problem stringlengths 19 983 | answer stringlengths 1 8.19k |
|---|---|
The area of triangle ABC is given by the expression \(a^2 - (b + c)^2\), where \(a\), \(b\), and \(c\) are the sides of triangle ABC. Calculate \( an A\). | \frac{8}{15} |
In triangle $PQR$, the medians $PS$, $QT$, and $RU$ intersect at the centroid $O$. The line through $O$ that is parallel to $QR$ intersects $PQ$ and $PR$ at $A$ and $B$, respectively. If the area of triangle $PQR$ is 100, find the area of triangle $OAB$. | \frac{100}{9} |
Two numbers, $a$ and $b$ are selected at random from the interval $(0, 5)$. What is the probability that a triangle with sides of length 2, $a$, and $b$ exists? | \frac{16}{25} |
Find all values of $y$ that satisfy the equation $y = \sqrt{13 - 2y} + 3$. | No solution |
In the circle with center P, radii AX and BX form a right angle. The two smaller regions are tangent semicircles, as shown. The radius of the circle with center P is 18 inches. What is the radius of the smaller semicircle? Express your answer as a common fraction. | 18(\sqrt{2} - 1) |
Let \(\mathbf{c}\) and \(\mathbf{d}\) be vectors such that the angle between \(\mathbf{c}\) and \(\mathbf{d}\) is \(30^\circ\), and the angle between \(\mathbf{d}\) and \(\mathbf{c} - \mathbf{d}\) is \(60^\circ\). Find the angle between \(\mathbf{c}\) and \(\mathbf{c} - \mathbf{d}\). | 60^\circ |
If \(2 \leq a \leq 4\) and \(2 \leq b \leq 3\), what is the greatest possible value of \((a + rac{1}{b})( rac{1}{b} - a)\)? Express your answer as a common fraction. | -\frac{15}{4} |
Three cyclists, A, B, and C, start from a point X on a circular track. Cyclist B rides twice as fast as cyclist A, and cyclist C rides twice as fast as cyclist B. An observer stands at point Y such that \(\overline{XY}\) is perpendicular to the track. Determine the maximum value of \(\angle AYC\), in degrees. | 120 |
If the domain of the function \(\log(x^2 - 4)\) is \(x < a\) or \(x > b\), find \(a + b\). | 0 |
How many ways can 6 people sit around a round table if no two of the 3 people Alice, Bob, and Carl can sit next to each other? (Seating arrangements that are rotations of each other are considered the same.) | 84 |
In how many ways can 7 people sit around a round table if no two of the 3 people Alice, Bob, and Carol can sit next to each other? (Seating arrangements which are rotations of each other are treated as the same.) | 144 |
Find all solutions to \(\sin \left( an^{-1} (2) + \cot^{-1} \left( rac{1}{2}
ight)
ight) = rac{\sqrt{3}}{2}\). Enter all the solutions, separated by commas. | \frac{\pi}{3} |
Joanne from South Korea, Maria from Portugal, and Ravi from Thailand are discussing their part-time jobs in the conference hall. Joanne makes 180 won per hour, Maria makes 20 euros per hour, and Ravi makes 490 baht per hour. If one US dollar is equivalent to 1145 South Korean won, 0.84 Portuguese euros, and 36.56 Thai ... | Maria |
Consider the function \(\log |x|^2\). Determine the domain of this function and express it as the union of two intervals. If the domain is written as \(x < a\) or \(x > b\), find \(a + b\). | 0 |
Calculate the value of \( rac{2+3i}{1+i}\). | \frac{5}{2} + \frac{1}{2}i |
Twenty-seven students attend a math club meeting. The number of girls at the meeting is a multiple of 9, and there are more girls than boys attending the meeting. How many boys are at the meeting? | 9 |
Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx) = 2\log(x+2)$ has exactly one real solution. | 1 |
A regular 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 + \sqrt{2} |
Let \( \mu \) be a constant, \( 0 \le \mu \le 3 \), and let \( g : [0,1] o [0,1] \) be defined by \( g(x) = \mu x(1 - x) \). Find the values of \( \mu \), \( 0 \le \mu \le 3 \), for which there exists an \( x \in [0,1] \) such that \( g(x)
eq x \) but \( g(g(x)) = x \). | 2 |
Two runners, $X$ and $Y,$ start at a point $Q$ on a circular track with radius $r$. Runner $Y$ runs twice as fast as runner $X$. An observer stands at point $R$ so that $\overline{QR}$ is perpendicular to the track. Find the maximum of $\angle XRY$, in degrees. | 90 |
Simplify \( an 20^\circ + 4 \sin 20^\circ \). | \sqrt{3} |
Find the sum of all integers $N$ such that $ N^2 + N + 3 $ is either an integer or irrational. | 0 |
An equiangular hexagon has three sides of length 2 and three sides of length 1, arranged so that no two consecutive sides have the same length. What is the area of the hexagon? | \frac{3\sqrt{3}}{2} |
Find all values of \( x \) that satisfy the equation \( x = \sqrt{10 - 3x} + 3 \). | \frac{3 + \sqrt{13}}{2} |
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 \((5,1)\) and \((6,4)\). What is the greatest possible value of the slope of the line containing points \(C\) and \(D\)? Expre... | 2 |
Let \( x \) and \( y \) be positive real numbers such that \( xy^3 = 3 \). Find the minimum value of \( x^3 + y^6 \). | 6 |
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