Unnamed: 0
int64 0
40.3k
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stringlengths 10
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stringlengths 1
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float64 0
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|---|---|---|---|
1,200 | The negation of the proposition "For all pairs of real numbers $a,b$, if $a=0$, then $ab=0$" is: There are real numbers $a,b$ such that | $a=0$ and $ab \ne 0$ | 0 |
1,201 | A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? | 24 | 92.1875 |
1,202 | Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? | 5 | 96.875 |
1,203 | When $x^{13}+1$ is divided by $x-1$, the remainder is: | 2 | 98.4375 |
1,204 | Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? | 80^{\circ} | 0.78125 |
1,205 | Consider the operation "minus the reciprocal of," defined by $a \diamond b = a - \frac{1}{b}$. What is $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$? | -\frac{7}{30} | 89.84375 |
1,206 | Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: | y =\frac{3}{4}x+1 | 28.90625 |
1,207 | Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? | 19 | 80.46875 |
1,208 | [asy]
draw((-7,0)--(7,0),black+linewidth(.75));
draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));
draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75));
[/asy]
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is | 12\sqrt{3} | 13.28125 |
1,209 | 4(299) + 3(299) + 2(299) + 298 = | 2989 | 63.28125 |
1,210 | If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is | 3f(x) | 96.875 |
1,211 | The function $f$ is given by the table
\[\begin{tabular}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \ \hline f(x) & 4 & 1 & 3 & 5 & 2 \ \hline \end{tabular}\]
If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find $u_{2002}$ | 2 | 87.5 |
1,212 | When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is | 420 | 93.75 |
1,213 | Rectangle $ABCD$ has $AB=8$ and $BC=6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $AB$ with $\overline{ME} \perp \overline{AC}$. What is the area of $\triangle AME$? | \frac{75}{8} | 82.03125 |
1,214 | The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is: | 6 | 83.59375 |
1,215 | The area of this figure is $100\text{ cm}^2$. Its perimeter is
[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]
[figure consists of four identical squares] | 50 cm | 0 |
1,216 | Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$, and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$? | \frac{10}{91} | 20.3125 |
1,217 | In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is | 5/6 | 0 |
1,218 | In a certain population the ratio of the number of women to the number of men is $11$ to $10$.
If the average (arithmetic mean) age of the women is $34$ and the average age of the men is $32$,
then the average age of the population is | $33\frac{1}{21}$ | 0 |
1,219 | What is the value of \[\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?\] | 21000 | 100 |
1,220 | $[x-(y-z)] - [(x-y) - z] = $ | 2z | 91.40625 |
1,221 | The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 - r}$ where $a$ denotes the first term and $- 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: | \frac {a^2}{1 - r^2} | 94.53125 |
1,222 | A house and store were sold for $12,000 each. The house was sold at a loss of 20% of the cost, and the store at a gain of 20% of the cost. The entire transaction resulted in: | -$1000 | 0 |
1,223 | The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is: | 20 | 49.21875 |
1,224 | 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} | 61.71875 |
1,225 | In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room.
What is the maximum number of people in the room that could have shaken hands with everyone else? | $N-1$ | 0 |
1,226 | There are integers $a, b,$ and $c,$ each greater than $1,$ such that
\[\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]
for all $N \neq 1$. What is $b$? | 3 | 26.5625 |
1,227 | Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 | 14.0625 |
1,228 | If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$ | 3 | 96.09375 |
1,229 | The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that | -\frac{4}{3}<x<\frac{1}{2} | 3.125 |
1,230 | The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? | \pi + 6\sqrt{3} | 0 |
1,231 | A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? | 6 | 17.1875 |
1,232 | The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^{\text{th}}$ term of this sequence? | 8041 | 53.90625 |
1,233 | The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is: | \frac{1}{5} | 59.375 |
1,234 | How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | 12 | 20.3125 |
1,235 | Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? | \frac{2}{3} | 1.5625 |
1,236 | Suppose that $\triangle{ABC}$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP=1$, $BP=\sqrt{3}$, and $CP=2$. What is $s$? | \sqrt{7} | 13.28125 |
1,237 | How many three-digit numbers have at least one $2$ and at least one $3$? | 52 | 81.25 |
1,238 | A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? | \frac{2}{3} | 27.34375 |
1,239 | Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? | 8 | 85.15625 |
1,240 | As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson? | 15 | 85.9375 |
1,241 | Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$? | \frac{1}{6} | 74.21875 |
1,242 | A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is: | \frac{81}{256} | 75 |
1,243 | The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? | 240 | 11.71875 |
1,244 | The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is: | 4 | 91.40625 |
1,245 | A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | 23 | 42.1875 |
1,246 | If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ | 1991 | 99.21875 |
1,247 | A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$? | \frac{1}{2} | 77.34375 |
1,248 | From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. The number of girls in the beginning was: | 40 | 92.1875 |
1,249 | The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is: | 16 | 90.625 |
1,250 | Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of: | 28\% | 96.09375 |
1,251 | Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? | 300 | 87.5 |
1,252 | Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table? | 30 | 75.78125 |
1,253 | The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? | \frac{281}{13} | 74.21875 |
1,254 | Twenty percent less than 60 is one-third more than what number? | 36 | 96.09375 |
1,255 | 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 | 96.875 |
1,256 | Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32d$ in base $n$ equals $263$, and the value of the numeral $324$ in base $n$ equals the value of the numeral $11d1$ in base six. What is $n + d$? | 11 | 85.9375 |
1,257 | Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? | 5 | 88.28125 |
1,258 | For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\div 14=\frac{12}{7}$.
What is $f(768)-f(384)?$ | \frac{1}{192} | 53.90625 |
1,259 | Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$ | 27a^2 | 94.53125 |
1,260 | Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is | 500 | 78.90625 |
1,261 | Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city? | 23 | 41.40625 |
1,262 | Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$, and the ratio of blue to green marbles in Jar $2$ is $8:1$. There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$? | 5 | 96.09375 |
1,263 | There is a unique positive integer $n$ such that $\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.$ What is the sum of the digits of $n?$ | 13 | 92.1875 |
1,264 | A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is | 15 | 89.84375 |
1,265 | The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical? | \frac{7}{27} | 82.03125 |
1,266 | An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used? | 191 | 0.78125 |
1,267 | If $x, y$, and $y-\frac{1}{x}$ are not $0$, then $\frac{x-\frac{1}{y}}{y-\frac{1}{x}}$ equals | \frac{x}{y} | 96.875 |
1,268 | How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) | 4 | 10.15625 |
1,269 | Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$ | 5 | 10.15625 |
1,270 | The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \text{ cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \text{ cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? | 663 | 96.09375 |
1,271 | If $S=1!+2!+3!+\cdots +99!$, then the units' digit in the value of S is: | 3 | 100 |
1,272 | What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ | \pi + 2 | 0 |
1,273 | It is now between 10:00 and 11:00 o'clock, and six minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? | 10:15 | 42.96875 |
1,274 | Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? | 20 | 80.46875 |
1,275 | What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, \dots, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.) | \frac{2}{3} | 32.8125 |
1,276 | If it is known that $\log_2(a)+\log_2(b) \ge 6$, then the least value that can be taken on by $a+b$ is: | 16 | 88.28125 |
1,277 | What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? | 7.5 | 53.90625 |
1,278 | When $(a-b)^n,n\ge2,ab\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: | 2k+1 | 36.71875 |
1,279 | In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is: | 3 | 82.03125 |
1,280 | If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is: | $10\frac{2}{3}$ | 0 |
1,281 | The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? | \frac{3}{2} | 96.875 |
1,282 | A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy] | (44,35) | 0 |
1,283 | Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? | \frac{3}{16} | 0 |
1,284 | Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. What is the probability that circles $A$ and $B$ intersect? | \frac {4 \sqrt {3} - 3}{4} | 0 |
1,285 | Given $\triangle PQR$ with $\overline{RS}$ bisecting $\angle R$, $PQ$ extended to $D$ and $\angle n$ a right angle, then: | \frac{1}{2}(\angle p + \angle q) | 0 |
1,286 | Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$? | \sqrt{3} | 11.71875 |
1,287 | Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$-mile bicycle ride? | \frac{120}{11} | 45.3125 |
1,288 | 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 | 86.71875 |
1,289 | A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? | 4 | 87.5 |
1,290 | Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | 280 | 68.75 |
1,291 | Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? | 0.4 | 87.5 |
1,292 | John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? | 30 | 75 |
1,293 | If $a=\log_8 225$ and $b=\log_2 15$, then | $a=2b/3$ | 0 |
1,294 | Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? | 120 | 50.78125 |
1,295 | Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $\left(T_n\right)$? | \frac{1509}{128} | 0 |
1,296 | A triangle with integral sides has perimeter $8$. The area of the triangle is | 2\sqrt{2} | 85.15625 |
1,297 | The number of the distinct solutions to the equation $|x-|2x+1||=3$ is | 2 | 78.90625 |
1,298 | Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=3$, and $DC=4$. What is the area of $\triangle ABC$? | $7\sqrt{3}$ | 0 |
1,299 | The ratio $\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$ is: | \frac{3}{2} | 100 |
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