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1,600 | If \(\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}\) for three positive numbers \(x,y\) and \(z\), all different, then \(\frac{x}{y}=\) | 2 | 50.78125 |
1,601 | What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest? | \frac{19}{15}<\frac{17}{13}<\frac{15}{11} | 0 |
1,602 | Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled? | 900 | 0 |
1,603 | In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $-9$ and $-1$ for the roots. The correct equation was: | x^2-10x+9=0 | 74.21875 |
1,604 | Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$? | 3 | 53.125 |
1,605 | Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then | 2700^{\circ} | 73.4375 |
1,606 | If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: | \frac {\sqrt {8d^2 - p^2}}{2} | 0 |
1,607 | A jar contains $5$ different colors of gumdrops. $30\%$ are blue, $20\%$ are brown, $15\%$ are red, $10\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? | 42 | 87.5 |
1,608 | Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is | \frac{1}{f(x)} | 0.78125 |
1,609 | Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together? | 5760 | 57.03125 |
1,610 | A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive $x$ and $y$ axes is shown in each figure I to IV below.
The inequalities
\(|x|+|y| \leq \sqrt{2(x^{2}+y^{2})} \leq 2\mbox{Max}(|x|, |y|)\)
are represented geometrically* by the figure numbered
* An inequality of the form $f(x, y) \leq g(x, y)$, for all $x$ and $y$ is represented geometrically by a figure showing the containment
$\{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\
\{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}$
for a typical real number $a$. | II | 28.90625 |
1,611 | Consider all triangles $ABC$ satisfying in the following conditions: $AB = AC$, $D$ is a point on $AC$ for which $BD \perp AC$, $AC$ and $CD$ are integers, and $BD^{2} = 57$. Among all such triangles, the smallest possible value of $AC$ is | 11 | 52.34375 |
1,612 | For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let $f(x) = \lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1)$. The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? | 1 | 78.125 |
1,613 | Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive,
$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ] | -1 < x < 11 | 15.625 |
1,614 | How many whole numbers are between $\sqrt{8}$ and $\sqrt{80}$? | 6 | 94.53125 |
1,615 | $ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$. $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$. $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$. Then $PR + PS$ is equal to: | $AF$ | 0 |
1,616 | If $y = 2x$ and $z = 2y$, then $x + y + z$ equals | $7x$ | 0 |
1,617 | In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? | 3 | 66.40625 |
1,618 | Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? | 2 | 7.03125 |
1,619 | A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes? | 3 | 48.4375 |
1,620 | Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic? | 9 | 95.3125 |
1,621 | Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ | 11 | 13.28125 |
1,622 | Let $O$ be an interior point of triangle $ABC$, and let $s_1=OA+OB+OC$. If $s_2=AB+BC+CA$, then | $s_2\ge 2s_1,s_1 \le s_2$ | 0 |
1,623 | For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? | 8 | 92.1875 |
1,624 | If $f(n)=\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals: | r(r+1) | 94.53125 |
1,625 | Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. | 240 | 0 |
1,626 | The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$? | 66 | 58.59375 |
1,627 | In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was: | 7 | 97.65625 |
1,628 | Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar? | 10 | 43.75 |
1,629 | How many integers between $1000$ and $9999$ have four distinct digits? | 4536 | 94.53125 |
1,630 | A merchant buys goods at $25\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\%$ on the marked price and still clear a profit of $25\%$ on the selling price. What percent of the list price must he mark the goods? | 125\% | 68.75 |
1,631 | A straight line passing through the point $(0,4)$ is perpendicular to the line $x-3y-7=0$. Its equation is: | y+3x-4=0 | 0 |
1,632 | Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$? | 150 | 11.71875 |
1,633 | What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$? | 127 | 97.65625 |
1,634 | In a circle of radius $5$ units, $CD$ and $AB$ are perpendicular diameters. A chord $CH$ cutting $AB$ at $K$ is $8$ units long. The diameter $AB$ is divided into two segments whose dimensions are: | 2,8 | 0 |
1,635 | Let $Q(z)$ and $R(z)$ be the unique polynomials such that $z^{2021}+1=(z^2+z+1)Q(z)+R(z)$ and the degree of $R$ is less than $2.$ What is $R(z)?$ | -z | 25 |
1,636 | Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? | 32 | 21.875 |
1,637 | In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ$ and $\angle BED = \angle BDE$, then $\angle CDE =$ | 95^\circ | 0 |
1,638 | For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer? | 9 | 88.28125 |
1,639 | The function $f(x)$ satisfies $f(2+x)=f(2-x)$ for all real numbers $x$. If the equation $f(x)=0$ has exactly four distinct real roots, then the sum of these roots is | 8 | 97.65625 |
1,640 | How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle? | 4 | 83.59375 |
1,641 | On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group? | 60 | 68.75 |
1,642 | Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | 11 | 79.6875 |
1,643 | Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid. What is $DE$? | 13 | 0 |
1,644 | Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible? | 4 | 29.6875 |
1,645 | Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time? | \frac{1}{2} | 76.5625 |
1,646 | If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is | \frac{3}{4} | 82.8125 |
1,647 | There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? | 40\% | 80.46875 |
1,648 | On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap? | \frac{7}{25} | 99.21875 |
1,649 | In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$? | \frac{5}{18} | 22.65625 |
1,650 | Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40$ dollars on Thursday? | 39.60 | 73.4375 |
1,651 | Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$? | \frac{4}{5} | 0.78125 |
1,652 | In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? | 2-\sqrt{3} | 0 |
1,653 | On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 | 17.96875 |
1,654 | In the following equation, each of the letters represents uniquely a different digit in base ten:
\[(YE) \cdot (ME) = TTT\]
The sum $E+M+T+Y$ equals | 21 | 67.1875 |
1,655 | A wooden cube with edge length $n$ units (where $n$ is an integer $>2$) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$? | 8 | 83.59375 |
1,656 | Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$? | \sqrt{13} | 46.875 |
1,657 | If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals | 12 | 91.40625 |
1,658 | In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN \perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$. | \frac{5}{2} | 3.90625 |
1,659 | The adjacent map is part of a city: the small rectangles are blocks, and the paths in between are streets.
Each morning, a student walks from intersection $A$ to intersection $B$, always walking along streets shown,
and always going east or south. For variety, at each intersection where he has a choice, he chooses with
probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. | \frac{21}{32} | 0 |
1,660 | $\overline{AB}$ is a diameter of a circle. Tangents $\overline{AD}$ and $\overline{BC}$ are drawn so that $\overline{AC}$ and $\overline{BD}$ intersect in a point on the circle. If $\overline{AD}=a$ and $\overline{BC}=b$, $a \not= b$, the diameter of the circle is: | \sqrt{ab} | 16.40625 |
1,661 | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top? | 367 | 0 |
1,662 | Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by | \frac{100p}{100+p}\%$ | 0 |
1,663 | Let $x_1, x_2, \ldots , x_n$ be a sequence of integers such that
(i) $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$
(ii) $x_1 + \cdots + x_n = 19$; and
(iii) $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$.
Let $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \cdots + x_n^3$, respectively. Then $\frac Mm =$ | 7 | 41.40625 |
1,664 | What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$ | 18 | 96.09375 |
1,665 | A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | \frac{50}{99} | 6.25 |
1,666 | A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home by bus in the afternoon $15$ times, and commuted by train (either morning or afternoon) $9$ times. Find $x$. | 16 | 55.46875 |
1,667 | The arithmetic mean (average) of the first $n$ positive integers is: | \frac{n+1}{2} | 82.03125 |
1,668 | A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is | $8r^2$ | 0 |
1,669 | Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? | \sqrt{3} | 44.53125 |
1,670 | If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is: | $x \le -2$ or $x \ge 3$ | 0 |
1,671 | Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is: | 10(q - 1) | 18.75 |
1,672 | Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ | 15 | 85.15625 |
1,673 | A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain? | 51.5 | 70.3125 |
1,674 | An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is: | 6a | 100 |
1,675 | Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is | 80 | 99.21875 |
1,676 | If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals | 1 | 96.875 |
1,677 | What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? | 10 | 100 |
1,678 | If $x$ is such that $\frac{1}{x}<2$ and $\frac{1}{x}>-3$, then: | x>\frac{1}{2} \text{ or } x<-\frac{1}{3} | 12.5 |
1,679 | Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? | 20 | 87.5 |
1,680 | If $a = -2$, the largest number in the set $\{ -3a, 4a, \frac{24}{a}, a^2, 1\}$ is | -3a | 0 |
1,681 | Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? | \frac{19}{81} | 11.71875 |
1,682 | A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is | 10 | 93.75 |
1,683 | The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? | \frac{\sqrt{5}-1}{2} | 12.5 |
1,684 | What is the value of ${\left[\log_{10}\left(5\log_{10}100\right)\right]}^2$? | 1 | 63.28125 |
1,685 | Let
\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\]
Then | T>2 | 0 |
1,686 | $\dfrac{10-9+8-7+6-5+4-3+2-1}{1-2+3-4+5-6+7-8+9}=$ | 1 | 87.5 |
1,687 | Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$ | 471 | 0 |
1,688 | In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters? | 5 | 10.9375 |
1,689 | Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period? | 55 | 82.8125 |
1,690 | A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.) | 4 | 78.125 |
1,691 | How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$? | 15 | 27.34375 |
1,692 | Define \(P(x) =(x-1^2)(x-2^2)\cdots(x-100^2)\). How many integers \(n\) are there such that \(P(n)\leq 0\)? | 5100 | 17.96875 |
1,693 | $\sqrt{\frac{8^{10}+4^{10}}{8^4+4^{11}}}=$ | $16$ | 0 |
1,694 | A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answered correctly? | 12 | 97.65625 |
1,695 | The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is | \frac{1}{3} | 79.6875 |
1,696 | Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is | 21 | 3.90625 |
1,697 | For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$? | 10 | 77.34375 |
1,698 | A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? | \frac{127}{924} | 0.78125 |
1,699 | The number of terms in the expansion of $[(a+3b)^{2}(a-3b)^{2}]^{2}$ when simplified is: | 5 | 78.90625 |
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