Unnamed: 0
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40.3k
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stringlengths 10
5.15k
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stringlengths 1
1.22k
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100
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|---|---|---|---|
2,000 | A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $\frac{B}{A}$? | \frac{4}{5} | 0 |
2,001 | If $i^2 = -1$, then the sum $\cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} + \cdots + i^{40}\cos{3645^\circ}$ equals | \frac{\sqrt{2}}{2}(21 - 20i) | 0 |
2,002 | How many different real numbers $x$ satisfy the equation $(x^{2}-5)^{2}=16$? | 4 | 100 |
2,003 | Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\frac{s}{S}$ equals | \sqrt{2} | 1.5625 |
2,004 | Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's cookies are parallelograms:
$\circ$ Trisha's cookies are triangles:
Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? | 40 | 3.90625 |
2,005 | Points $( \sqrt{\pi} , a)$ and $( \sqrt{\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$? | 2 | 73.4375 |
2,006 | A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is: | x < \frac{1}{2} | 0 |
2,007 | Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have? | 8 | 90.625 |
2,008 | What is the area of the shaded figure shown below? | 6 | 4.6875 |
2,009 | Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans? | 1:2 | 0 |
2,010 | What expression describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points? | a>\frac{1}{2} | 52.34375 |
2,011 | Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is: | $s\sqrt{2}$ | 0 |
2,012 | If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is: | 32 | 100 |
2,013 | A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$? | $\frac{3\sqrt{7}-\sqrt{3}}{2}$ | 0 |
2,014 | If the radius of a circle is increased $100\%$, the area is increased: | 300\% | 97.65625 |
2,015 | The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to: | \sqrt{2} | 82.03125 |
2,016 | Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$ | 2 | 34.375 |
2,017 | A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures? | 5 | 85.9375 |
2,018 | An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins? | 48\% | 81.25 |
2,019 | Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear? | 11 | 4.6875 |
2,020 | Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then | 120 < n \leq 130 | 0 |
2,021 | A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn? | 76 | 89.0625 |
2,022 | The three row sums and the three column sums of the array \[
\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]
\]
are the same. What is the least number of entries that must be altered to make all six sums different from one another? | 4 | 9.375 |
2,023 | If $\log_2(\log_2(\log_2(x)))=2$, then how many digits are in the base-ten representation for x? | 5 | 96.09375 |
2,024 | If $x$, $y$, and $z$ are real numbers such that
$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$,
then $x + y + z =$ | $12$ | 0 |
2,025 | If $x>y>0$ , then $\frac{x^y y^x}{y^y x^x}=$ | {\left(\frac{x}{y}\right)}^{y-x} | 0 |
2,026 | In their base $10$ representations, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. What is the sum of the digits of the base $10$ representation of $9ab$? | 17865 | 46.09375 |
2,027 | A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? | \frac{37}{35} | 8.59375 |
2,028 | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | 2 | 65.625 |
2,029 | Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? | 18 | 57.8125 |
2,030 | Azar and Carl play a game of tic-tac-toe. Azar places an \(X\) in one of the boxes in a \(3\)-by-\(3\) array of boxes, then Carl places an \(O\) in one of the remaining boxes. After that, Azar places an \(X\) in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third \(O\). How many ways can the board look after the game is over? | 148 | 0 |
2,031 | The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$? | 85 | 91.40625 |
2,032 | Suppose that $p$ and $q$ are positive numbers for which $\log_{9}(p) = \log_{12}(q) = \log_{16}(p+q)$. What is the value of $\frac{q}{p}$? | \frac{1+\sqrt{5}}{2} | 93.75 |
2,033 | Let $x = .123456789101112....998999$, where the digits are obtained by writing the integers $1$ through $999$ in order. The $1983$rd digit to the right of the decimal point is | 7 | 89.84375 |
2,034 | For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$
What is the maximum value of $P(a)?$ | 2 - \sqrt{2} | 0 |
2,035 | The fraction $\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}$ is equal to | \frac43 | 38.28125 |
2,036 | How many different prime numbers are factors of $N$ if
$\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11?$ | 1 | 99.21875 |
2,037 | How many distinct triangles can be drawn using three of the dots below as vertices?
[asy]
dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));
[/asy] | 18 | 71.09375 |
2,038 | 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 | 0 |
2,039 | How many of the numbers, $100,101,\cdots,999$ have three different digits in increasing order or in decreasing order? | 168 | 14.84375 |
2,040 | The degree measure of angle $A$ is | 30 | 3.90625 |
2,041 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) | 7 | 35.9375 |
2,042 | Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? | \frac{1}{2} | 60.15625 |
2,043 | Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\frac{m}{n}?$ | \frac{\sqrt{2}}{2} | 8.59375 |
2,044 | A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$? | 20 | 72.65625 |
2,045 | How many three-digit numbers are divisible by 13? | 69 | 97.65625 |
2,046 | Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish? | 5 | 75.78125 |
2,047 | A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$. The line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$? | - \frac {1}{6} | 37.5 |
2,048 | Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded? | 20 | 25 |
2,049 | Consider the non-decreasing sequence of positive integers
\[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots\]
in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is | 3 | 90.625 |
2,050 | The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is | $x+2\sqrt{x}+1$ | 0 |
2,051 | If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: | \sqrt{50} | 0 |
2,052 | Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then: | 1 | 78.125 |
2,053 | For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide? | 48 | 100 |
2,054 | Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? | 96 | 95.3125 |
2,055 | George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today? | 6 | 91.40625 |
2,056 | Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? | 12\pi | 33.59375 |
2,057 | If $2x-3y-z=0$ and $x+3y-14z=0, z \neq 0$, the numerical value of $\frac{x^2+3xy}{y^2+z^2}$ is: | 7 | 90.625 |
2,058 | Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles? | \frac{\sqrt{3}}{3} | 92.1875 |
2,059 | How many non-congruent triangles have vertices at three of the eight points in the array shown below?
[asy]
dot((0,0)); dot((.5,.5)); dot((.5,0)); dot((.0,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));
[/asy] | 7 | 3.125 |
2,060 | What is the value in simplest form of the following expression?
\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7} | 10 | 58.59375 |
2,061 | If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: | -3 | 99.21875 |
2,062 | When $x^5$, $x+\frac{1}{x}$ and $1+\frac{2}{x} + \frac{3}{x^2}$ are multiplied, the product is a polynomial of degree. | 6 | 70.3125 |
2,063 | One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain? | 9 | 81.25 |
2,064 | A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The per cent of the list price at which he should mark them is: | 125 | 92.1875 |
2,065 | The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? | 6 | 34.375 |
2,066 | The expression $(81)^{-2^{-2}}$ has the same value as: | 3 | 0.78125 |
2,067 | At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ | 60 | 89.84375 |
2,068 | For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always | -6[-W] | 0 |
2,069 | David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? | 210 | 68.75 |
2,070 | If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\angle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is | \frac{\sqrt{3}}{3} | 50.78125 |
2,071 | A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\text{th}$ row is
[asy]
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(6,0)--(6,2)--(1,2)--cycle);
draw((2,0)--(5,0)--(5,3)--(2,3)--cycle);
draw((3,0)--(4,0)--(4,4)--(3,4)--cycle);
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black);
fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black);
fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black);
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black);
fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black);
[/asy] | 36 | 86.71875 |
2,072 | A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? | 45 | 25.78125 |
2,073 | On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? | 28 | 56.25 |
2,074 | Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: | 1 | 79.6875 |
2,075 | A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is | 2601 | 79.6875 |
2,076 | A fair die is rolled six times. The probability of rolling at least a five at least five times is | \frac{13}{729} | 40.625 |
2,077 | $\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =$ | $\frac{4}{3}$ | 0 |
2,078 | When simplified, $(- rac{1}{125})^{-2/3}$ becomes: | 25 | 86.71875 |
2,079 | How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$? | 469 | 92.1875 |
2,080 | Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting average was $40\%$. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\%$. How many of these $10$ shots did she make? | 8 | 56.25 |
2,081 | The number of values of $x$ satisfying the equation
\[\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\]is: | 0 | 96.09375 |
2,082 | The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are: | 0, 1, 2 and 4 | 0 |
2,083 | Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? | 26 | 99.21875 |
2,084 | Let $c$ be a constant. The simultaneous equations
\begin{align*}x-y = &\ 2 \\
cx+y = &\ 3 \\
\end{align*}have a solution $(x, y)$ inside Quadrant I if and only if | $-1<c<\frac{3}{2}$ | 0 |
2,085 | A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? | \sqrt{65} | 0 |
2,086 | If $\log 2 = .3010$ and $\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately | 1.47 | 11.71875 |
2,087 | .4 + .02 + .006 = | .426 | 64.84375 |
2,088 | A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy] | 5 | 21.875 |
2,089 | For positive numbers $x$ and $y$ the operation $\spadesuit (x,y)$ is defined as
\[\spadesuit (x,y) = x-\dfrac{1}{y}\]
What is $\spadesuit (2,\spadesuit (2,2))$? | \frac{4}{3} | 91.40625 |
2,090 | David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? | 210 | 75 |
2,091 | The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$? | 6 | 44.53125 |
2,092 | As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | 170 | 0 |
2,093 | A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q?$ (The figure shows a sample path.) | 28 | 18.75 |
2,094 | Let S be the statement "If the sum of the digits of the whole number $n$ is divisible by $6$, then $n$ is divisible by $6$." A value of $n$ which shows $S$ to be false is | 33 | 87.5 |
2,095 | Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is | 1150 | 96.09375 |
2,096 | For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers? | 12 | 21.875 |
2,097 | A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is: | 83.33 | 33.59375 |
2,098 | Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$? | 154 | 94.53125 |
2,099 | If $x < a < 0$ means that $x$ and $a$ are numbers such that $x$ is less than $a$ and $a$ is less than zero, then: | $x^2 > ax > a^2$ | 0 |
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