problem stringlengths 19 983 | answer stringlengths 1 8.19k |
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In the diagram below, \( \overline{DE} \) is parallel to the segment through \( F \), and \( DF = EF \). What is the number of degrees represented by \( y \)? | 60 |
The graphs of \(x^2 + y^2 - 4x - 12y + 35 = 0\) and \(x^2 - y^2 - 4x + 12y - 35 = 0\) intersect at four points. Compute the sum of the distances from these four points to the point \((2,6)\). | 4\sqrt{5} |
For \(0 \le x \le 10\) and \(0 \le y \le 15\), find the minimum value of \(\sqrt{x^2 + 25} + \sqrt{y^2 + 64} + \sqrt{x^2 + y^2 - 20x - 24y + 481}\). | 20 |
Simplify \( an 110^\circ + 4 \sin 110^\circ\). | 0 |
Find the greatest integer less than \((\sqrt{11} + \sqrt{7})^6\). | 44495 |
Find the number of ordered pairs \((c, d)\) of integers such that \(|c + di| \le 3\). | 29 |
In triangle \(XYZ\), the medians \(XD\), \(YE\), and \(ZF\) intersect at the centroid \(G\). The line through \(G\) that is parallel to \(YZ\) intersects \(XY\) and \(XZ\) at \(M\) and \(N\), respectively. If the area of triangle \(XYZ\) is 96, find the area of triangle \(YNG\). | 16 |
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 |
If \(c\) and \(d\) are positive integers such that \(\gcd(c, d) = 42\) and \(\mathop{ ext{lcm}}[c, d] = 42^3\), how many possible values are there for \(c\)? | 8 |
Consider an equilateral triangle inscribed in the parabola \(x^2 = 16y\), such that one of the vertices of the triangle coincides with the vertex of the parabola. Find the side length of this equilateral triangle. | 32\sqrt{3} |
A truck travels upstream at a neighbor's rate, and I have the ability to reset. This change I read the error categories now anywhere nearby ≠. Now, please clarify whether the truck's speed is constant or if it is affected by the direction it is traveling relative to the neighbors. Additionally, consider the impact of v... | The truck's speed is not constant and is affected by the direction of travel and external factors such as weather and road conditions |
There is a total of 70 squares of three sizes whose vertices are points on this rectangular 3 × n grid of points. What is the value of n? | 13 |
An equilateral triangle is inscribed in the parabola \(y = x^2 + 4\), such that one of the vertices of the triangle coincides with the vertex of the parabola. Find the side length of this equilateral triangle. | 2\sqrt{3} |
In triangle \(PQR\), \(PQ = 13\), \(PR = 14\), and \(QR = 15\). Let \(E\) be the foot of the altitude from \(Q\) to \(PR\). Find the area of triangle \(PQE\). | 30 |
A circle is inscribed in an equilateral triangle with a side length of 16√3. Find the radius of the circle. | 8 |
Find the number of ordered pairs $(a, b)$ of integers such that $|a + bi| \le 5.$ | 81 |
Let \( Q(x) \) be a monic polynomial of degree 3. Suppose that \( Q(x) \) has remainder \( T(x) \) when it is divided by \( (x - 2)(x - 5) \), and remainder \( 3T(x) \) when it is divided by \( (x - 3)(x - 4) \). Given that \( Q(0) = 7 \), find \( Q(6) \). | 73 |
In triangle \( DEF \), segment \( DF \) is parallel to the segment through \( E \). Given that \( DE = EF \) and the angle at \( D \) is \( 80^\circ \), what is the number of degrees represented by \( y \)? | 20 |
<If \(u\) and \(v\) are positive integers such that \(\gcd(u, v) = 27\), \(\mathop{ ext{lcm}}[u, v] = 27^3\), and \(u < v\), how many possible values are there for \(u\)?> | 3 |
If \(\alpha\) is a primitive 12th root of unity and \(\alpha
eq 1\), then evaluate \[ rac{1}{1 + \alpha} + rac{1}{1 + \alpha^2} + \dots + rac{1}{1 + \alpha^{12}}.\] | 6 |
The graphs of \(x^2 + y^2 + 8x - 16y + 64 = 0\) and \(x^2 - y^2 + 8x + 20y - 80 = 0\) intersect at four points. Compute the sum of the distances from these four points to the point \((-4,6)\). | 16 |
In a $3 imes n$ grid of points, there are 70 squares that can be formed. Determine the value of $n$. | 12 |
A rectangle with an area of 24 square centimeters is divided into 6 smaller rectangles. Each of these rectangles has an area of 4 square centimeters. What is the area of one of the corner rectangles? | 4 |
Let \( C \) be the circle with equation \( x^2 + y^2 = r^2 \) and let \( P \) be the parabola with equation \( y = ax^2 \). Given that the circle passes through two points on the \( x \)-axis and is tangent to the parabola at two points, find the value of \( a \) if the radius \( r \) of the circle is equal to the dist... | \frac{1}{2} |
In a small town, three friends—Rita from India, Carlos from Argentina, and Aisha from Nigeria—are discussing their weekend jobs. Rita earns 900 rupee per hour, Carlos earns 1,440 pesos per hour, and Aisha earns 800 naira per hour. If one US dollar is equivalent to 75 rupee, 73.02 pesos, and 75.75 naira, who would earn ... | Carlos |
To serve as a table | A table is a structured collection of data in a relational database, consisting of rows and columns, where each row represents a record and each column represents a field. Tables are used to store and organize data efficiently |
The product of integers 360 and $k$ is a perfect cube. What is the smallest possible positive value of $k$? | 75 |
An equiangular nonagon has four sides of length 1 and five sides of length \(\frac{\sqrt{2}}{2}\), arranged so that no two consecutive sides have the same length. What is the area of the nonagon? | 2 |
The coordinates of a parallelogram are (3, 2), (4, 7), (5, 3) and $(x, y)$ and $x > 5$. What is the value of $x + y$? | 14 |
In a game, two players, Alex and Jamie, take turns rolling a fair six-sided die. Alex rolls first. If the result is even, Alex wins. If the result is odd, Jamie rolls the die. If Jamie rolls a number less than 4, Jamie wins. If Jamie rolls a number 4 or greater, it becomes Alex's turn again. What is the probability tha... | \frac{2}{3} |
A bicycle wheel makes 200 revolutions per minute. Another bicycle wheel makes 250 revolutions per minute. Initially, a mark on each wheel is pointing due north. After how many seconds will the two wheels next have both their marks pointing due north? | 1.2 |
A certain unfair six-sided die has faces numbered 1 through 6. The probability of obtaining face 3 is greater than \(\frac{1}{6}\), the probability of obtaining the face opposite to 3 (which is 4) is less than \(\frac{1}{6}\), the probability of obtaining each of the other faces is \(\frac{1}{6}\), and the sum of the n... | 79 |
In how many ways can 5 people sit around a round table if no two of the 3 people Anna, Bella, and Chris can sit next to each other? (Seating arrangements which are rotations of each other are treated as the same.) | 4 |
Two players, Mia and Leo, each have a box containing one marble of each color: white, black, yellow, green, and purple. Mia randomly picks one marble from her box and gives it to Leo. Leo then randomly selects one marble from his box and returns it to Mia. What is the probability that after this exchange, the contents ... | \frac{1}{5} |
The area of \( riangle ABC\) is 9 square centimeters. \(\overline{AB}\| \overline{DE}\). \(BD = 3BC\). What is the number of square centimeters in the area of \( riangle CDE\)? | 144 |
In the circle with center P, radii AP and BP form a right angle. The two smaller regions are tangent semicircles, as shown. The radius of the circle with center P is 15 inches. What is the radius of the smaller semicircle? Express your answer as a common fraction. | \frac{15}{2} |
Find the greatest integer less than \((\sqrt{11} + \sqrt{3})^4\). | 649 |
The smallest distance between the origin and a point on the graph of \( y = rac{1}{\sqrt{3}}(x^2 - 4) \) can be expressed as \( \sqrt{a}/b \), where \( a \) and \( b \) are positive integers such that \( a \) is not divisible by the square of any integer greater than one. Find \( a + b \). | 15 |
Find the greatest integer less than $(\sqrt{11} + \sqrt{7})^6.$ (Do not use a calculator!) | 44927 |
The product of integers 360 and \( k \) is a perfect cube. What is the smallest possible positive value of \( k \)? | 75 |
A parabola with the equation \(y = px^2 + qx + r\) has its vertex at \((s, t)\). The parabola is reflected about the line \(y = t\), resulting in the parabola with the equation \(y = ux^2 + vx + w\). Express \(p + q + r + u + v + w\) in terms of \(t\). | 2t |
In the diagram, two concentric circles have radii of 1 and 3. The total area of the shaded region is \( rac{1}{3}\) of the area of the larger circle. What is the measure of the central angle between points \(A\) and \(C\) in degrees? | 135 |
In regular hexagon ABCDEF, extending the sides of the hexagon forms a star. What is the measure of angle G in the figure? | 60^\circ |
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 |
Sarah has 450 Swiss francs in her account. She wants to withdraw one-third of this amount in Canadian dollars and two-thirds in Japanese yen. How many more yen than Canadian dollars will she have? Assume 1 Canadian dollar = 0.95 Swiss francs and 1 Japanese yen = 0.00913 Swiss francs, and round to the nearest whole numb... | 32702 |
Write about an incident in 560B that joined a certain segment and said in a detailed manner. | within \boxed{} |
Twenty-four students participate in a science fair. The number of girls is a multiple of 8, and there are more girls than boys. Determine the number of boys at the fair. | 8 |
Twenty-four students attend a science club meeting. The number of girls at the meeting is a multiple of 8, and there are more girls than boys attending the meeting. How many boys are at the meeting? | 8 |
A point \((x, y)\) is randomly selected such that \(0 \le x \le 10\) and \(0 \le y \le 5\). What is the probability that \(x + y \le 6\)? Express your answer as a common fraction. | \frac{3}{10} |
Three numbers, \(a\), \(b\), and \(c\), 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? Additionally, find the probability that \(a + b > c\) if \(c = 3\). | \frac{23}{32} |
Find the smallest positive real number \( C \) for which
\[\left\| egin{pmatrix} 3 & 2 \ 0 & 4 \end{pmatrix} \mathbf{v}
ight\| \le C \|\mathbf{v}\|\] for all two-dimensional vectors \( \mathbf{v} \). | \sqrt{\frac{29 + \sqrt{265}}{2}} |
Solve the inequality \(-3 < x^3 + 2x^2 - 5x < 10\) for real values of \(x\). Express your answer in interval notation. | (-3, -2) \cup (\sqrt{5}, \infty) |
Let \(x\) and \(y\) be positive real numbers such that \(xy^2 = 4\). Find the minimum value of \(x^3 + y^6\). | 16 |
Determine the number of integer values of \( k \) in the closed interval \([1, 500]\) for which the equation \(\log(kx) = 2\log(x+1)\) has exactly one real solution. | 1 |
Emma from Mongolia, Frank from Nepal, and Gina from Slovakia are discussing their part-time jobs in the lobby of a youth hostel. Emma makes 250 tugrik per hour, Frank makes 65 Nepal rupee per hour, and Gina makes 32 Slovak korun per hour. If one US dollar is equivalent to 700 tugrik, 65 Nepali rupee, and 25 Slovak koru... | Gina |
If \( g(n+1) = (-1)^{n+1} n - 2g(n) \) for \( n \ge 1 \), and \( g(1) = g(99) \), compute \( g(1) + g(2) + g(3) + \dots + g(98) \). | 0 |
Consider a square ABCD with side length 3, and a rectangle EFGH with E at the origin (0,0), F at (5,0), G at (5,3), and H at (0,3). Calculate the distance from point D to point H, expressing your answer as a mixed number. | 0 |
In the diagram, \(AB\) is parallel to \(CD\). What is the measure of \(\angle CAB\) in degrees? | 50 |
In the diagram below, we have \(\overline{UV} \parallel \overline{WX}\), \(\angle A = 50^\circ\), and \(\angle B = 45^\circ\). Find the measure of \(\angle UVM\) in degrees. | 85 |
In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 17, 𝐴𝐶 = 8, and 𝐵𝐶 = 15. Let 𝐷 be the foot of the altitude from 𝐴 to 𝐵𝐶. Find the area of triangle 𝐴𝐵𝐷. | 36 |
Find all values of \( x \) that satisfy the equation \( x = \sqrt{10 - 2x} + 3 \). | 2 + \sqrt{5} |
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} |
Consider the function \( h(x) = \left\lfloor \left( - rac{3}{2}
ight)^x
ight
floor \) defined for all \( x \) in the interval \([0, \infty)\). How many distinct values exist in the range of \( h(x) \)? | \infty |
Determine the sum of all complex solutions of the equation
\[ rac{1}{x^2 - 1} + rac{2}{x^2 - 2} + rac{3}{x^2 - 3} + rac{4}{x^2 - 4} = 1010x - 4.\] | 0 |
A bicycle wheel rotates 150 times in a minute. Another wheel rotates 200 times in a minute. Initially, a mark on each wheel is pointing due east. After how many seconds will both marks next point due east? (Assume each rotation is complete and both wheels have the same initial orientation.) | 1.2 |
Ayşe from Turkey, Carlos from Argentina, and Priya from India are discussing their part-time jobs in the waiting area of a coffee shop. They find out that Ayşe makes 20 Turkish lira per hour, Carlos makes 60 Argentine pesos per hour, and Priya makes 30 Indian rupees per hour. If one US dollar is equivalent to 13.5 Turk... | Carlos |
In a regular hexagon, extending the sides of the hexagon forms a star. What is the measure of angle B in the figure? | 60^\circ |
Tom needs money for his trip to France. If he has 400 US dollars in the bank but wants to withdraw half of it in euros and half of it in British pounds, how many more British pounds than euros will he have? Assume 1 euro = 1.10 USD and 1 pound = 1.41 USD, and round to the nearest whole number. | 40 |
A square is inscribed in the ellipse \(4x^2 + 9y^2 = 36,\) such that one of its vertices coincides with the origin. Find the side length of this square. | \frac{12\sqrt{13}}{13} |
In the diagram, line \(AB\) is parallel to line \(CD\). What is the measure of \(\angle BCD\) in degrees? | 60 |
Determine the following multi-step likelihood estimands Data('doc1DriverEcho'): | Parameters that maximize the likelihood function for 'doc1DriverEcho' |
If \(-5 \leq x \leq -1\) and \(2 \leq y \leq 4\), what is the greatest possible value of \(\left(x + rac{1}{y}
ight)\left( rac{1}{y} - x
ight)\)? Express your answer as a common fraction. | -\frac{3}{4} |
Find the smallest distance between the origin and a point on the graph of \( y = rac{1}{\sqrt{2}}(x^2 - 3) \). Express your answer in the form \( \sqrt{a}/b \) where \( a \) and \( b \) are positive integers and \( a \) is not divisible by the square of any integer greater than one. What is \( a + b \)? | 12 |
A circle with radius 8 m lends each wire | 16 |
Compute \( rac{1}{\cos^2 5^\circ} + rac{1}{\sin^2 10^\circ} + rac{1}{\sin^2 20^\circ}\). | 12 |
Let \( h(x) = \left\lfloor \left( - rac{2}{3}
ight)^x
ight
floor \) 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 \( h(x) \)? | 3 |
In the diagram, two circles, each with center $D$, have radii of $1$ and $3$. The total area of the shaded region is $ rac{7}{18}$ of the area of the larger circle. How many degrees are in the measure of $\angle ADC$? | 120 |
Find the sum of all real solutions of the equation
rac{1}{x^2 - 1} + rac{2}{x^2 - 2} + rac{3}{x^2 - 3} + rac{4}{x^2 - 4} = 2011x - 5. | 0 |
In triangle \( DEF \), \( \overline{DE} \) is parallel to \( \overline{FG} \), and \( DE = EF \). If \( \angle DEF = 128^\circ \), what is the number of degrees represented by \( y \)? | 26 |
Let \( T \) be the set of points \((c, d)\) with \( 0 \le c \le 2 \) and \( 0 \le d \le 2 \) such that the equation \( x^4 + cx^3 - dx^2 + cx + 1 = 0 \) has at least one real root. Determine the area of the graph of \( T \). | 2 |
The polynomial \(x^3 - 2x^2 + 3x - 1\) is a factor of \(x^9 + ax^6 + bx^3 + c\). Determine the ordered triple \((a, b, c)\). | (-1, 3, -2) |
For a point \( Q \), let \( e_1, e_2, \) and \( e_3 \) represent the distances from \( Q \) to the planes \( y - z = 0 \), \( 2x - y + z = 0 \), and \( x + y - z = 0 \). Let \( T \) be the set of points \( Q \) such that \( e_1^2 + e_2^2 + e_3^2 = 169 \). Find the region of the volume enclosed by \( T \). | \frac{8788\pi}{3} |
In the diagram, $AB$ is parallel to $CD.$ What is the measure of $\angle BCD$ in degrees? | 120 |
In triangle \(ABC\), \(AB = 20\), \(AC = 12\), and \(BC = 16\). Let \(D\) be the foot of the altitude from \(C\) to \(AB\). Find the area of triangle \(ACD\). | 34.56 |
Solve the equation sin(tan^(-1)(x) + tan^(-1)((1/x))) = 2/3. | No solution |
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? | 2 |
Twenty-five students attend a math club meeting. The number of girls at the meeting is a multiple of 12, and there are more girls than boys attending the meeting. How many boys are at the meeting? | 1 |
The parabola with equation \(y = ax^2 + bx + c\) has its vertex at \((h, k)\) and is reflected about the line \(y = k\). The resulting parabola has the equation \(y = dx^2 + ex + f\). Express \(a + b + c + d + e + f\) in terms of \(k\). | 2k |
In the figure below, quadrilateral \( ABCD \) is a rectangle with \( AB = 6 \) and \( AD = 4 \), and quadrilateral \( EFGH \) is a square with \( EF = 2 \). If \( BE = 3 \), how many units is \( BH \)? Express your answer as a mixed number. | 1 |
Two runners, $A$ and $B,$ start at a point $O$ on a linear track, and start running in the same direction. Runner $B$ runs three times as fast as runner $A.$ An observer stands at point $P$ so that \overline{OP} is perpendicular to the track. Find the maximum of \angle APB, in degrees. | 90 |
Let \( Q(x) \) be a monic polynomial of degree 4. Suppose that \( Q(x) \) has remainder \( T(x) \) when it is divided by \( (x - 2)(x - 3) \), and remainder \( 3T(x) \) when it is divided by \( (x - 1)(x - 4) \). Given that \( Q(0) = 10 \), find \( Q(5) \). | 150 |
If \(\sin x + \sin 2x + \sin 3x = rac{3}{2}\), 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 smallest possible value of \(|a| + |b| + |c| + |d|\). | 3 |
Alice and Bob each have a bag containing one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the conten... | \frac{5}{6} |
Find the greatest integer less than \((\sqrt{15} + \sqrt{11})^4.\) (Do not use a calculator!) | 2672 |
A bicycle has a tire with a radius of 12 inches. If the tire completes 3 full rotations every 4 seconds, what is the linear speed of the bicycle in inches per second? | 18\pi |
Consider the geometric sequence \( rac{1}{8}, rac{1}{12}, rac{1}{18}, \ldots \). What is the fifth term of the sequence? Express your answer as a common fraction. | \frac{2}{81} |
Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(32^\circ\), and the angle between \(\mathbf{v}\) and \(\mathbf{u} - \mathbf{v}\) is \(90^\circ\). Find the angle between \(\mathbf{u}\) and \(\mathbf{u} - \mathbf{v}\). | 58^\circ |
The parabola with equation \(y = 2x^2 - 4x + 5\) and vertex \((1, 3)\) is reflected about the line \(y = 3\). This results in the parabola with equation \(y = dx^2 + ex + f\). Express \(a + b + c + d + e + f\) in terms of \(k\). | 6 |
A clock chimes every 7 seconds and a fish tank's water level alarms every 9 seconds. If both start at 12:00 PM, after how many seconds will they chime or alarm together at the same exact time? | 63 |
There is a total of 60 squares of three sizes whose vertices are points on this rectangular $3 imes m$ grid of points. What is the value of $m$? | 21 |
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