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2,500 | The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class? | 39 | 98.4375 |
2,501 | A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. | \frac{5}{6} | 40.625 |
2,502 | A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: | 400 | 100 |
2,503 | Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. What is the minimum possible value of $N$? | 1 | 54.6875 |
2,504 | Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done? | 20 | 96.09375 |
2,505 | If $r$ is positive and the line whose equation is $x + y = r$ is tangent to the circle whose equation is $x^2 + y^2 = r$, then $r$ equals | 2 | 100 |
2,506 | Consider the sequence
$1,-2,3,-4,5,-6,\ldots,$
whose $n$th term is $(-1)^{n+1}\cdot n$. What is the average of the first $200$ terms of the sequence? | -0.5 | 6.25 |
2,507 | Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy] | 3 | 50.78125 |
2,508 | The percent that $M$ is greater than $N$ is: | \frac{100(M-N)}{N} | 0 |
2,509 | In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$, $14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of points she could have scored in the tenth game? | 29 | 57.8125 |
2,510 | If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$? | $2$ | 0 |
2,511 | The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August? | 2.5 | 1.5625 |
2,512 | Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.
What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class? | 4.36 | 0 |
2,513 | If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: | 1 | 92.1875 |
2,514 | In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | 7 | 95.3125 |
2,515 | If the product $\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{a}{b} = 9$, what is the sum of $a$ and $b$? | 37 | 0 |
2,516 | 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 | 93.75 |
2,517 | If $x=\frac{a}{b}$, $a\neq b$ and $b\neq 0$, then $\frac{a+b}{a-b}=$ | \frac{x+1}{x-1} | 99.21875 |
2,518 | Simplify $\left(\sqrt[6]{27} - \sqrt{6 \frac{3}{4} }\right)^2$ | \frac{3}{4} | 55.46875 |
2,519 | A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is: | 2519 | 77.34375 |
2,520 | What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\cdots2^{(2n+1)/7}$ is greater than $1000$?
(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.) | 9 | 81.25 |
2,521 | Ten points are selected on the positive $x$-axis, $X^+$, and five points are selected on the positive $y$-axis, $Y^+$. The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant? | 450 | 94.53125 |
2,522 | Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? | \frac{\pi}{3} | 69.53125 |
2,523 | Given the areas of the three squares in the figure, what is the area of the interior triangle? | 30 | 17.96875 |
2,524 | If $y=a+\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals: | 11 | 74.21875 |
2,525 | Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals | \sqrt{4q+1} | 46.875 |
2,526 | A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? | 200 | 0 |
2,527 | Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is
[asy]
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<4; j=j+1) {
dot((i,j));
}}
draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));
[/asy] | 6 | 71.09375 |
2,528 | Harry has 3 sisters and 5 brothers. His sister Harriet has $\text{S}$ sisters and $\text{B}$ brothers. What is the product of $\text{S}$ and $\text{B}$? | 10 | 85.9375 |
2,529 | When simplified $\sqrt{1+ \left (\frac{x^4-1}{2x^2} \right )^2}$ equals: | \frac{x^2}{2}+\frac{1}{2x^2} | 11.71875 |
2,530 | A man walked a certain distance at a constant rate. If he had gone $\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\frac{1}{2}$ mile per hour slower, he would have been $2\frac{1}{2}$ hours longer on the road. The distance in miles he walked was | 15 | 80.46875 |
2,531 | The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac{y}{x}$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$? | \frac{7}{2} | 50.78125 |
2,532 | If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: | a > 1 | 33.59375 |
2,533 | Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\[f \left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right)?\] | 0 | 96.09375 |
2,534 | A line $x=k$ intersects the graph of $y=\log_5 x$ and the graph of $y=\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$? | 6 | 83.59375 |
2,535 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 | 51.5625 |
2,536 | Four whole numbers, when added three at a time, give the sums $180, 197, 208$ and $222$. What is the largest of the four numbers? | 89 | 89.84375 |
2,537 | Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have? | 6 | 6.25 |
2,538 | In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? | 2 | 86.71875 |
2,539 | The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: | \frac{\sqrt{3}}{6} | 98.4375 |
2,540 | Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? | 21 | 96.875 |
2,541 | The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares? | 77 | 82.03125 |
2,542 | If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$ | $f(-x)=f(x)$ | 0 |
2,543 | A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$? | 63 | 2.34375 |
2,544 | If 5 times a number is 2, then 100 times the reciprocal of the number is | 50 | 1.5625 |
2,545 | A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? | 10 | 84.375 |
2,546 | The coordinates of $A, B$ and $C$ are $(5,5), (2,1)$ and $(0,k)$ respectively.
The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is: | 2\frac{1}{7} | 0 |
2,547 | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$? | 37 | 67.1875 |
2,548 | If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ | \frac{2a+b}{1-a} | 77.34375 |
2,549 | $A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is: | 5/6 | 0 |
2,550 | Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is | 13 | 42.1875 |
2,551 | In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$? | 13 | 60.15625 |
2,552 | Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | \frac{n}{2n+1} | 1.5625 |
2,553 | Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is: | 6\% | 50 |
2,554 | Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is | 3 | 85.15625 |
2,555 | The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income? | 32000 | 93.75 |
2,556 | If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals: | 4 | 99.21875 |
2,557 | There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$? | 3 | 86.71875 |
2,558 | If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the $xy$-plane, then $x$ is equal to | -6 | 92.96875 |
2,559 | If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: | $y = \frac{1}{3} x + 8$ | 0 |
2,560 | What is $10 \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}?$ | \frac{25}{2} | 43.75 |
2,561 | Define $[a,b,c]$ to mean $\frac {a+b}c$, where $c \neq 0$. What is the value of $\left[[60,30,90],[2,1,3],[10,5,15]\right]?$ | 2 | 92.96875 |
2,562 | 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 | 92.1875 |
2,563 | The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of | 10.3 | 0.78125 |
2,564 | For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then
$M(M(a,m(b,c)),m(d,m(a,e)))=$ | b | 96.875 |
2,565 | $\triangle ABC$ has a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then $\angle BDC =$ | 55^{\circ} | 72.65625 |
2,566 | A circle has center $(-10, -4)$ and has radius $13$. Another circle has center $(3, 9)$ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$? | 3 | 87.5 |
2,567 | Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? | \frac{3}{2} | 96.09375 |
2,568 | A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive? | 1925 | 3.125 |
2,569 | If a worker receives a $20$% cut in wages, he may regain his original pay exactly by obtaining a raise of: | 25\% | 92.96875 |
2,570 | Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is $h + m$? | 46 | 87.5 |
2,571 | In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.
Of the three equations
\begin{equation*} \label{eq:1} d-s=1, \qquad ds=1, \qquad d^2-s^2=\sqrt{5} \end{equation*}those which are necessarily true are | I, II and III | 0 |
2,572 | The statement $x^2 - x - 6 < 0$ is equivalent to the statement: | -2 < x < 3 | 13.28125 |
2,573 | The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? | 20 | 67.96875 |
2,574 | The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: | 0, 12 | 0.78125 |
2,575 | Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other.
The man at $R$ walks uniformly at the rate of $4\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant
rate of $3\tfrac{1}{4}$ miles per hour for the first hour, at $3\tfrac{3}{4}$ miles per hour for the second hour,
and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is: | 4 | 32.03125 |
2,576 | Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA=1$. The area of the "bat wings" (shaded area) is | 3 \frac{1}{2} | 0 |
2,577 | A right circular cone has for its base a circle having the same radius as a given sphere.
The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: | 2/1 | 0 |
2,578 | The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? | 2001 | 32.8125 |
2,579 | In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle? | 12 + 12\sqrt{2} | 27.34375 |
2,580 | For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | 4.6875 |
2,581 | The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$? | 502 | 74.21875 |
2,582 | Distinct planes $p_1, p_2, \dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? | 20 | 0 |
2,583 | 1990-1980+1970-1960+\cdots -20+10 = | 1000 | 57.03125 |
2,584 | Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? | \frac{10-7\sqrt{2}}{3} | 11.71875 |
2,585 | Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$ | 12\sqrt2 | 0 |
2,586 | If $2^{1998} - 2^{1997} - 2^{1996} + 2^{1995} = k \cdot 2^{1995},$ what is the value of $k$? | 3 | 99.21875 |
2,587 | Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BC$ at points $D$ and $E$, respectively. The length of $AE$ is | \sqrt{3} | 37.5 |
2,588 | The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full? | 125 | 90.625 |
2,589 | Let the product $(12)(15)(16)$, each factor written in base $b$, equals $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is | 44 | 20.3125 |
2,590 | What is the tens digit of $2015^{2016}-2017?$ | 0 | 39.0625 |
2,591 | Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? | \sqrt{95} | 46.09375 |
2,592 | If $i^2=-1$, then $(i-i^{-1})^{-1}=$ | -\frac{i}{2} | 96.875 |
2,593 | The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? | 90 | 79.6875 |
2,594 | Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$? | \frac{35}{12} | 45.3125 |
2,595 | \frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} equals | \sqrt{2}+\sqrt{3}-\sqrt{5} | 52.34375 |
2,596 | Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$ | 2x-3y=0 | 15.625 |
2,597 | Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy? | 36 | 90.625 |
2,598 | Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$? | \frac{85\pi}{8} | 0 |
2,599 | The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares? | 77 | 86.71875 |
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