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|---|---|---|---|
2,600 | Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5? | 32 | 94.53125 |
2,601 | When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even? | \frac{5}{8} | 54.6875 |
2,602 | Let $W,X,Y$ and $Z$ be four different digits selected from the set
$\{ 1,2,3,4,5,6,7,8,9\}.$
If the sum $\dfrac{W}{X} + \dfrac{Y}{Z}$ is to be as small as possible, then $\dfrac{W}{X} + \dfrac{Y}{Z}$ must equal | \frac{25}{72} | 32.8125 |
2,603 | The product of the 9 factors $\left(1 - \frac12\right)\left(1 - \frac13\right)\left(1 - \frac14\right) \cdots \left(1 - \frac {1}{10}\right) =$ | \frac{1}{10} | 98.4375 |
2,604 | In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A+43_B=69_{A+B}.\]What is $A+B$? | 13 | 76.5625 |
2,605 | Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1 to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1? (For example, Raashan and Ted may each decide to give $1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0, Sylvia will have $2, and Ted will have $1, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1 to, and the holdings will be the same at the end of the second round.) | \frac{1}{4} | 38.28125 |
2,606 | How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits is a perfect square? | 8 | 98.4375 |
2,607 | A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half? | 34 | 24.21875 |
2,608 | Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? | 0 | 0 |
2,609 | The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is: | 65 | 77.34375 |
2,610 | If $N$, written in base $2$, is $11000$, the integer immediately preceding $N$, written in base $2$, is: | 10111 | 74.21875 |
2,611 | Let $f(x)=|x-2|+|x-4|-|2x-6|$ for $2 \leq x \leq 8$. The sum of the largest and smallest values of $f(x)$ is | 2 | 67.96875 |
2,612 | In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? | \frac{366}{31 \times 24} | 0 |
2,613 | How many ordered pairs $(m,n)$ of positive integers are solutions to
\[\frac{4}{m}+\frac{2}{n}=1?\] | 4 | 97.65625 |
2,614 | In the non-decreasing sequence of odd integers $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b, c$, and $d$ such that for all positive integers $n$, $a_n=b\lfloor \sqrt{n+c} \rfloor +d$, where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals | 2 | 20.3125 |
2,615 | Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$? | \frac{1}{32} | 74.21875 |
2,616 | Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is | $2\pi-\pi \sqrt{2}-1$ | 0 |
2,617 | Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | 13 | 25.78125 |
2,618 | Students from three middle schools worked on a summer project.
Seven students from Allen school worked for 3 days.
Four students from Balboa school worked for 5 days.
Five students from Carver school worked for 9 days.
The total amount paid for the students' work was 744. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether? | 180.00 | 0 |
2,619 | Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: | 5 | 99.21875 |
2,620 | 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 | 100 |
2,621 | Let $S_1 = \{(x, y)|\log_{10}(1 + x^2 + y^2) \le 1 + \log_{10}(x+y)\}$ and $S_2 = \{(x, y)|\log_{10}(2 + x^2 + y^2) \le 2 + \log_{10}(x+y)\}$. What is the ratio of the area of $S_2$ to the area of $S_1$? | 102 | 97.65625 |
2,622 | Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | 37 | 84.375 |
2,623 | How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? | 45 | 100 |
2,624 | A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end? | 4:40 a.m. | 0 |
2,625 | The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is | 20 | 0 |
2,626 | Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$? | 10 | 96.09375 |
2,627 | In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then what is the value of $C$? | 1 | 2.34375 |
2,628 | The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? | 160 | 19.53125 |
2,629 | Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$? | 10 | 25 |
2,630 | For integers $a,b,$ and $c$ define $\fbox{a,b,c}$ to mean $a^b-b^c+c^a$. Then $\fbox{1,-1,2}$ equals: | 2 | 97.65625 |
2,631 | For $p=1, 2, \cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\cdots+S_{10}$ is | 80200 | 97.65625 |
2,632 | In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$? | 19 | 89.0625 |
2,633 | How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$? | 6 | 55.46875 |
2,634 | Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? | \frac{147}{1024} | 0 |
2,635 | If $x<0$, then $|x-\sqrt{(x-1)^2}|$ equals | 1-2x | 42.96875 |
2,636 | Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round? | \frac{5}{36} | 71.875 |
2,637 | Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was | 1920 | 98.4375 |
2,638 | When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, what is the smallest difference that is possible?
\[\begin{array}{cccc} & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{array}\] | 149 | 96.09375 |
2,639 | Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is: | $x >\frac{10}{3}$ | 0 |
2,640 | An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old? | 134 | 0 |
2,641 | Mindy made three purchases for $1.98, $5.04, and $9.89. What was her total, to the nearest dollar? | 17 | 100 |
2,642 | A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor? | 361 | 26.5625 |
2,643 | A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game? | 37 | 1.5625 |
2,644 | In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression | \frac{37b}{27} | 0.78125 |
2,645 | A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number which are white or blue is at least $55$. The minimum number of red chips is | 57 | 49.21875 |
2,646 | The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? | 35\% | 95.3125 |
2,647 | If $y=(\log_23)(\log_34)\cdots(\log_n[n+1])\cdots(\log_{31}32)$ then | y=5 | 95.3125 |
2,648 | A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What is the remainder when $N$ is divided by $100$? | 95 | 78.125 |
2,649 | There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that $\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.$ What is $k?$ | 137 | 33.59375 |
2,650 | What is the value of $(625^{\log_5 2015})^{\frac{1}{4}}$? | 2015 | 64.84375 |
2,651 | Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$? | \frac{13}{27} | 64.84375 |
2,652 | Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$ | \sqrt[6]{3} | 11.71875 |
2,653 | A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is ? | -13.5 | 3.90625 |
2,654 | Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}{2}$. The length of the hypotenuse is | \frac{3\sqrt{5}}{5} | 67.96875 |
2,655 | A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? | \frac{10b}{t} | 96.875 |
2,656 | 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} | 0.78125 |
2,657 | If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: | -9 | 96.09375 |
2,658 | Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? | 5 | 8.59375 |
2,659 | Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$? | (13,-22) | 0 |
2,660 | The medians of a right triangle which are drawn from the vertices of the acute angles are $5$ and $\sqrt{40}$. The value of the hypotenuse is: | 2\sqrt{13} | 31.25 |
2,661 | How many integer values of $x$ satisfy $|x|<3\pi$? | 19 | 97.65625 |
2,662 | A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? | \frac{1}{2} | 3.125 |
2,663 | Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$? | 66 | 0.78125 |
2,664 | How many even integers are there between $200$ and $700$ whose digits are all different and come from the set $\{1,2,5,7,8,9\}$? | 12 | 71.875 |
2,665 | Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time? | 48 | 48.4375 |
2,666 | ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between | 7 and 8 | 0 |
2,667 | Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed? | 150 | 98.4375 |
2,668 | When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is: | 1: 2 | 0 |
2,669 | The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ | 9 | 78.125 |
2,670 | A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$? | 8078 | 74.21875 |
2,671 | For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$? | 97 | 1.5625 |
2,672 | What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? | 6 | 93.75 |
2,673 | A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | $\frac{15}{8}$ | 0 |
2,674 | The formula expressing the relationship between $x$ and $y$ in the table is:
\begin{tabular}{|c|c|c|c|c|c|}
\hline x & 2 & 3 & 4 & 5 & 6\
\hline y & 0 & 2 & 6 & 12 & 20\
\hline
\end{tabular} | $y = x^{2}-3x+2$ | 0 |
2,675 | Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by
\[f=\{(x,y)\quad |\quad y=x \text{ when } 0 \le x \le 5, y=2x-5 \text{ when } 5 \le x \le 8\}.\]
Then $K$ is: | 36.5 | 3.90625 |
2,676 | When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. How long does it take Jack to get to school? | 18 minutes | 0 |
2,677 | Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select? | \frac{25}{63} | 4.6875 |
2,678 | A boat has a speed of $15$ mph in still water. In a stream that has a current of $5$ mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is: | \frac{8}{9} | 95.3125 |
2,679 | Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was | 36 | 99.21875 |
2,680 | Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is | 36 | 78.90625 |
2,681 | Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? | 17 | 0 |
2,682 | Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is | 3 | 79.6875 |
2,683 | If $x \neq 0$, $\frac{x}{2} = y^2$ and $\frac{x}{4} = 4y$, then $x$ equals | 128 | 100 |
2,684 | Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$.
If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then: | $a>9, b>6$ | 0 |
2,685 | At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? | 25\% | 0.78125 |
2,686 | Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? | \frac{3}{16} | 4.6875 |
2,687 | For how many $n$ in $\{1, 2, 3, ..., 100 \}$ is the tens digit of $n^2$ odd? | 20 | 96.875 |
2,688 | The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$? | 761 | 82.8125 |
2,689 | If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then $c$ is | $5$ | 0 |
2,690 | Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? | 4 | 60.9375 |
2,691 | The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are: | \frac{3}{5} < x \le 2 | 0 |
2,692 | When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to: | \frac{xy}{x+y} | 96.875 |
2,693 | Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | 4 | 2.34375 |
2,694 | Which one of the following is not equivalent to $0.000000375$? | $\frac{3}{8} \times 10^{-7}$ | 0 |
2,695 | Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2));
draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1));
draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8));
draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1));
draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy] | \frac{5}{9} | 50.78125 |
2,696 | How many polynomial functions $f$ of degree $\ge 1$ satisfy
$f(x^2)=[f(x)]^2=f(f(x))$ ? | 1 | 85.15625 |
2,697 | In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161.$ What is the correct value of the product of $a$ and $b$? | 224 | 61.71875 |
2,698 | 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 |
2,699 | What is the sum of the two smallest prime factors of $250$? | 7 | 91.40625 |
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