images images listlengths 1 1 | problem stringlengths 22 1.13k | answer stringlengths 0 174 |
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<image>Question: As shown, what is the area of the square? | 22 + 12√2 | |
<image>Question: m n 75m = n^3 m + n? | 15 | |
<image>Question: Consider the 12-sided polygon ABCDEFGHIJKL, as shown. Each of its sides has length 4, and each two consecutive sides form a right angle. Suppose that AG and CH meet at M. What is the area of quadrilateral ABCM? | 88/5 | |
<image>Question: A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? | 2√2 + 2 | |
<image>Question: A sphere is inscribed in a cube that has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? | 8 | |
<image>Question: 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 ... | 37 | |
<image>Question: How many ordered pairs \((m,n)\) of positive integers, with \(m \geq n\), have the property that their squares differ by 96? | 4 | |
<image>Question: Quadrilateral \(ABCD\) satisfies \(\angle ABC = \angle ACD = 90^\circ\), \(AC = 20\), and \(CD = 30\). Diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at point \(E\) and \(AE = 5\). What is the area of quadrilateral \(ABCD\)? | \(500\) | |
<image>Question: 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 $s$? | $\sqrt{5+\sqrt{5}}$ | |
<image>Question: A big L is formed as shown. What is its area? | 22 | |
<image>Question: A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | 3 | |
<image>Question: Lucky Larry's teacher asked him to substitute numbers for \(a, b, c, d,\) and \(e\) in the expression \(a - (b - (c - (d + e)))\) and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for \(... | 3 | |
<image>Question: At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at t... | 60 | |
<image>Question: Shelby drives her scooter at a speed of miles per hour if it is not raining, and miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of miles in minutes. How many minutes did she drive in the rain? | 24 | |
<image>Question: Every high school in the city of Euclid sent a team of 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 37th and 64th, respective... | 23 | |
<image>Question: Let n be the smallest positive integer such that n is divisible by , n^2 is a perfect cube, and n^3 is a perfect square. What is the number of digits of n? | 5 | |
<image>Question: The average of the numbers 1,2,3,...,98,99 and x is 100x. | 50/101 | |
<image>Question: Triangle $ABC$ has vertices $A=(3,0)$, $B=(0,3)$, and $C$, where $C$ is on the line $x+y=7$. What is the area of $\triangle ABC$? | $6$ | |
<image>Question: Triangle \(ABC\) has \(AB = 13\) and \(AC = 15\), and the altitude to \(\overline{BC}\) has length. What is the sum of the two possible values of \(BC\)? | 20 | |
<image>Question: Suppose that cows give \( a \) gallons of milk in \( b \) days. At this rate, how many gallons of milk will \( c \) cows give in \( d \) days? | \(\frac{a d}{b c}\) | |
<image>Question: Square \( EFGH \) is inside the square \( ABCD \) so that each side of \( EFGH \) can be extended to pass through a vertex of \( ABCD \). Square \( ABCD \) has a side length of \( \sqrt{50} \) and \( BE = 1 \). What is the area of the inner square \( EFGH \)? | 20 | |
<image>Question: The regular 5-point star \(A B C D E\) is drawn and in each vertex, there is a number. Each \(A, B, C, D,\) and \(E\) are chosen such that all 5 of them came from set \(\{3, 5, 6, 7, 9\}\). Each letter is a different number (so one possible way is \(A=3, B=5, C=6, D=7, E=9\)). Let \(AB\) be the sum of ... | 12 | |
<image>Question: What is the probability that the top face has an odd number of dots? | \(\frac{11}{21}\) | |
<image>Question: A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex W? | 1/12 | |
<image>Question: Call a number prime-looking if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? | 100 | |
<image>Question: A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | 1462 | |
<image>Question: The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers? | 4 | |
<image>Question: Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically? | If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong | |
<image>Question: Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip... | 30% | |
<image>Question: At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? | 245 | |
<image>Question: Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3 cm, 7 cm, and 15 cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she c... | 17 | |
<image>Question: Define a function on the positive integers recursively by \( f(1) = 2 \), \( f(n) = f(n-1) + 2 \) if \( n \) is even, and \( f(n) = f(n-2) + 2 \) if \( n \) is odd and greater than 1. What is \( f(2017) \)? | 2018 | |
<image>Question: The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume 216π. What is the length AB? | 20 | |
<image>Question: At her usual speed, but after driving 1/3 of the way, she hits a bad snowstorm and reduces her speed by 20 miles per hour. This time the trip takes her a total of 276 minutes. How many miles is the drive from Sharon's house to her mother's house? | 135 | |
<image>Question: Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions? | 28 | |
<image>Question: Let \( f(x) = \sin x + 2 \cos x + 3 \tan x \), using radian measure for the variable \( x \). In what interval does the smallest positive value of \( x \) for which \( f(x) = 0 \) lie? | (3,4) | |
<image>Question: Let S(n) equal the sum of the digits of positive integer n. For example, S(1507)=13. For a particular positive integer n, S(n)=1274. Which of the following could be the value of S(n+1)? | 1239 | |
<image>Question: z such that z^24=1. How many distinct values of z^6 are there? | 12 | |
<image>Question: A square with side length \( x \) is inscribed in a right triangle with sides of length 3, 4, and 5 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \( y \) is inscribed in another right triangle with sides of length 3, 4, and 5 so that o... | \( \frac{37}{35} \) | |
<image>Question: How many ordered pairs \((a,b)\) such that \(a\) is a positive real number and \(b\) is an integer between 2 and 200 \((\log_{b}a)^{2017}=\log_{b}(a^{2017})?\) | 597 | |
<image>Question: On circle Oi, points C and D are on the same side of diameter AB, ∠AOC=30∘, and ∠DOB=45∘. What is the ratio of the area of the smaller sector COD to the area of the circle? | \(\frac{1}{3}\) | |
<image>Question: Points B and C lie on \(\overline{AD}\). The length of \(\overline{AB}\) is \(\frac{1}{3}\) times the length of \(\overline{BD}\), and the length of \(\overline{AC}\) is \(\frac{1}{4}\) times the length of \(\overline{CD}\). The length of \(\overline{BC}\) is what fraction of the length of \(\overline{... | \(\frac{1}{10}\) | |
<image>Question: Points A and B are on a circle of radius \( r \) and \( AB = 6 \). Point C is the midpoint of the minor arc \( AB \). What is the length of the line segment \( AC \)? | \( \sqrt{14} \) | |
<image>Question: A function \( f \) is defined by \( f(z) = (4+i)z^2 + \alpha z + \gamma \) for all complex numbers, where \( \alpha \) and \( \gamma \) are complex numbers and \( i^2 = -1 \). Suppose that \( f(1) \) and \( f(i) \) are both real. What is the smallest possible value of \( |\alpha| + |\gamma| \)? | \(\sqrt{2}\) | |
<image>Question: 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? | None of the above | |
<image>Question: Let \(a,\ b,\) and \(c\) be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation \((x-a)(x-b)+(x-b)(x-c)=0\)? | \(\frac{281}{13}\) | |
<image>Question: Quadrilateral \(ABCD\) is inscribed in a circle with \(\angle BAC = 70^\circ\), \(\angle ADB = 40^\circ\), \(AD = 4\), and \(BC = 6\). What is \(AC?\) | \(6\) | |
<image>Question: At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. Sh... | \(\frac{1}{3}\) | |
<image>Question: A plane contains points \(A\) and \(B\) with \(AB = 1\). Let \(S\) be the union of all disks of radius \(\frac{1}{2}\) in the plane that cover \(\overline{AB}\). What is the area of \(S?\) | \(3\pi - \frac{\sqrt{3}}{2}\) | |
<image>Question: If \( QR = 3\sqrt{3} \), and \( \angle QPR = 60^\circ \), then the area of \( \triangle PQR \) is \( \frac{a\sqrt{b}}{c} \) where \( a \) and \( c \) are relatively prime positive integers, and \( b \) is a positive integer not divisible by the square of any prime. What is \( a + b + c \)? | 122 | |
<image>Question: What is the value of \( 3 + \cfrac{1}{3 + \cfrac{1}{3 + \cfrac{1}{3}}} \)? | 109 | |
<image>Question: Five rectangles, A, B, C, D, and E, are arranged in a square as shown below. These rectangles have dimensions 1×6, 2×4, 5×6, 2×7, and 2×3, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle? | B | |
<image>Question: A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color—red, orange, yellow, blue, or green—so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? | 540 | |
<image>Question: A circle with integer radius \( r \) is centered at \( (r,r) \). Distinct line segments of length \( c_{i} \) connect points \( (0, a_{i}) \) to \( (b_{i}, 0) \) for \( 1 \leq i \leq 14 \) and are tangent to the circle, where \( a_{i}, b_{i} \), and \( c_{i} \) are all positive integers and \( c_{1} \l... | 17 | |
<image>Question: Juan rolls a fair regular octahedral die marked with the numbers through . Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? | 1/2 | |
<image>Question: A point \( P \) is randomly selected from the rectangular region with vertices \((0,0), (2,0), (2,1), (0,1)\). What is the probability that \( P \) is closer to the origin than it is to the point \((3,1)\)? | \( \frac{3}{4} \) | |
<image>Question: A convex quadrilateral \(ABCD\) with area contains a point \(P\) in its interior such that \(PA = 24\), \(PB = 32\), \(PC = 28\), \(PD = 45\). Find the perimeter of \(ABCD\). | \(4(36+\sqrt{113})\) | |
<image>Question: Let \( f(x) = x^2 + 6x + 1 \), and let \( R \) denote the set of points \( (x,y) \) in the coordinate plane such that
\[ f(x) + f(y) \leq 0 \quad \text{and} \quad f(x) - f(y) \leq 0 \]
The area of \( R \) is closest to | 8\pi | |
<image>Question: Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | \(\frac{2}{5}\) | |
<image>Question: Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? | 1+\(\sqrt{3}\) | |
<image>Question: Let S be the set of ordered triples (x,y,z) of real numbers for which log_{10}(x+y)=z and log_{10}(x^{2}+y^{2})=z+1. There are real numbers a and b such that for all ordered triples (x,y.z) in S we have x^{3}+y^{3}=a⋅10^{3z}+b⋅10^{2z}. What is the value of a+b? | 29/2 | |
<image>Question: Square AXYZ is inscribed in equiangular hexagon ABCDEF with X on BC, Y on DE, and Z on EF. Suppose that AB = 40, and EF = 41(√3 - 1). What is the side-length of the square? | 29√3 | |
<image>Question: Let S denote all points in the complex plane of the form \( a + b\omega + c\omega^2 \), where \( 0 \leq a \leq 1 \), \( 0 \leq b \leq 1 \), \( 0 \leq c \leq 1 \). What is the area of S? | \( \frac{3}{2} \sqrt{3} \) | |
<image>Question: Circles with centers \( O \) and \( P \) have radii 2 and 4, respectively, and are externally tangent. Points \( A \) and \( B \) are on the circle centered at \( O \), and points \( C \) and \( D \) are on the circle centered at \( P \), such that \( \overline{AD} \) and \( \overline{BC} \) are common... | \( 36 \) | |
<image>Question: Let \( x \) be chosen at random from the interval \( (0,1) \). What is the probability that \( \left\lfloor \log_{10} 4x \right\rfloor - \left\lfloor \log_{10} x \right\rfloor = 0 \)? Here \( \lfloor x \rfloor \) denotes the greatest integer that is less than or equal to \( x \). | \( \frac{1}{6} \) | |
<image>Question: Let \( S \) be the set of all points \((x,y)\) in the coordinate plane such that \( 0 \leq x \leq \frac{\pi}{2} \) and \( 0 \leq y \leq \frac{\pi}{2} \). What is the area of the subset of \( S \) for which
\[ \sin^{2}x - \sin x \sin y + \sin^{2}y \leq \frac{3}{4}? \] | \(\frac{\pi^{2}}{6}\) | |
<image>Question: Let \(ABCD\) be a square. Let \(E, F, G,\) and \(H\) be the centers, respectively, of equilateral triangles with bases \(\overline{AB}, \overline{BC}, \overline{CD},\) and \(\overline{DA}\) each exterior to the square. What is the ratio of the area of square \(EFGH\) to the area of square \(ABCD?\) | \(\frac{2+\sqrt{3}}{3}\) | |
<image>Question: How many sets of three teams {A,B,C} were there in which A beat B, B beat C, and C beat A? | 6 | |
<image>Question: Kymbrea's comic book collection currently has 30 comic books in it, and she is adding to her collection at the rate of 2 comic books per month. LaShawn's collection currently has 10 comic books in it, and he is adding to his collection at the rate of 6 comic books per month. After how many months will ... | 25 | |
<image>Question: Real numbers \( x, y, \) and \( z \) satisfy the inequalities \( 0 < x < 1 \), \( -1 < y < 0 \), and \( 1 < z < 2 \). Which of the following numbers is necessarily positive? | \( y + z \) | |
<image>Question: Supposed that \( x \) and \( y \) are nonzero real numbers such that \( \frac{3x+y}{x-3y} = -2 \). What is the value of \( \frac{x+3y}{3x-y} \)? | 2 | |
<image>Question: Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. If the total distance to her friend's house is 8.8 kilomet... | 28 | |
<image>Question: The data set [6,19,33,33,39,41,41,43,51,57] has median \(Q_{2}=40\), first quartile \(Q_{1}=33\), and \(Q_{3}=43\). Below the first quartile \((Q_{1})\) or more than 1.5 times the interquartile range above the third quartile \((Q_{3})\), where the interquartile range is defined as \(Q_{3}-Q_{1}\). How ... | 1 | |
<image>Question: The functions sin(x) and cos(x) are periodic with least period 2π. What is the least period of the function cos(sin(x))? | π | |
<image>Question: 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? | (sqrt(5)-1)/2 | |
<image>Question: (-10,-4) (3,9) √65 two circles has equation x+y=c. What is c; | 3 | |
<image>Question: 60% 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% | |
<image>Question: An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom... | 44π/3 | |
<image>Question: Let \(ABC\) be an equilateral triangle. Extend side \(\overline{AB}\) beyond \(B\) to a point \(B'\) so that \(BB' = 3AB\). Similarly, extend \(\overline{BC}\) beyond \(C\) to a point \(C'\) so that \(CC' = 3BC\), and extend \(\overline{CA}\) beyond \(A\) to a point \(A'\) so that \(AA' = 3CA\). What i... | 37:1 | |
<image>Question: The number \(21! = 51,090,942,171,709,440,000\) has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | \(\frac{1}{19}\) | |
<image>Question: The diameter \(AB\) of a circle of radius 2 is extended to a point \(D\) outside the circle so that \(BD = 3\). Point \(E\) is chosen so that \(ED = 5\) and line \(ED\) is perpendicular to line \(AD\). Segment \(AE\) intersects the circle at a point \(C\) between \(A\) and \(E\). What is the area of \(... | \(\frac{140}{37}\) | |
<image>Question: Let \( N = 123456789101112\ldots.4344 \) be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when \( N \) is divided by 45? | 9 | |
<image>Question: The graph of \( y = f(x) \), where \( f(x) \) is a polynomial of degree 3, contains points \( A(2,4) \), \( B(3,9) \), and \( C(4,16) \). Lines \( AB \), \( AC \), and \( BC \) intersect the graph again at points \( D \), \( E \), and \( F \), respectively, and the sum of the \( x \)-coordinates of \( ... | \(\frac{24}{5}\) | |
<image>Question: Quadrilateral \(ABCD\) has right angles at \(B\) and \(C\). \(\triangle ABC\) is similar to \(\triangle BCD\), and \(AB > BC\). There exists a point \(E\) in the interior of \(ABCD\) such that \(\triangle ABC\) is similar to \(\triangle CEB\) and the area of \(\triangle AED\) is 17 times the area of \(... | \(2 + \sqrt{5}\) | |
<image>Question: Alex, Mel, and Chelsea play a game that has 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 1/2, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins t... | 5/36 | |
<image>Question: Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had $43 dollars. What is the original (before discount) price of the most expensive shoes h... | $50 | |
<image>Question: What is the length of the longest interior diagonal connecting two vertices of P? | 9/4 | |
<image>Question: A regular pentagon with area \(\sqrt{5} + 1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? | \(\sqrt{5} - 1\) | |
<image>Question: An integer \( x \), with \( 10 \leq x \leq 99 \), is to be chosen. If all choices are equally likely, what is the probability that at least one digit of \( x \) is a 7? | \(\frac{1}{5}\) | |
<image>Question: In $\triangle ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 10$. The values of $\angle CBE$, $\angle DBE$, and $\angle ABE$ form a geometric progression, and the values of $\angle DBE$, $\angle CBE$, and $\angle DBC$ form an arithm... | $30$ | |
<image>Question: Line segment \(\overline{AB}\) is a diameter of a circle with \(AB = 24\). Point \(C\), not equal to \(A\) or \(B\), lies on the circle. As point \(C\) moves around the circle, the centroid (center of mass) of \(\triangle ABC\) traces out a closed curve missing two points. To the nearest positive integ... | 565 | |
<image>Question: Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent intersects the circles at C and D, respectively. Lines AB and CD intersect at E, and AE=5. What is CD? | 9 | |
<image>Question: Square ABCD has side length, a circle centered at E has radius, and r and s are both rational. The circle passes through D, and D lies on BE. Point F lies on the circle, on the same side of BE as | 1 | |
<image>Question: A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the... | 2/243 | |
<image>Question: In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? | 10 | |
<image>Question: Rectangle $A B C D$, pictured below, shares $50\%$ of its area with square $E F G H$. Square shares $20\%$ of its area with rectangle $A B C D$. What is ${\frac{A B}{A D}}$? | $\frac{3}{2}$ | |
<image>Question: If \( x < 0 \), then which of the following must be positive? | \( -x^{-1} \) | |
<image>Question: Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of triangle ACE is 70% of the area of the hexagon. What is the sum of all possible values of r? | \( \frac{10}{3} \) | |
<image>Question: Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $20 | |
<image>Question: Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is \(4:3\). The horizontal length of a \(27\)-inch television screen is closest, in inches, to which of the following? | 21 |
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