question stringlengths 20 2.99k |
|---|
If the function $y=ax^2-2ax$ ($a\neq 0$) has a maximum value of 3 in the interval $[0,3]$, then the value of $a$ is. |
Annie has 100 barrettes, four times as many scrunchies as barrettes, and seventeen fewer bobby pins than three times the number of barrettes. What percentage of her hair decorations are bobby pins, rounded to the nearest percent? |
If the wheel has a diameter of 41 cm, how many full revolutions and remaining distance in centimeters will it take to cover a distance of 1367 cm? |
Find the largest integer value of \( n \) such that \( \frac{2019! \pi}{12^n} \) is an integer. |
What is the range of the function \( y = \log_2(\sqrt{\sin x}) \) for \( 0 \leq x \leq 180^\circ \)? |
Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and
\[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\ri... |
Let \((X, Y)\) have a uniform distribution on the set \((x, y) : 0 < x < 2\) and \(0 < y < 8\) and \(x < y\). Find the probability \(P(Y < X^3)\). |
Consider the surface defined by \(x^2 + y^2 + z^2 = 1\). For a point \((a, b, c)\) on this surface, let \(f(a, b, c)\) be the area of the tangent plane to the surface at \((a, b, c)\) intersected with the cube centered at the origin with side length 2. Calculate \(f\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}... |
7. \( 2y''' - 7y'' - 10y' - 7y = 0 \) |
Find an irreducible polynomial in $\mathbb{Z}[X]$ that is reducible over $\mathbb{Z}/p \mathbb{Z}$ for every prime number $p$. |
Given $\theta_1 + \theta_2 + \cdots + \theta_n = 2\pi$, $\theta_1, \theta_2, \ldots, \theta_n > 0$, and $\theta_i + \theta_{i+1} < \pi$ (with $\theta_{n+1} = \theta_1$), find the minimum value of
\[
\frac{\sin (\theta_1 + \theta_2) + \sin (\theta_2 + \theta_3) + \cdots + \sin (\theta_n + \theta_1)}{\sin \theta_1 + \si... |
The arithmetic mean of an even number of consecutive even integers is \(z\). Find the sum of the smallest and largest of the integers in terms of \(z\). |
Compute $\sqrt{3^4 \cdot 2^3}$.
This problem requires a different approach to solve as it involves multiplying two powers with different exponents. The solution would involve using the rule of exponents, which states that when multiplying two powers with the same base, you add the exponents. Therefore, the solution wo... |
Find the area enclosed by the curve defined by the equation \(x^2 + (y + \sqrt[3]{|x|})^2 = 1\). |
Compute \( \mathbb{E} \left( \int_0^1 \left| B_s \right| dB_s \right)^2 \) using Ito's formula, where \(B_s\) is a Brownian motion. |
A 40°F roast is placed in a 350°F oven. After 2 hours, the temperature of the roast is 125°F. Assuming the temperature of the roast follows Newton’s Law of Warming, find the formula for the temperature of the roast \( T \) as a function of its time in the oven, \( t \), in hours. |
Solve the system of equations: $-7x - 14y - 10z = -17$, $2x + 4y + 2z = 4$, and $2x + 4y - 7z = -6$. |
A building has an elevator. If there are 5 passengers on the ground floor, and the probability of each passenger getting off at the 10th floor is $\dfrac{1}{3}$, let $\xi$ represent the number of passengers getting off at the 10th floor. Then, the expected value of the random variable $\xi$, $E(\xi)=$ ______. |
What is the cardinality of the set of automorphisms of the complex field $\mathbf{C}$? |
In a regular hexagon, each side is 6 units long. Draw a circle tangent to all six sides of the hexagon. What is the radius of the circle in units? |
Find the value of the expression given $$\frac{\sqrt{3}\left(2\sqrt3-1\right)}{6}$$ |
In triangle \( ABC \), let \( I \) be the incenter, and \( B_1 \) and \( C_1 \) be the midpoints of sides \( AC \) and \( AB \), respectively. If \( \angle BIC_1 + \angle CIB_1 = 180^\circ \), calculate the value of \( \frac{AB + AC}{BC} \). |
Let \( S = \{1, 2, \cdots, 1990\} \). If the sum of all the numbers in a 31-element subset of \( S \) is a multiple of 5, it is called a "good subset". Find the number of good subsets of \( S \). |
Find the least possible value of \(a + b\), where \(a\) and \(b\) are positive integers, such that 11 divides \(a + 13b\) and 13 divides \(a + 11b\). |
Simpson's rule can be used to approximate \(\log 2\), since \(\log 2 = \int_{1}^{2} \frac{1}{x} \, dx\). The Simpson's rule approximation for \(\int_{a}^{b} f(x) \, dx\) with \(n\) subintervals is given by:
\[
S_n = \frac{\Delta x}{3} \left[ f(a) + 4f(a + \Delta x) + 2f(a + 2\Delta x) + 4f(a + 3\Delta x) + \cdots + f(b... |
What is the positive number $x$ that satisfies the equation $(\log_2 x)(\log_x 7) = \log_2 7$?
**A)** All positive $x$ \\
**B)** Only $x = 7$ and $x = 2$ \\
**C)** All positive $x \neq 1$ \\
**D)** Only $x = 7$ |
Solve the inequality
\[\frac{1}{x - 1} - \frac{4}{x - 2} + \frac{4}{x - 3} - \frac{1}{x - 4} < \frac{1}{30}.\]
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
The values of \(x\) and \(y\) are always positive, and \(x^3\) and \(y\) vary inversely. If \(y = 5\) when \(x = 2\), find \(x\) when \(y = 2000\). |
Find the antiderivative of \( \frac{1}{\sqrt{x}} \) with respect to \( x \), including the constant of integration. |
Solve the recurrence relation:
$$
T(n) = \begin{cases}1 & \text { if } n=1 \\ T(n / 2)+n & \text { otherwise. }\end{cases}
$$ |
A tank is one metre high and filled with watery mud that has settled to be denser at the bottom than at the top. At height \( h \) metres above the bottom of the tank, the cross-section of the tank has the shape of the finite region bounded by the two curves \( y = x^2 \) and \( y = 2 - h - 3x^2 \). At height \( h \) m... |
What is the smallest value of $x$ that satisfies the inequality \[|12-6x| < 18.\] |
Given that
\[g(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x>4, \\
x^2 &\text{ if }x \le 4,
\end{array}
\right.\]
determine the domain of the real function $F$ defined by $F(x)=g(g(g(g(x))))$. |
Given a sequence \\(\{a_n\}\) where \\(a_1=1\\) and \\(a_{n+1}=2a_n+4(n\in \mathbb{N}^*)\\), find the general formula for \\(a_n=\\) . |
The base of a pyramid is a regular pentagon with a perimeter of 50 cm. The pyramid's height is 24 cm. Find the volume of the pyramid. |
1. Calculate:
$$
\sqrt[3]{1134^{3}+1135^{3}+\cdots+2133^{3}}=
$$
$\qquad$ |
Find the standard form of the solution to the system of equations given below:
\begin{align*}x &= y - 8\\2x + y &= -7\end{align*} |
A man lost 4% by selling oranges at a certain rate per rupee. To gain 44%, he must sell them at the rate of 8 a rupee. At what rate per rupee was he initially selling the oranges? |
Into how many parts can $n$ planes divide space?
(Each group of three planes intersects at exactly one point, no group of four planes shares a common point.) |
The value of $\sin 600^\circ \cos 330^\circ + \cos 120^\circ \sin 150^\circ$ is:
(a) $-1$
(b) $1$
(c) $\frac{1}{\sqrt{2}}$
(d) $\frac{\sqrt{3}}{2}$ |
A right hexagonal prism has a regular hexagon base with side length of 2 cm. Its height is 6 cm. Calculate the prisim's surface area. |
Find all pairs of natural numbers $(m,n)$ such that $m^2-2mn-n^2$ and $m^2+2mn-n^2$ are both perfect squares. |
Express the MAX 2SAT problem as an integer quadratic program in which the only constraints are \( y_i \in \{-1,1\} \) and the objective function is quadratic in the \( y_i \). (Hint: Introduce a variable \( y_0 \) to indicate whether \(-1\) or \(1\) represents "TRUE.") |
Evaluate the surface integral \( \int_{S} \vec{F} \cdot d\vec{S} \) where \( \vec{F} = y\hat{i} + 2x\hat{j} - z\hat{k} \) and \( S \) is the surface of the plane \( 2x + y = 4 \) in the first octant, cut off by the plane \( z = 4 \). |
Combine like terms: 4x + 2 + 3x -10. |
A triangle with side lengths $x$, $x + 3$, and $2x + 1$ has the property that its perimeter and the sum of the squares of its sides are in a ratio of 1:3. What is $x$? |
if the function q is defined by the formula q = 7w / (6mx (z^3)), by what factor will q be multiplied if w is quadrupled, m is doubled, x is reduced to half, and z is raised to the power of 4? |
A particle moves along a straight line with velocity given by \( v(t) = 3t^2 - 12t + 8 \) meters per second. What is the total distance traveled by the particle between times \( t = 2 \) seconds and \( t = 4 \) seconds? |
Find an entire function \( f \neq \text{const} \) such that \( f(z) = 0 \) for all \( z \) on the unit circumference \( S^1 \). |
Find the sum of the series \( \sum_{1 \leq x < y < z}^\infty \frac{1}{3^x4^y5^z} \). |
As a Ford Mustang enthusiast, you've decided to customize your car's engine for optimal performance. You plan to install a new turbocharger that increases the engine's power output by 40%. However, due to a glitch in the current mod, the turbocharger only functions 75% of the time. If your Mustang's original power outp... |
Solve the linearized Klein-Gordon equation in one spatial dimension:
\[ u_{tt} - c^2 u_{xx} + m^2 u = 0. \] |
Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). |
1A. Solve the irrational equation
$$
\frac{\sqrt{x^{2}+x+6}+\sqrt{x^{2}-x-4}}{\sqrt{x^{2}+x+6}-\sqrt{x^{2}-x-4}}=5
$$ |
A triangular region is enclosed by the lines with equations \( y = 2x + 4 \), \( y = -x + 3 \), and \( y = 0 \). Calculate the area of this triangular region. |
Solve the system \(ay + bx = c; cx + az = b; bz + cy = a\). |
In triangle \(ABC\) with incenter \(I\), let \(A'\) be the incenter of triangle \(IBC\), \(B'\) be the incenter of triangle \(IAC\), and \(C'\) be the incenter of triangle \(IAB\). If \(O\) is the circumcenter of triangle \(ABC\), calculate the length of \(OO'\), where \(O'\) is the circumcenter of triangle \(A'B'C'\). |
The point corresponding to the complex number $z= \frac {2-i}{1+i}$ is located in the \_\_\_\_\_ quadrant of the complex plane. |
Find the categorical sequential limit of the sequence \(... \to \mathbb{Q} \to \mathbb{Q} \to \mathbb{Q}\) in the category of abelian groups, where each arrow is multiplication by a fixed natural number \( n > 1 \). |
On a plane, vectors $\mathbf{A}$ and $\mathbf{B}$ lie the same length but are perpendicular. Their difference has magnitude 16. Calculate the length of vector $\mathbf{A}$. |
Given \(\alpha \in (0, 1)\), construct a set \( A \) whose density at \( x_0 \in \mathbb{R} \) is equal to \(\alpha\). |
Solve the inequality: $$\frac{t-3}{t+6} \ge 0 \qquad (\text{Give the answer using interval notation})$$ |
Find the envelope of each of the following families of surfaces:
(a) \( \alpha(x - y) - \alpha^2 z = 1 \)
(b) \( (x - \alpha)^2 + y^2 = 2\alpha z \). |
4. (15 points) A tanker is being filled with oil at a rate of 2 barrels per half minute. Given that 1 barrel equals 159 liters, determine the filling rate of the tanker in m ${ }^{3} /$ hour. |
Determine the radius and center of the circle given by the equation \(x^2 + y^2 + 2x - 4y = 20\). |
To find \( x \) such that \(\sin(x - \alpha)\cos x\) is maximum \( \alpha \) being fixed. |
A cylindrical jar has a base circumference of 10 inches and a height of 8 inches. Two spiral strips are painted on the jar: one starts from the bottom and winds around the jar exactly once, reaching the top directly above its start point, while another does the same but starts from the top and winds downward to the bot... |
Compute the exact value of the expression $\left| 3\pi - | 3\pi - 11 | \right|$. Write your answer using only integers and $\pi$, without any absolute value signs. |
In acute $\triangle ABC$ with centroid \( G \), \( AB = 22 \) and \( AC = 19 \). Let \( E \) and \( F \) be the feet of the altitudes from \( B \) and \( C \) to \( AC \) and \( AB \) respectively. Let \( G' \) be the reflection of \( G \) over \( BC \). If \( E, F, G \), and \( G' \) lie on a circle, compute \( BC \).... |
What is the factored form of $x^4-16$? |
Let $x,y$ be real numbers with $x \geqslant \sqrt{2021}$ such that
\[ \sqrt[3]{x+\sqrt{2021}}+\sqrt[3]{x-\sqrt{2021}} = \sqrt[3]{y}\]
Determine the set of all possible values of $y/x$. |
Let triangle \(ABC\) with point \(M\) moving on segment \([BC]\). Points \(I\) and \(J\) are the incenters of triangles \(MAB\) and \(MAC\), respectively. \(PQ\) are the points of tangency of the common external tangents to the circles \(\odot(I)\) and \(\odot(J)\). Let \(K\) be the fixed point through which the circle... |
\[\n \int x^4 J_3(x) \, dx\n\] |
Return your final response within \boxed{}. Suppose that \( x, y \), and \( z \) are complex numbers of equal magnitude that satisfy
\[ x + y + z = -\frac{\sqrt{3}}{2} - i \sqrt{5} \]
and
\[ x y z = \sqrt{3} + i \sqrt{5}. \]
If \( x = x_{1} + i x_{2}, y = y_{1} + i y_{2} \), and \( z = z_{1} + i z_{2} \) for real \( x_... |
Solve the equation $[x^3] - 4x + 2 = 0$, where $[x]$ denotes the floor function of $x$. |
Let \( g:[0,1] \rightarrow [0,1] \) be a continuous function. Determine conditions on \( g \) which are equivalent to the property that \( \lim_{n \rightarrow \infty} ||g^n f||_2 = 0 \) for all \( f \in L^2[0,1] \). |
Show by means of the Fourier transform an explicit form of the Duhamel formula for wave equation in \(\mathbb{R}^n\),\n\n\[u_{tt} - \Delta u = f \quad (x, t) \in \mathbb{R}^n \times \mathbb{R}_+\]\n\nif \( u(x, 0) = u_t(x, 0) = 0 \). |
If all vertices of a cube with volume 8 are on the surface of a sphere, then the volume of this sphere is \_\_\_\_\_\_ . (Keep the result in terms of π) |
Find the general solution of the equation \(\frac{dy}{dt} + y \cos t = 0\). |
Suppose P is a point of intersection of the hyperbola C₁: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ and the circle C₂: $x^2 + y^2 = a^2 + b^2$, and ∠PF₂F₁ = 2∠PF₁F₂, where F₁ and F₂ are the two foci of the hyperbola C₁. Then, the eccentricity of the hyperbola C₁ is \_\_\_\_\_\_. |
10. (20 points) How many zeros does the number $4^{5^{6}}+6^{5^{4}}$ end with in its decimal representation?
# |
In an acute non-isosceles triangle \(ABC\) with altitudes \(AD\) and \(BE\), and angle bisector \(CF\) such that \(AE + BD = AB\), denote by \(I_A\), \(I_B\), and \(I_C\) the incenters of triangles \(AEF\), \(BDF\), and \(CDE\) respectively. If the radius of the circle passing through points \(D\), \(E\), \(F\), \(I_A\... |
Identify the finite abelian group of order 8 that has three elements of order two and four elements of order four, up to isomorphism. |
Find a closed form for the expression \(1 + \sin x + \sin 2x + \sin 3x + \cdots + \sin nx\). |
Determine the eigenvalues of the adjacency matrix of a bipartite graph $K_{r,s}$ with two sets of nodes containing $r$ and $s$ nodes, respectively. |
Determine whether the series \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \) is absolutely convergent, conditionally convergent, or divergent. |
Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and centroid $G$. If $2\sin A\overrightarrow{GA}+\sqrt{3}\sin B\overrightarrow{GB}+3\sin C\cdot \overrightarrow{GC}=\vec{0}$, then $\cos B=$_______.
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematica... |
Find the value of $a$ for which the area of the triangle whose vertices are the points $(0, 0)$, $(0, a)$, and $(a, a^2)$ is 5 square units. |
Consider a finite noncyclic group \( G \) in which every proper subgroup is abelian. Let \( N \) be a normal subgroup of \( G \) which is distinct from \( G \) and \( 1 \). If \( |G| = p^aq^b \) where \( p \) and \( q \) are distinct primes and \( a, b \) are positive integers, find the number of possible orders for \(... |
What is the sum of all ordinals up to \(\omega\), where \(\omega\) is the first transfinite ordinal? |
The solution to the equation $$\sqrt {x^{2}+6x+10}+ \sqrt {x^{2}-6x+10}=8$$ is \_\_\_\_\_\_. |
Three balls are placed inside a hemispherical tent. The first two balls have a radius of $1$, touch each other, the roof, and the diameter of the tent base. The third ball of radius $r$ touches the other two, the roof, and the base of the tent. Find the radius $r$ of the third ball. |
In a curved space, a vector \( A^\mu \) is parallel-transported along a segment of a closed path \( C \). Express the line integral \( I \) for this segment in terms of the affine connection coefficients \( \Gamma^\mu_{\nu\rho} \), the vector components \( A^\rho \), and the coordinate differentials \( dx^\nu \). |
Given the original letter arrangement "abcd", how many different rearrangements are there such that no two originally adjacent letters are adjacent in the new arrangement? Express your answer as a whole number. |
Let $b_1, b_2, \dots$ be a sequence for which $b_1=1$, $b_2=2$, and $b_n = b_{n-1} + b_{n-2}$ if $n$ is odd, while $b_n = b_{n-1} - b_{n-2}$ if $n$ is even. Determine the period of this sequence. |
Find all real positive numbers $a$ such that the series $\sum_{n=1}^{+ \infty} \sin(n^a)$ converges. |
Why is the series of partition numbers \( p(n) \) not C-finite? |
Let $G$ be a group with $n$ elements. Fix an element $x \in G$. Calculate the probability that the product of three randomly chosen elements from $G$ equals $x$. Express your answer as a fraction or a decimal. |
Find \( F^+(z) \), if \( C \) is a closed contour, the points \( a \) and \( b \) lie within it and \( \phi(\zeta) \) is a single-valued branch of a many-valued function defined outside the cut which connects the points \( a \) and \( b \) and lies within the contour \( C \).\n\n\( \phi(\zeta) = \sqrt{[(\zeta-a)/(\zeta... |
Let $a$, $b$, $c$ be positive real numbers such that $a+b+c=33$ and $\sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c} = 42$. Find $a^2+b^2+c^2$. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.