problem stringlengths 3 604 | answer stringlengths 1 304 |
|---|---|
Alice and Bob are playing a game. Alice starts first. On Alice's turn, she flips a coin. If she gets a heads, she wins. If not, it becomes Bob's turn. On Bob's turn, he flips a coin. If she gets a tails, she wins. If not, it becomes Alice's turn. What is the probability that Alice wins the game? | \dfrac{2}{3} |
Given the original problem statement with error, it is important to devise a new problem example, that is adjustment can be performed. Newton's Method for Differentiable Functions Conflict: Comparison of Fourfold Computational Error Solution Errors Confuter.
If ∛(9) = | The value of ∛(9) is approximately 2 |
If $\omega^{2023} = 1$ and $\omega
eq 1,$ then evaluate
\[ rac{1}{1 + \omega} + rac{1}{1 + \omega^2} + \dots + rac{1}{1 + \omega^{2023}}.\] | -\frac{1}{2} |
Let \( E \) be the ellipse given by \( 2x^2 + y^2 = 1 \), with foci \( F_1 \) and \( F_2 \). Suppose there is a circle that passes through \( F_1 \) and \( F_2 \) and is tangent to the ellipse at two points on the \( x \)-axis. Compute the value of the constant \( 2 \). | 2 |
Find the sum of all complex solutions of the equation \[ rac{1}{x^2 - 1} + rac{2}{x^2 - 2} + rac{3}{x^2 - 3} + rac{4}{x^2 - 4} = 2010x - 4.\] | 0 |
If \( \alpha \) is a complex number such that \( \alpha^{1009} = 1 \) and \( \alpha
eq 1 \), then evaluate \[ rac{1}{1 + \alpha} + rac{1}{1 + \alpha^2} + \dots + rac{1}{1 + \alpha^{1008}}. \] | 504 |
A first gear rotates \(25 rac{1}{5}\) times per minute, while a second gear rotates 30 times per minute. Initially, a mark on each gear is pointing due north. After how many seconds will the two gears next have both their marks pointing due north? | 50 |
Solve the equation \(\sqrt{x + \sqrt{4x + 8}} + \sqrt{x - \sqrt{4x + 8}} = 8.\) | 17.2 |
Given the recurrence relation \( g(n + 1) = (-1)^{n + 1} n - 2g(n) \) for \( n \ge 1 \), and the initial condition \( g(1) = g(1986) \), compute the sum \( g(1) + g(2) + g(3) + \dots + g(1985) \). | 0 |
In the diagram below, we have \(\overline{AB} \parallel \overline{CD}\), \(\angle E = 60^\circ\), and \(\angle F = 30^\circ\). Find the measure of \(\angle BFC\) in degrees. | 90 |
Let \(x, y, z, w\) be positive real numbers such that \(x + y + z + w = 1\). Find the minimum value of \( rac{1}{x} + rac{4}{y} + rac{9}{z} + rac{16}{w}\). | 100 |
Consider two cyclists, \(X\) and \(Y\), starting at a common point \(Q\) on a circular track and cycling in the same direction. Cyclist \(Y\) cycles twice as fast as cyclist \(X\). A spectator stands at point \(R\) on the track such that \(\overline{QR}\) is perpendicular to the track. Determine the maximum value of \(... | 90 |
Find all solutions to \[\sin \left( an^{-1} (x) + \cot^{-1} (x)
ight) = rac{1}{2}.\] | No solution |
The area of \( riangle ABC\) is 8 square centimeters. \(\overline{AB} \parallel \overline{DE}\). \(BD = 5BC\). What is the number of square centimeters in the area of \( riangle CDE\)? | 288 |
In triangle ABC, angle A is trisected by lines AD and AE, and angle B is trisected by lines BF and BG. If angle A is \(90^\circ\) and angle B is \(60^\circ\), what is the measure of angle DFG? | 80^\circ |
A point \((x,y)\) is randomly selected such that \(0 \le x \le 5\) and \(0 \le y \le 3\). What is the probability that \(x + y \le 3\)? Express your answer as a common fraction. | \frac{3}{10} |
An equiangular octagon has four sides of length $1$ and four sides of length $ rac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon? | 1 + \sqrt{2} |
Consider the function \( g(x) = rac{1}{|x^2 - 5x + 6| + |x^2 - 7x + 10|} \). What real values of \( x \) are not in the domain of \( g(x) \)? | 2 |
A hexagon is inscribed in a circle with one of its angles measuring \(120^\circ\) and another angle measuring \(130^\circ\). If the measure of one of the remaining angles is \(100^\circ\), what is the measure of the angle \( heta\) opposite the \(120^\circ\) angle? | 60^\circ |
In triangle \( riangle DEF \), \( DE = EF \), and the angle at \( D \) is \( 120^\circ \). Find the number of degrees in the angle \( \angle EDF \). | 30 |
An investor deposits \$15,000 in a savings account that compounds quarterly at an annual interest rate of 8%. Find the total amount of money in the account after 3 years, rounded to the nearest dollar. | 19024 |
Let \( p(x) \) be a polynomial of degree 4 such that \$ p(n) = rac{n}{n^2 - 1} \$ for \( n = 2, 3, 4, 5, 6 \). Find \( p(7) \). | \frac{7}{48} |
An investment of $30,000 is made in a government bond that pays 1.5% monthly interest. At the end of four years, what is the total number of dollars in this investment? Express your answer to the nearest whole number. | 63415 |
Let \( x, y, z, w \) be positive real numbers such that \( x + y + z + w = 1 \). Find the minimum value of \( rac{1}{x} + rac{1}{y} + rac{9}{z} + rac{81}{w} \). | 196 |
The polynomial \(x^3 - 3x^2 + 4x - 1\) is a factor of \(x^9 + px^6 + qx^3 + r\). Find the ordered triple \((p, q, r)\). | (-27, -36, 27) |
Let \( G_1 \) and \( G_2 \) be the foci of the ellipse \( 3x^2 + y^2 = 12 \). Suppose there is a circle which passes through \( G_1 \) and \( G_2 \) and which is tangent to the ellipse at two points on the \( x \)-axis. Compute the distance between the foci, and use it to determine the value of \( 3 \). | 3 |
在一个由许多大小相等的圆形区域进行SOVO修订的控制器损伤超过的车辆数字选择的问题，群集部分方法。但该句子构成大量朋友的文化文化。 | in Chinese |
Alex needs money for his vacation. If he has 400 US dollars in his bank account but wants to withdraw one-third of it in British pounds and two-thirds in euros, how many fewer pounds than euros will he have? Assume 1 pound = 1.75 USD and 1 euro = 1.25 USD, and round to the nearest whole number. | 137 |
In quadrilateral \(ABCD\), angle \(BAD\) is trisected into three equal parts, and angle \(CDA\) is also trisected into three equal parts. If \( \angle BAD = 120^\circ \) and \( \angle CDA = 150^\circ \), what is the degree measure of angle \(AFD\)? | 90 |
Triangle \(DEF\) is isosceles, with \(DE = DF\) and altitude \(EG = 15\). Suppose that there is a point \(H\) on \(\overline{EG}\) with \(EH = 12\) and \(\angle FGH = 2\angle DEF\). Find the perimeter of triangle \(DEF\). | 60 |
In triangle \(ADC\), \(AD = 8\), \(CD = 5\), and \(AC = 9\). Let \(E\) be the foot of the altitude from \(A\) to \(CD\). Find the area of triangle \(AEC\). | \frac{36}{5} |
The superfactorial \(n\$ is defined as \(n!^{{n!}^{{\cdot}^{{\cdot}^{{\cdot}^{n!}}}}}\) with \(n!\) levels of exponentiation. What is the units digit of \(3\$\)? | 6 |
For \(0 \le x \le 10\) and \(0 \le y \le 15\), find the minimum value of \[\sqrt{x^2 + 25} + \sqrt{y^2 + 81} + \sqrt{x^2 + y^2 - 10x - 18y + 101}.\] | 14\sqrt{2} |
Assume \(0 < r < 2\). Below are five equations for \(x\). Which equation has the largest solution \(x\)?
extbf{(A)} \(2(1 + r)^x = 5\)
extbf{(B)} \(2(1 + r/10)^x = 5\)
extbf{(C)} \(2(1 + 2r)^x = 5\)
extbf{(D)} \(2(1 + \sqrt{r})^x = 5\)
extbf{(E)} \(2(1 + 1/r)^x = 5\) | E |
Find the minimum value of the expression √(x² + 400) + √(y² + 900) + √(x² + y² - 80x - 100y + 4100) for 0 ≤ x ≤ 40 and 0 ≤ y ≤ 50. | 50 |
The area of triangle \(ABC\) is equal to \(a^2 - (b - c)^2\), where \(a\), \(b\), and \(c\) are the sides of triangle \(ABC\). Compute \( an A\). | \frac{8}{15} |
The trapezoid with sides $3x, 2x, y, 8-y$ has an area of $10x^2$. Given that $x$ and $y$ are positive integers, what is the value of $x + y$? | 6 |
A historical tourist guide in Istanbul earns 150 Turkish Liras per hour, a customer service representative in Paris earns 30 Swiss Francs per hour, and a sushi chef in Tokyo earns 450 Japanese Yen per hour. If one US dollar is equivalent to 5.20 Turkish Liras, 1.40 Swiss Francs, and 115.50 Japanese Yen, who would earn ... | historical tourist guide in Istanbul |
Find all values of \( y \) that satisfy the equation \( y = \sqrt{20 - 2y} + 3 \). | No real solutions |
Let \( g(x) = x^6 + 4x^5 - 6x^4 + 2x^3 - 8x^2 + 3x - 7 \). Without using long division, find the remainder when \( g(x) \) is divided by \( x^2 - 1 \). | 9x - 20 |
Let \( \lambda \) be a constant, \( 0 \le \lambda \le 4 \), and let \( f : [0,1] o [0,1] \) be defined by \[ f(x) = \lambda x(1 - x). \] Find the values of \( \lambda \) for which there exists an \( x \in [0,1] \) such that \( f(x)
eq x \) and \( f(f(x)) = x \) but \( f(x)
eq f(f(x)) \). | 2 |
在一个半径为5m的圆形区域内，一只小鸟以恒定速度2m/s绕圆心飞行。当它到达圆周上的任意一点时，立即反向飞回原点。假设小鸟开始时位于圆周上，求在t=144s时小鸟与圆心的距离。请详细解答并说明每一步的推理过程。 | 5 米 |
<Your JSON output here> | within \boxed{} |
Let \( T \) be the set of points \((p, q)\) with \( 0 \le p \le 1 \) and \( 0 \le q \le 1 \) such that the equation \( x^4 + px^3 - qx^2 + px + 1 = 0 \) has at least one real root. Determine the area of the graph of \( T \). | \frac{1}{2} |
In triangle \( ABC \), angle \( A \) is trisected into three equal parts, and angle \( C \) is also trisected into three equal parts. If angle \( A \) measures \( 90^\circ \) and angle \( C \) measures \( 50^\circ \), what is the degree measure of the angle formed at the intersection of the trisectors of angle \( A \) ... | 70^\circ |
For \(0 \le x \le 30\), find the minimum value of \(\sqrt{x^2 + 900} + \sqrt{y^2 + 16} + \sqrt{x^2 + y^2 - 60x - 8y + 916}\). | 34\sqrt{2} |
Let \( g(x) = x^{10} + 7x^9 - 6x^8 + 9x^7 - 2x^6 - 10x^5 + 3x^4 - 9x^3 + 13x^2 - 4x - 6 \). Without using long division, find the remainder when \( g(x) \) is divided by \( x^2 - 1 \). | -2x + 8 |
What is the new problem for this dissatisfied posing id internal datasource? | The internal data source for the posing ID is outdated or inaccurate, causing discrepancies and inefficiencies in identification processes |
小明和小红在玩一个游戏。小明先开始。在他第一次轮流时，他会掷一枚硬币。如果他掷出正面，他就赢了。如果不是，轮到小红。小红掷硬币时，如果她掷出反面，她就赢了。如果不是，轮到小明再次掷硬币。那么小明赢得这个游戏的概率是多少？ | \dfrac{2}{3} |
Let \( x, y, z, \) and \( w \) be positive real numbers such that \( x + 2y + 3z + 4w = 15. \) Find the maximum value of \( x^1 y^2 z^3 w^4. \) | \frac{50625}{256} |
James Edward knew to calculate \( rac{-5+7i}{2+3i}\), but he missed the minus sign in the numerator, finding \( rac{5+7i}{2+3i} = rac{31}{13} - rac{1}{13}i\). What correct answer should he have obtained? | \frac{11}{13} + \frac{29}{13}i |
Consider the equation \( 2(1 + 0.5s)^t = 4 \). Which of the following values of \( s \) results in the smallest solution for \( t \)?
(A) \( s = 2 \)
(B) \( s = 1 \)
(C) \( s = 0.5 \)
(D) \( s = 0.25 \)
(E) \( s = 0.1 \) | A |
The distances from a point $P$ to five of the vertices of a regular octahedron are 5, 6, 8, 10, and 12. Find the distance from $P$ to the sixth vertex. | 9 |
A point \((x, y)\) is randomly selected such that \(0 \le x \le 6\) and \(0 \le y \le 3\). What is the probability that \(x + y \le 3\)? Express your answer as a common fraction. | \frac{1}{4} |
A point $(x,y)$ is randomly selected such that $0 \le x \le 7$ and $0 \le y \le 5$. What is the probability that $x + y \le 5$? Express your answer as a common fraction. | \frac{5}{14} |
A pentagon is inscribed in a circle. The measures of four of its interior angles are \(100^\circ\), \(110^\circ\), \(120^\circ\), and \(130^\circ\). What is the measure of the fifth angle, in degrees? | 80 |
In triangle \(ABC\), \( \overline{BC} \) is parallel to the line through \(A\), and \( AB = BC \). If the angle at \( A \) is \( 128^\circ \), what is the measure of angle \( x \)? | 26 |
In the circle with center P, radii PA and PB form a right angle. The two smaller regions are tangent semicircles, as shown. The radius of the circle with center P is 12 inches. What is the radius of the smaller semicircle? Express your answer as a common fraction. | 6 |
A \(4 imes n\) grid contains a total of 84 squares of various sizes. Determine the value of \(n\). | 9 |
The product of integers 1800 and \( k \) is a perfect cube. What is the smallest possible positive value of \( k \)? | 15 |
Evaluate the expression \[\sin \left( an^{-1} (x) + \cot^{-1} (x)
ight) - \sin \left( an^{-1} (x) + \cot^{-1} \left( rac{1}{x}
ight)
ight)\] and find the value when \(x = 2\). | \frac{1}{5} |
In the diagram below, we have \(\overline{BC}\parallel\overline{AD}\), \(\angle A = 50^\circ\), and \(\angle D = 60^\circ\). Find the measure of \(\angle CBA\) in degrees. | 60 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the maximum value of \( |2 + w| + |w - 2 + w^2| \). | 2\sqrt{5} |
If \( z^{2020} = 1 \) and \( z
eq 1 \), then evaluate \( rac{1}{1 + z} + rac{1}{1 + z^2} + \dots + rac{1}{1 + z^{2019}}.\) | 1009 |
Find the greatest integer less than \((\sqrt{10} + \sqrt{6})^6\). (Do not use a calculator!) | 31231 |
Two friends, Mike and Lisa, each have a backpack with one ball of each of the colors blue, green, orange, red, and violet. Mike randomly selects one ball from his backpack and puts it into Lisa's backpack. Lisa then randomly selects one ball from her backpack and puts it into Mike's backpack. What is the probability th... | \frac{1}{6} |
In triangle DEF, the area is given by the expression \( d^2 - (e - f)^2 \), where \( d \), \( e \), and \( f \) are the sides of the triangle. Compute \( an D \). | 2 |
In triangle \( DEF \), \( DE = EF \) and \( \angle DEF = 124^\circ \). Find the measure of \( \angle DFE \). | 28^\circ |
Given positive real numbers \(x, y, z, w\) such that \(x + y + z + w = 1\). Find the minimum value of \( rac{1}{x} + rac{1}{y} + rac{9}{z} + rac{25}{w}\). | 100 |
If the domain of the function \(\log (x - 1)^2\) is \(x < a\) or \(x > b\), for some \(a\) and \(b\), find \(a + b\). | 2 |
Two friends, Jack and Jill, each have a box containing one stone of each of the colors, blue, green, orange, red, and violet. Jill randomly picks one stone from her box and places it in Jack's box. Jack then randomly picks one stone from his box and places it in Jill's box. What is the probability that after these acti... | \frac{1}{6} |
Assume \(0 < s < 2\). Below are five equations for \(y\). Which equation has the smallest solution \(y\)?
\[ extbf{(A)}\ 2(1 + s)^y = 5\qquad extbf{(B)}\ 2(1 + s/10)^y = 5\qquad extbf{(C)}\ 2(1 + 2s)^y = 5\]\[ extbf{(D)}\ 2(1 + \sqrt{s})^y = 5\qquad extbf{(E)}\ 2(1 + 1/s)^y = 5\] | E |
In triangle \( DEF \), the medians \( DG \), \( EH \), and \( FI \) intersect at the centroid \( K \). The line through \( K \) that is parallel to \( DE \) intersects \( DF \) and \( EF \) at \( P \) and \( Q \), respectively. If the area of triangle \( DEF \) is 100, find the area of triangle \( KPQ \). | \frac{100}{9} |
An investment of $15,000 is made in a savings account that pays 2% weekly interest. How much will be in the account after four years? Express your answer to the nearest whole number. | 29505 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of \[ rac{(a + b + c)[(2a)^2 + (3a)^2]}{abc}.\] | 39 |
Determine the maximum value of \( rac{2x - 3y}{4x^2 + 9y^2 + k} \) over all real numbers \(x\) and \(y\), where \(k\) is a positive constant. | \frac{\sqrt{13}}{2} |
Let \(a\), \(b\), and \(c\) be positive real numbers. Find the minimum value of \( rac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}\). | 120 |
If \(\arccos x + \arccos 2x + \arccos 3x = \pi\), then \(x\) satisfies a cubic polynomial of the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are integers, and \(a
eq 0\). Find the value of \(x\) such that the polynomial equation \(6x^3 + 12x^2 + 2x - 1 = 0\) holds. | \frac{1}{3} |
There are 240 people in a school. 10 take mathematics, science, and history, and 10 don't take any of them. 160 take mathematics. Twice as many students take history as take science. 80 take both mathematics and history, and 80 take both science and history. Only 40 take both science and mathematics. How many stu... | 80 |
If the domain of the function \(\log (x^2 - 4)\) is \(x < a\) or \(x > b\), for some \(a\) and \(b\), find \(a + b\). | 0 |
Find the greatest integer less than \((2 + \sqrt{3})^6.\) | 2701 |
If $-5 \leq a \leq -1$ and $4 \leq b \leq 7$, what is the greatest possible value of $\left(a + rac{1}{b}
ight)\left( rac{1}{b} - a
ight)$? Express your answer as a common fraction. | -\frac{399}{16} |
Find the sum of all complex solutions of the equation
\( rac{1}{x^2 - 1} + rac{2}{x^2 - 2} + rac{3}{x^2 - 3} + rac{4}{x^2 - 4} = 2010x - 40\). | 0 |
If \(a\) and \(b\) are positive integers such that \(\gcd(a, b) = 30\), \(\mathop{ ext{lcm}}[a, b] = 30^3\), and \(a < b\), how many possible values are there for \(a\)? | 4 |
Assume \(0 < t < 5\). Below are five equations for \(y\). Which equation has the smallest solution \(y\)? | n(n+1) |
Find the product of the \( y \)-coordinates of all the distinct solutions \((x, y)\) for the two equations \( y = x^2 - 4 \) and \( y^2 = -3x + 36 \). | 16 |
A regular quadrilateral (square) and a regular hexagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of angle $BAC$? Express your answer as a common fraction. | 30 |
Two cars, \(C_1\) and \(C_2\), start from the same point \(O\) on a straight road and move in the same direction. Car \(C_2\) travels twice as fast as car \(C_1\). An observer stands at point \(P\) such that \(\overline{OP}\) is perpendicular to the road. Find the maximum value of \(\angle PC_1C_2\), in degrees. | 90 |
The distances from a point \( Q \) to five of the vertices of a regular tetrahedron are 6, 10, 12, 14, and 18. Find the distance from \( Q \) to the sixth vertex. | 16 |
Five points, \(P\), \(Q\), \(R\), \(S\), and \(T\), are chosen randomly and independently on the circumference of a circle. What is the probability that segments \(PR\) and \(QS\) intersect? | \frac{1}{3} |
Compute \( rac{1}{\cos^2 15^\circ} + rac{1}{\sin^2 30^\circ} + rac{1}{\sin^2 60^\circ}\). | \frac{40}{3} - 4\sqrt{3} |
In the diagram below, we have \(\overline{UV} \parallel \overline{YZ}\), \(\angle A = 60^\circ\), and \(\angle B = 20^\circ\). Find the measure of \(\angle UXY\) in degrees. | 60 |
Let \( q(x) = (x^2 - 1)p(x) - x \) be a polynomial of degree 7 such that \( q(n) = 0 \) for \( n = 2, 3, 4, 5, 6, 7, 8 \). Find the value of \( p(8) \). | \frac{8}{63} |
Calculate the value of \(\frac{2+3i}{4+i}\) and simplify the result. | \frac{11}{17} + \frac{10}{17}i |
An investment of \$30,000 is made in a government bond that will pay 2\% monthly interest (meaning that the investment will increase by 2\% every month). At the end of three years, what is the total number of dollars in this investment? Express your answer to the nearest whole number. | 61197 |
A point \((x, y)\) is randomly selected such that \(0 \le x \le 5\) and \(0 \le y \le 3\). What is the probability that \(x + y \le 4\)? Express your answer as a common fraction. | \frac{2}{5} |
A trader from Venezuela, Ana from Ecuador, and Carlos from Chile are discussing their part-time jobs. Ana makes 120 bolivares per hour, Carlos makes 150 pesos per hour, and the trader makes 2000 manat per hour. If one US dollar is equivalent to 3.5 venus bolivar, 6.00 ecuadorian sucre, and 7.50 chilean guarani, and all... | Trader |
Solve \(\sqrt{x + \sqrt{4x + 4}} + \sqrt{x - \sqrt{4x + 4}} = 8.\) | \frac{257}{15} |
For \(0 \le x \le 30\) and \(0 \le y \le 40\), find the minimum value of \(\sqrt{x^2 + 900} + \sqrt{y^2 + 1600} + \sqrt{x^2 + y^2 - 60x - 80y + 2500}\). | 120 |
Find the greatest integer less than \((\sqrt{11} + \sqrt{7})^6\). | 12870 |
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