
1,705 | Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | \frac{2}{5} | 76.5625 |
1,706 | What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? | 10 | 100 |
1,707 | On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days? | 400 | 92.96875 |
1,708 | The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word.
The last letter of the $86$th word in this list is: | E | 35.9375 |
1,709 | Two equilateral triangles are contained in a square whose side length is $2\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? | 8\sqrt{3} - 12 | 0 |
1,710 | If $\frac{a}{10^x-1}+\frac{b}{10^x+2}=\frac{2 \cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is: | \frac{4}{3} | 98.4375 |
1,711 | The 2-digit integers from 19 to 92 are written consecutively to form the integer \(N=192021\cdots9192\). Suppose that \(3^k\) is the highest power of 3 that is a factor of \(N\). What is \(k\)? | 1 | 77.34375 |
1,712 | How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ | 100 | 96.09375 |
1,713 | If $\sqrt{x+2}=2$, then $(x+2)^2$ equals: | 16 | 100 |
1,714 | From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by | $\left(i+j+\frac{ij}{100}\right)\%$ | 0 |
1,715 | If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals: | $b$ | 0 |
1,716 | If $a<b<c<d<e$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e$ is a perfect cube, what is the smallest possible value of $c$? | 182 | 0 |
1,717 | Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?
[asy]
size(150);
defaultpen(linewidth(0.8));
path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);
for(int i=0;i<=2;i=i+1) {
for(int j=0;j<=3-i;j=j+1) {
filldraw(shift((i,j))*h,black);
filldraw(shift((j,i))*v,black);
}
}
[/asy] | 22 | 32.8125 |
1,718 | What is the value of $1234 + 2341 + 3412 + 4123$ | 11110 | 54.6875 |
1,719 | The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | 17 | 68.75 |
1,720 | Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides $5$ times as fast as Bella walks. The distance between their houses is $2$ miles, which is $10,560$ feet, and Bella covers $2 \frac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella? | 704 | 90.625 |
1,721 | Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*unitcircle; pair[] A; pen grey1 = rgb(100/256, 100/256, 100/256); pen grey2 = rgb(183/256, 183/256, 183/256); for (int i=0; i<7; ++i) { A[i] = rotate(60*i)*(1,0);} path hex = A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--cycle; fill(p,grey1); draw(scale(1.25)*hex,black+linewidth(1.25)); pair S = 6A[0]+2A[1]; fill(shift(S)*p,grey1); for (int i=0; i<6; ++i) { fill(shift(S+2*A[i])*p,grey2);} draw(shift(S)*scale(3.25)*hex,black+linewidth(1.25)); pair T = 16A[0]+4A[1]; fill(shift(T)*p,grey1); for (int i=0; i<6; ++i) { fill(shift(T+2*A[i])*p,grey2); fill(shift(T+4*A[i])*p,grey1); fill(shift(T+2*A[i]+2*A[i+1])*p,grey1); } draw(shift(T)*scale(5.25)*hex,black+linewidth(1.25)); [/asy] | 37 | 89.84375 |
1,722 | An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? | \frac{1}{5} | 10.15625 |
1,723 | Amelia has a coin that lands heads with probability $\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$? | 4 | 69.53125 |
1,724 | The larger root minus the smaller root of the equation $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$ is | 6-3\sqrt{3} | 29.6875 |
1,725 | A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? | \frac{3}{5} | 88.28125 |
1,726 | If $x$ and $y$ are non-zero real numbers such that
\[|x|+y=3 \qquad \text{and} \qquad |x|y+x^3=0,\]
then the integer nearest to $x-y$ is | -3 | 88.28125 |
1,727 | What is the median of the following list of $4040$ numbers?
\[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] | 1976.5 | 10.15625 |
1,728 | Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? | \frac{1}{8} | 98.4375 |
1,729 | Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10444$ and $3245$, and LeRoy obtains the sum $S = 13689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? | 25 | 74.21875 |
1,730 | In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? | 3 | 83.59375 |
1,731 | A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$? | \frac{32}{5} | 0 |
1,732 | (1+11+21+31+41)+(9+19+29+39+49)= | 200 | 0.78125 |
1,733 | If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals | -1 | 89.0625 |
1,734 | Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | \frac{195}{8} | 3.125 |
1,735 | Suppose that $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? | 3 | 76.5625 |
1,736 | The smaller root of the equation $\left(x- \frac{3}{4} \right) \left(x - \frac{3}{4} \right) + \left(x - \frac{3}{4} \right) \left(x - \frac{1}{2} \right) = 0$ is: | \frac{5}{8} | 83.59375 |
1,737 | In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}[t]{r} ABBCB \\ + \\ BCADA \\ \hline DBDDD \end{array}$ | 7 | 31.25 |
1,738 | The factors of $x^4+64$ are: | (x^2-4x+8)(x^2+4x+8) | 56.25 |
1,739 | An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$? | \frac{1}{3} | 65.625 |
1,740 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ | 85 | 0 |
1,741 | A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result? | 120 | 70.3125 |
1,742 | Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? | 20201 | 95.3125 |
1,743 | How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 | 98.4375 |
1,744 | Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: | \frac{1}{\sqrt{6}} | 10.9375 |
1,745 | How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.) | 15 | 73.4375 |
1,746 | Let $(a_1, a_2, \dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there? | 512 | 60.15625 |
1,747 | How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. | 17 | 96.09375 |
1,748 | 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 | 89.84375 |
1,749 | A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X$ + $Y$ + $Z$? | 22 | 84.375 |
1,750 | Each valve $A$, $B$, and $C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. The number of hours required with only valves $A$ and $B$ open is | 1.2 | 70.3125 |
1,751 | How many whole numbers between $100$ and $400$ contain the digit $2$? | 138 | 96.09375 |
1,752 | LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.$ How many dollars must LeRoy give to Bernardo so that they share the costs equally? | \frac{B-A}{2} | 81.25 |
1,753 | A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle? | \frac{2}{5}x^2 | 0 |
1,754 | In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ becomes arbitrarily close to: | \frac{1}{\sqrt{2}}+\frac{1}{2} | 0 |
1,755 | A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy]
fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);
draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));
fill(circle((1,5/4),1/2),white);
draw(circle((1,5/4),1/2),linewidth(1));
[/asy] | 5 | 96.875 |
1,756 | Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win? | 1 | 71.875 |
1,757 | A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? | \frac{2}{243} | 0.78125 |
1,758 | The sum of two numbers is $10$; their product is $20$. The sum of their reciprocals is: | \frac{1}{2} | 93.75 |
1,759 | A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? | 6 | 95.3125 |
1,760 | If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals: | \frac{2S}{V+V_0} | 0.78125 |
1,761 | To complete the grid below, each of the digits 1 through 4 must occur once
in each row and once in each column. What number will occupy the lower
right-hand square?
\[\begin{tabular}{|c|c|c|c|}\hline 1 & & 2 &\ \hline 2 & 3 & &\ \hline & &&4\ \hline & &&\ \hline\end{tabular}\] | 1 | 10.9375 |
1,762 | There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? | \frac{22}{45} | 94.53125 |
1,763 | What is $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?$ | \frac{7}{12} | 91.40625 |
1,764 | If $a = \log_8 225$ and $b = \log_2 15$, then $a$, in terms of $b$, is: | \frac{2b}{3} | 94.53125 |
1,765 | Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$. What is the minimum possible value of $m + n$? | 60 | 93.75 |
1,766 | The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square? | 50 | 84.375 |
1,767 | How many subsets containing three different numbers can be selected from the set \(\{ 89,95,99,132, 166,173 \}\) so that the sum of the three numbers is even? | 12 | 80.46875 |
1,768 | What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$? | 4 | 84.375 |
1,769 | Triangle $ABC$ has a right angle at $B$, $AB=1$, and $BC=2$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. What is $BD$? | \frac{\sqrt{5} - 1}{2} | 78.90625 |
1,770 | The centers of two circles are $41$ inches apart. The smaller circle has a radius of $4$ inches and the larger one has a radius of $5$ inches.
The length of the common internal tangent is: | 40 \text{ inches} | 87.5 |
1,771 | Given the following six statements:
(1) All women are good drivers
(2) Some women are good drivers
(3) No men are good drivers
(4) All men are bad drivers
(5) At least one man is a bad driver
(6) All men are good drivers.
The statement that negates statement (6) is: | (5) | 0 |
1,772 | Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet? | \frac{12}{13} | 56.25 |
1,773 | Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has 28 solutions in positive integers $x$, $y$, and $z$, then $n$ must be either | 17 or 18 | 0 |
1,774 | Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\$10500$ for both taxes. How many dollars was his inheritance? | 37500 | 98.4375 |
1,775 | In $\triangle ABC$, $\angle A=55^\circ$, $\angle C=75^\circ$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle{BED} =$ | 65^\circ | 85.15625 |
1,776 | 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} | 93.75 |
1,777 | A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is: | 7 | 55.46875 |
1,778 | Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? | 336 | 56.25 |
1,779 | Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive? | 9 | 92.96875 |
1,780 | On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$? | 1 | 5.46875 |
1,781 | Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? | 11:00 | 92.1875 |
1,782 | For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is: | $R+R'D$ | 0 |
1,783 | The diagonal of square $I$ is $a+b$. The area of square $II$ with twice the area of $I$ is: | $(a+b)^2$ | 0 |
1,784 | According to the standard convention for exponentiation,
\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible? | 1 | 88.28125 |
1,785 | $(4^{-1} - 3^{-1})^{-1} = $ | -12 | 92.1875 |
1,786 | Let $s$ be the limiting sum of the geometric series $4 - \frac{8}{3} + \frac{16}{9} - \dots$, as the number of terms increases without bound. Then $s$ equals: | 2.4 | 10.15625 |
1,787 | Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | 3 | 96.09375 |
1,788 | Which digit of $.12345$, when changed to $9$, gives the largest number? | 1 | 32.8125 |
1,789 | If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11? | \frac{1}{5} | 17.1875 |
1,790 | Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins? | \frac{5}{192} | 0 |
1,791 | The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is | 80 | 96.09375 |
1,792 | For how many integers $n$ is $\frac n{20-n}$ the square of an integer? | 4 | 25.78125 |
1,793 | Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:
\[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\]
What is $a-d$? | 10 | 40.625 |
1,794 | As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring? | 810 | 0 |
1,795 | Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice? | \frac{11}{30} | 89.0625 |
1,796 | A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is | 3n+1 | 0 |
1,797 | If $x = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$, then: | 1 < x < 2 | 0 |
1,798 | A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? | 17 | 7.8125 |
1,799 | Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is: | 9 | 80.46875 |
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