problem stringlengths 19 983 | answer stringlengths 1 8.19k |
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John and Mary each have a bag that contains one ball of each of the colors: blue, green, orange, red, and violet. Mary randomly selects one ball from her bag and puts it into John's bag. John then randomly selects one ball from his bag and puts it into Mary's bag. What is the probability that after this process the con... | \frac{1}{6} |
Given that \( ABCD \) is a square with side length 4, and \( EFGH \) is a rectangle with \( EF = 8 \). If the point \( B \) is at \( (0, 0) \) and the point \( E \) is at \( (4, 4) \), find the length \( AH \). Express your answer as a simplified radical. | 4\sqrt{10} |
If \(a\) and \(b\) are positive integers such that \(\gcd(a,b) = 3675\) and \(\mathop{ ext{lcm}}[a,b] = 3675^2\), and \(a < b\), how many possible values are there for \(a\)? | 9 |
Simplify \( an 120^\circ + 3 \sin 120^\circ\). | \frac{\sqrt{3}}{2} |
Let \( P(x) \) be a monic polynomial of degree 3. Suppose that \( P(x) \) has remainder \( R(x) \) when it is divided by \( (x - 1)(x - 4) \), and remainder \( 2R(x) \) when it is divided by \( (x - 2)(x - 3) \). Given that \( P(0) = 5 \), find \( P(5) \). | 125 |
The equation \(x^6 + (5x - 1)^6 = 0\) has 6 complex roots \(r_1, \overline{r}_1, r_2, \overline{r}_2, r_3, \overline{r}_3\), where the bar denotes complex conjugation. Find the value of \( rac{1}{r_1 \overline{r}_1} + rac{1}{r_2 \overline{r}_2} + rac{1}{r_3 \overline{r}_3}\). | 3 |
A point \((x, y)\) is randomly selected such that \(0 \le x \le 3\) and \(0 \le y \le 2\). What is the probability that \(x + y \le 2\)? Express your answer as a common fraction. | \frac{1}{3} |
Mary wants to buy a new laptop priced at 1000 US dollars. She has 1500 euros in her bank account and needs some Japanese Yen to cover the difference. The current exchange rates are: 1 US dollar = 0.95 euros, 1 US dollar = 110 Japanese Yen, and her bank charges a 2% fee on all transactions. How many more Japanese yen do... | 0 |
Consider a flat field with points \(A\), \(B\), \(C\), \(D\), and \(O\) arranged such that \(A\) is directly north of \(O\), \(B\) is directly west of \(O\), \(C\) is directly south of \(O\), and \(D\) is directly east of \(O\). The distance between \(C\) and \(D\) is 160 m. A hot-air balloon is positioned above \(O\) ... | 120 |
Simplify $ an 200^\circ + 4 \sin 200^\circ$. | \tan 20^\circ - 4 \sin 20^\circ |
For what real values of \(x\) is \(-9 < x^4 + 4x^2 < 10\) satisfied? Express your answer in interval notation. | (-\sqrt{-2 + \sqrt{14}}, \sqrt{-2 + \sqrt{14}}) |
Ahmad tried to solve the expression $ rac{2+i}{3-i}$. Unfortunately, he made a mistake in the final simplification, getting $ rac{2+i}{3-i}= rac{7-5i}{10}$. What is the correct answer Ahmad should have obtained? | \frac{1}{2} + \frac{1}{2}i |
Find the value of \(\sin 10^\circ \sin 20^\circ \sin 30^\circ \sin 40^\circ \sin 50^\circ \sin 60^\circ \sin 70^\circ \sin 80^\circ \sin 90^\circ \sin 100^\circ \sin 110^\circ \sin 120^\circ \sin 130^\circ \sin 140^\circ \sin 150^\circ \sin 160^\circ \sin 170^\circ \sin 180^\circ\). | 0 |
리커버런스 랑셔션 \( S_n + 1 = S_n^2 - S_n \)에 대한 계수 \( S_1 + S_2 + S_3 + \dots + S_n \)를 구하기위해, \( S_1 = S_{100} \)이며, \( S_1 = 2 \)인 경우의 전 도출 깨어내 현실 큰 비 으베메노력을 제공하고 다양한 방법 협력해야 하습니다. | 2n |
The polynomial \(x^3 - 2x^2 + 3x - 4\) is a factor of \(x^9 + cx^6 + dx^3 + e\). Enter the ordered triple \((c,d,e)\). | (4, 0, 4) |
Solve \[\sqrt{y + \sqrt{5y + 4}} + \sqrt{y - \sqrt{5y + 4}} = 8.\] | 17 |
For \(0 \le x \le 30\) and \(0 \le y \le 40,\) find the minimum value of \[\sqrt{x^2 + 1600} + \sqrt{y^2 + 1600} + \sqrt{x^2 + y^2 - 60x - 80y + 2600}.\] | 80 |
The polynomial \(x^3 - 3x^2 + 4x - 1\) is a factor of \(x^9 + px^6 + qx^5 + r\). Find the ordered triple \((p, q, r)\). | (27, -48, -12) |
In triangle \( ABC \), \( AB = AC \) and the exterior angle at \( C \) is \( 130^\circ \). Find the measure of angle \( x \) at \( B \). | 50 |
Solve \(\sqrt{x + \sqrt{2x + 4}} + \sqrt{x - \sqrt{2x + 4}} = 8\). | \frac{514}{31} |
If \(-4 \leq a \leq -1\) and \(2 \leq b \leq 4\), 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{3}{4} |
<Generate your problem here> | 正确 |
What is the value of the expression $(2x-3)(3x+2)-(2x-3)3x+2$ when $x=5$? | 16 |
Let \( z \) be a complex number such that \( |z| = 1 \). Find the maximum value of \( |2 + z| + |2 - z + z^2| \). | 5 |
Two runners, X and Y, start at a point Q on a straight road and run in the same direction. Runner Y runs twice as fast as runner X. An observer stands at point R so that \(\overline{QR}\) is perpendicular to the road. Find the maximum of \angle QRX, in degrees. | 90 |
Consider a square and a regular hexagon that are coplanar and share a common side \( \overline{BC} \). What is the degree measure of angle \( \angle ABF \), where \( F \) is a vertex of the hexagon adjacent to \( B \)? Express your answer as a common fraction. | 30 |
A hexagon is inscribed in a circle with radii extending from the center to the vertices. If one interior angle is 110° and another is 120°, what is the measure of the angle labeled \(\alpha\) at the third vertex? | 130^\circ |
Carla is riding on her unicycle. If the unicycle's wheel has a radius of 12 inches and makes 3 revolutions every 4 seconds, what is the unicycle's speed in inches per second? | 18\pi |
For what real values of \(a\) is \(-2 < a^2 + 2a < 5\) satisfied? Express your answer in interval notation. | (-1 - \sqrt{6}, -1 + \sqrt{6}) |
A wheel makes 100 revolutions in 2 minutes. Another wheel makes 150 revolutions in 3 minutes. If both wheels start from rest and are initially aligned at the topmost point, after how many seconds will both wheels be at their starting position again? | 1.2 |
Quadrilateral $PQRS$ is a square with an area of 25 square units. The pieces of a mathematical puzzle (known as tangram) are illustrated in which all triangles are isosceles and piece 'f' is a square. If the area of the gray piece in the diagram is represented by $A$, find $A$. Calculate your answer as a common fractio... | \frac{25}{4} |
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 cubic polynomial and the sum \(|a| + |b| + |c| + |d|\). | 15 |
The product of integers 125 and $k$ is a perfect cube. What is the smallest possible positive value of $k$? | 1 |
The medians \(AD\), \(BE\), and \(CF\) of triangle \(ABC\) intersect at the centroid \(G\). The line through \(G\) that is parallel to \(BC\) intersects \(AB\) and \(AC\) at \(M\) and \(N\), respectively. If the area of triangle \(ABC\) is 90, find the area of triangle \(ENG\). | 10 |
Quadrilateral \(ABCD\) is a square with area 16 square inches. The figure represents the pieces of a Chinese tangram in which all the triangles are isosceles and piece 'e' is a square. What is the area of the gray piece, in square inches? | 4 |
Let \( a \) and \( b \) be positive real numbers such that \( ab = 3 \). Find the minimum value of \( a^3 + b^6 \). | 18 |
Find all solutions to \[\cos \left( \sin^{-1} (x) + \cos^{-1} \left( rac{1}{x}
ight)
ight) = rac{\sqrt{2}}{2}.\] | \pm \frac{1}{\sqrt{2}} |
An equilateral triangle is inscribed in the parabola $x^2 = 64y,$ such that one of the vertices of the triangle coincides with the vertex of the parabola. Find the side length of this equilateral triangle. | 128\sqrt{3} |
Given a triangle \(ABC\) with sides \(a\), \(b\), and \(c\) and circumradius \(R\), let \(G\) be the centroid and \(H\) the orthocenter. Define \(F\) as the midpoint of segment \(\overline{GH}\). Express \(AF^2 + BF^2 + CF^2\) in terms of \(a\), \(b\), \(c\), and \(R\). | \frac{1}{4} (a^2 + b^2 + c^2 + 6R^2) |
A strictly increasing sequence of positive integers \(b_1, b_2, b_3, \dots\) has the property that for every positive integer \(k\), the subsequence \(b_{2k-1}, b_{2k}, b_{2k+1}\) is geometric with a common ratio of 2, and the subsequence \(b_{2k}, b_{2k+1}, b_{2k+2}\) is arithmetic with a common difference equal to \(... | 1 |
The superfactorial of a number \( n \), denoted as \( n\$, is defined as \( n!^{{n!}^{{\cdot}^{{\cdot}^{{\cdot}^{n!}}}}}\) with \( n! \) levels of exponentiation. What is the units digit of \( 5\$\)? | 0 |
Let \( Q(x) \) be a monic polynomial of degree 2. Suppose that \( Q(x) \) has remainder \( R(x) \) when it is divided by \( (x - 1)(x - 3) \), and remainder \( 2R(x) \) when it is divided by \( (x - 2)(x - 4) \). Given that \( Q(0) = 5 \), find \( Q(5) \). | 20 |
Find the value of \( rac{1}{r_1 \overline{r}_1} + rac{1}{r_2 \overline{r}_2} + rac{1}{r_3 \overline{r}_3} + rac{1}{r_4 \overline{r}_4} + rac{1}{r_5 \overline{r}_5}\) where \(r_1, r_2, r_3, r_4, r_5\) are the roots of the equation \(x^5 + (3x - 1)^5 = 0\). The roots are given by \(r_k = rac{\omega_k}{3\omega_k - 1}... | 5 |
The equation \(x^{10} + (3x - 1)^{10} = 0\) has 10 complex roots \(r_1, \overline{r}_1, r_2, \overline{r}_2, r_3, \overline{r}_3, r_4, \overline{r}_4, r_5, \overline{r}_5\), where the bar denotes complex conjugation. Find the value of \( rac{1}{r_1 \overline{r}_1} + rac{1}{r_2 \overline{r}_2} + rac{1}{r_3 \overline{r... | 5 |
If \(p\) and \(q\) are positive integers such that \(\gcd(p, q) = 60\), \(\mathop{ ext{lcm}}[p, q] = 60^2\), and \(p < q\), how many possible values are there for \(p\)? | 4 |
The polynomial \(x^3 - 3x^2 + 4x - 1\) is a factor of \(x^9 + px^6 + qx^3 + r\). Find the ordered triple \((p, q, r)\). | (-3, 27, 1) |
Let \( f(x) = \left\lfloor \left( - rac{2}{3}
ight)^x
ight
floor \) be a function defined for all \( x \) in the interval \([0, \infty)\). How many distinct values exist in the range of \( f(x) \)? | 3 |
Find the smallest positive real number \( D \) for which \[\left\| egin{pmatrix} 1 & 4 \ 0 & -1 \end{pmatrix} \mathbf{w}
ight\| \le D \|\mathbf{w}\|\] for all two-dimensional vectors \(\mathbf{w}\). | \sqrt{17} |
A fair eight-sided die has faces numbered 1, 2, 3, 4, 5, 6, 7, and 8. When rolling two such dice, the probability of obtaining a sum of 7 is \( rac{47}{368} \). If the probability of obtaining the number 3 is \( rac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, find \( m+n \). | 33 |
Let \( x \) and \( y \) be positive real numbers such that \( xy^3 = 3 \). Find the minimum value of \( x^2 + y^6 \). | 6 |
Simplify \( an 135^\circ + 4 \sin 135^\circ\). | -1 + 2\sqrt{2} |
For \(0 \le x \le 40\) and \(0 \le y \le 50\), find the minimum value of \(\sqrt{x^2 + 400} + \sqrt{y^2 + 900} + \sqrt{(x-40)^2 + (y-50)^2 - 100}.\) | 50\sqrt{2} + 10\sqrt{7} |
Simplify the expression \(\sin 200^\circ + 2 an 200^\circ\). | -\sin 20^\circ + 2 \tan 20^\circ |
The product of integers 180 and \( k \) is a perfect cube. What is the smallest possible positive value of \( k \)? | 150 |
Calculate \[ rac{1}{\cos^2 5^\circ} + rac{1}{\sin^2 10^\circ} + rac{1}{\sin^2 20^\circ}.\] | 11 |
Solve the equation \(\sqrt{2x + \sqrt{6x + 4}} + \sqrt{2x - \sqrt{6x + 4}} = 8\). | 8 |
The sum of a number and its square root equals 15. Find the number. | 9 |
Let \( h(x) = \left\lfloor \left( - rac{3}{4}
ight)^x
ight
floor \) be a function that is defined for all values of \( x \) in \( [0, \infty) \) such that \( h(x) \) is a real number. How many distinct values exist in the range of \( h(x) \)? | 2 |
Find the value of \(\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ \sin 90^\circ \sin 110^\circ \sin 130^\circ \sin 150^\circ\). | \frac{1}{16} |
In the diagram, \( \overline{DE} \) is parallel to the segment through \( F \), and \( DF = EF \). What is the number of degrees represented by \( y \)?
[asy]
size(150);
pair A,D,E,F;
A = (0,0);
D = (10,0);
E = (8,0);
F = (4,3);
draw(A--D);
draw(F--(8,3));
draw(F--(4,0));
filldraw(Circle((4,0),0.2));
filldraw(Circle(... | 25 |
If $c$ and $d$ are positive integers such that $\gcd(c, d) = 3360$, $\mathop{ ext{lcm}}[c, d] = 3360^2$, and $c < d$, how many possible values are there for $c$? | 24 |
Calculate the value of the complex fraction $ rac{2+3i}{1+i}$. Henry mistakenly forgot to include the minus sign when the numerator starts with a minus, and he found $ rac{2+3i}{1+i} = rac{5}{2} + rac{1}{2}i$. What is the correct answer he should have obtained? | \frac{5}{2} + \frac{1}{2}i |
A strictly increasing sequence of positive integers \(b_1\), \(b_2\), \(b_3\), \dots\) has the property that for every positive integer \(k\), the subsequence \(b_{2k-1}\), \(b_{2k}\), \(b_{2k+1}\) is geometric with ratio 3 and the subsequence \(b_{2k}\), \(b_{2k+1}\), \(b_{2k+2}\) is arithmetic with common difference ... | 1 |
What real values of \( x \) are not in the domain of \( g(x) = rac{1}{|x^2 + 2x - 3| + |x^2 + 8x + 15|} \)? | -3 |
In the diagram below, we have \(\overline{ST} \parallel \overline{QR}\), \(\angle P = 50^\circ\), and \(\angle Q = 40^\circ\). Find the measure of \(\angle STR\) in degrees. | 40 |
Solve \(\sqrt{x + \sqrt{2x + 3}} + \sqrt{x - \sqrt{2x + 3}} = 4\). | \frac{67}{14} |
A young athlete, Geeta from Australia, Tom from India, and Alex from Japan are considering part-time jobs in their spare time. They learn that Geeta earns 180 Australian dollars per hour, Tom earns 1600 Indian rupees per hour, and Alex earns 2200 Japanese yen per hour. If one US dollar is equivalent to 1.70 Australian ... | Geeta |
Compute \(2! + 24\) and divide by 8. | 3.25 |
Point \(C\) lies somewhere within or on the square which has opposite corners at \((0,0)\) and \((3,3)\). Point \(D\) lies somewhere within or on the square which has opposite corners at points \((6,0)\) and \((7,1)\). What is the greatest possible value of the slope of the line containing points \(C\) and \(D\)? Expre... | \frac{1}{3} |
Let \(\mathbf{c}\) and \(\mathbf{d}\) be vectors such that the angle between \(\mathbf{c}\) and \(\mathbf{d}\) is \(120^\circ\). If the angle between \(\mathbf{d}\) and \(\mathbf{c} - \mathbf{d}\) is \(30^\circ\), find the angle between \(\mathbf{c}\) and \(\mathbf{c} - \mathbf{d}\). | 30^\circ |
Find the number of ordered pairs \((a, b)\) of integers such that \(|a + bi| \le 5.\) | 113 |
Let \(\mathbf{u}\) and \mathbf{v}\) be vectors such that the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(30^\circ\), and the angle between \(\mathbf{v}\) and \(\mathbf{u} - \mathbf{v}\) is \(75^\circ\). Find the angle between \(\mathbf{u}\) and \(\mathbf{u} - \mathbf{v}\). | 45^\circ |
Let \(D\), \(E\), and \(F\) be the midpoints of the sides \(BC\), \(CA\), and \(AB\) of triangle \(ABC\), respectively. If the circumradius of triangle \(ABC\) is \(R\) and the side lengths are \(a\), \(b\), and \(c\), express \(AD^2 + BE^2 + CF^2\) in terms of \(a\), \(b\), \(c\), and \(R\). | \frac{3(a^2 + b^2 + c^2)}{4} |
In triangle \( DEF \), \( DE = DF \). The angle at \( E \) is \( 110^\circ \). What is the number of degrees represented by \( y \)? | 35 |
Twenty-eight students attend a science club meeting. The number of girls at the meeting is a multiple of 7, and there are more girls than boys attending the meeting. How many boys are at the meeting? | 7 |
A rectangular prism has maximum and minimum points | within \boxed{} |
A cyclist covers a distance of 120 miles in 3 hours. Another cyclist covers the same distance in \(4 rac{1}{2}\) hours. After what time will both cyclists have traveled the same number of miles if they start from the same point at the same time? | 0 |
When rolling an unfair die with faces numbered 1, 2, 3, and 4, the probability of obtaining face \( F = 1 \) is \( rac{2}{3} \), and the probability of obtaining each of the other faces is \( rac{1}{12} \). Given that the sum of the numbers on each pair of opposite faces is 5, what is the probability of obtaining a s... | \frac{1}{8} |
An equiangular octagon has four sides of length \( rac{\sqrt{2} + 1}{2} \) 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 |
The area of triangle PQR is given by \(5x^2 - 2x + 3\). If \(x = 2\), compute \( an P\). | 19 |
Square \(EFGH\) has a center \(I\) and \(EF/EH = m\). A point is randomly chosen from the interior of square \(EFGH\). What is the probability that it is closer to \(I\) than to any of the four vertices? | \frac{1}{2} |
A biased six-sided die is rolled. The probability of rolling a number \( n \) is \( rac{n}{21} \). What is the probability of rolling a number greater than 4? Express your answer as a common fraction. | \frac{11}{21} |
A hexagon is inscribed in a circle. The angles at vertices \(A\), \(B\), and \(D\) are \(120^\circ\), \(100^\circ\), and \(\alpha\) respectively. Given that the sum of the interior angles of a hexagon is \(720^\circ\), find the measure of \(\alpha\). | 140^\circ |
Find the greatest integer less than \((\sqrt{11} + \sqrt{7})^6\). (Do not use a calculator!) | 44927 |
Two numbers, \(a\) and \(b\) are selected at random from the interval \((0,4)\). What is the probability that a triangle with sides of length 2, \(a\), and \(b\) exists? | \frac{3}{4} |
A regular hexagon and a regular octagon are coplanar and share a common side $\overline{BC}$, as shown. What is the degree measure of angle $ABD$? Express your answer as a common fraction. | 105 |
If \(c\) and \(d\) are positive integers such that \(\gcd(c, d) = 420\), \(\mathop{ ext{lcm}}[c, d] = 420^3\), and \(c < d\), how many possible values are there for \(c\)? | 67 |
In the diagram, two circles, each with center $E$, have radii of $1$ and $3$. The total area of the shaded region is $ rac{1}{12}$ of the area of the larger circle. How many degrees are in the measure of $\angle AEC$? | 60 |
Let \( w \) be a complex number such that \( |w| = 1 \). Find the maximum value of \[ |1 + w| + |1 - w + w^2| \]. | 3 |
In the diagram, \(PT\) is parallel to \(QR\). What is the measure of \(\angle PQR\) in degrees? | 90 |
In the diagram, \(AB\) is parallel to \(CD\). What is the measure of \(\angle ABD\) in degrees? | 60 |
For \(0 \le x \le 10\) and \(0 \le y \le 20\), find the minimum value of \(\sqrt{x^2 + 100} + \sqrt{y^2 + 200} + \sqrt{(x-10)^2 + (y-20)^2 - 100}.\) | 30 |
A student pays a monthly rental fee of 2700人民币, 1500kr, and 5400 Cuban Peso for renting a flat in Haysburg, Ukraine, and Cairo, Egypt. If 1 kuna equals 6 Ugandan shillings, 10 Cuban Peso equals 100 Egyptian pounds, and 1 Ugandan shilling equals 5 Egyptian pounds, who pays the highest rental fee per Egyptian pound when ... | Cairo, Egypt |
Find the value of \( an B\) in triangle \(DEF\) where the area of the triangle is given by \(d^2 - (e - f)^2\), and the sides \(d\), \(e\), and \(f\) are the lengths of the triangle. The area of the triangle can also be expressed using the formula \( rac{1}{2}ef \sin B\). | 2 |
The area of triangle $PQR$ is equal to $(p^2 - (q - r)^2),$ where $p,$ $q,$ and $r$ are the sides of triangle $PQR.$ Compute $ an P.$ | 2 |
In the ellipse \( 2x^2 + y^2 = 1 \), the foci \( F_1 \) and \( F_2 \) are given. A circle that passes through these foci and is tangent to the ellipse at two points on the \( x \)-axis has its center on the \( y \)-axis. Compute the coordinates of the center of this circle. | \left(0, \frac{\sqrt{2}}{2}\right) |
Point \(C\) lies somewhere within or on the circle with center at \((1,1)\) and radius 1. Point \(D\) lies somewhere within or on the square which has opposite corners at \((3,2)\) and \((4,3)\). What is the greatest possible value of the slope of the line containing points \(C\) and \(D\)? Express your answer as a com... | \frac{3}{2} |
Consider the geometric sequence \( rac{81}{25}, rac{27}{5}, 9, 3, \ldots\). What is the eighth term of the sequence? Express your answer as a common fraction. | \frac{3125}{27} |
If the domain of the function \(\log x^2\) is \(x < a\) or \(x > b\), for some \(a\) and \(b\), find \(a + b\). | 0 |
A parabola with equation \(y = 2x^2 - 4x + 3\) and vertex \((1, 1)\) is reflected about the line \(y = 1\). This results in the parabola with equation \(y = dx^2 + ex + f\). Express \(a + b + c + d + e + f\) in terms of \(k\). | 2 |
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