question stringlengths 20 2.99k |
|---|
A storybook costs 18 yuan. The class wants to buy 12 storybooks. Is 200 yuan enough to buy them? |
How many ways are there to remove $2$ diagonally non-adjacent vertices from an $n$-gon? |
9. Find the equations of the $n$ lines passing through the origin and perpendicular to the $n$ lines represented by $a_{0} x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\cdots+a_{n} y^{n}=0$. |
Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be three vectors such that the angle between any two of them is $\frac{\pi}{4}$. If the volume of the parallelepiped generated by $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ is $\frac{1}{\sqrt{2}}$, find the volume of the parallelepiped generated by $\mathbf{u} + \mathb... |
Determine whether the following statement is true or false: If a function $g$ is discontinuous, then the sequence $(g_n)_n$, where $g_n(x) = g(x + 1/n)$, does not converge pointwise to $g$ on $\mathbb{R}$. |
Example 2. Find the logarithmic residue of the function
$$
f(z)=\frac{\operatorname{ch} z}{e^{i z}-1}
$$
with respect to the contour $C:|z|=8$. |
6. Find all values of the parameter $a$, for each of which there exists a number $b$ such that the system
$$
\left\{\begin{array}{l}
x=|y+a|+\frac{4}{a} \\
x^{2}+y^{2}+24+b(2 y+b)=10 x
\end{array}\right.
$$
has at least one solution $(x ; y)$. |
If \( r_1, r_2, r_3 \) are the roots of the equation \( x^3 + 4x^2 + 6x - 22 \), find the value of \[ \frac{1}{r_1^2 + r_2^2} + \frac{1}{r_1^2 + r_3^2} + \frac{1}{r_2^2 + r_3^2}. \] |
The value of the integral\n\n\[\n2 \int_{-\infty}^{\infty} \left(\frac{\sin 2\pi t}{\pi t}\right) \, dt\n\] \n\nis equal to\n\n(a) 0 \n(b) 0.5 \n(c) 1 \n(d) 2 \n\n[EE GATE 2016] |
Verify that the function
\[ u(x, y, t) = 2 \sin\left(\frac{x}{3}\right) \sin\left(\frac{y}{4}\right) e^{-25t/16} \]
is a solution to the heat equation
\[ u_t = 9(u_{xx} + u_{yy}). \] |
Let \( G \) be a simply connected plane domain with smooth boundary which contains \( z=0 \). Let \( h = \frac{x}{r^2} + h_1 \) where \( h_1 \) is a regular harmonic function in \( G \). Let \( f \) be the analytic function whose real part is \( h \). What is the form of \( f \) near \( z=0 \)? |
What is the range of the function \( f(x) = \frac{1}{(x-1)^2} + 1 \)? |
Let $\Phi$ and $\Psi$ denote the Euler totient and Dedekind‘s totient respectively. Determine all $n$ such that $\Phi(n)$ divides $n +\Psi (n)$ . |
Solve the system\n\n\[\n\mathbf{X}' = \begin{pmatrix} -3 & 1 \\ 2 & -4 \end{pmatrix} \mathbf{X} + \begin{pmatrix} 3t \\ e^{-t} \end{pmatrix}\n\]\n\non \((-\infty, \infty)\). |
The perimeter of a regular hexagon and the perimeter of a rectangle are equal. The length and width of the rectangle are in the ratio 3:4. If the length of one side of the hexagon is 5 cm, find the area of the rectangle in square centimeters. |
Let \(a\), \(b\), and \(c\) be positive real numbers. Calculate the minimum value of the expression
\[
\sum \frac{a(2a + b)}{(a+b)^2} + \frac{4abc}{(a+b)(b+c)(c+a)}.
\] |
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i] |
If three-fourths of a number is equal to three-sixths of another number, what percent of the larger number is the smaller number? |
Let \(ABC\) be a triangle with \(I\) as its incenter, and the incircle touching \(BC\) at \(P\). Let \(\omega\) be a circle touching the incircle at \(K\) and passing through \(B\) and \(C\). Let \(AK\) meet \(\omega\) again at \(J\) (where \(K \neq J\)). If the circumradius of \(KIPJ\) is \(R\), and the radius of the ... |
Natural numbers \(a, b, c,\) and \(d\) satisfy \(ab = cd\). Can the number \(a + b + c + d\) be prime? |
(E. van Douwen) For which \(\lambda\) are there compact homogeneous spaces of cellularity \(\lambda\)? \(\lambda = \aleph_0\) trivially is possible and \(\lambda = \aleph_1\) is possible if \(\diamondsuit\). |
60 is at most 88% higher than some number. What's the smallest integer that is greater than 2-fifths of the number that is at most 88% higher than 60? |
Find the radius of convergence of the series \( \sum_{k=1}^{\infty}\frac{(-5)^k}{(k+1)!}\left(\frac x2\right)^{2k-1} \). |
Given a smooth function \( h \), its gradient in Cartesian coordinates is expressed as \( \nabla h = \frac{\partial h}{\partial x} \frac{\partial}{\partial x} + \frac{\partial h}{\partial y} \frac{\partial}{\partial y} \). Express the gradient \( \nabla h \) with respect to the polar basis vectors \( \frac{\partial}{\p... |
A curve is described parametrically by
\[(x,y) = (t^2 - 2t, t^3 - 3t^2).\]
The graph of the curve can be expressed in the form
\[y = f(x).\]
Enter the function $f(x)$. |
Determine the radius of convergence of the power series \( \sum_{n=1}^{\infty} \frac{n^{n!}}{(n^n)!} x^n \). Use the hint: \( \forall n \in \mathbb{N}, 3^n (n!) > n^n \). |
Function $f$ is defined such that $f\begin{pmatrix} a \\ b \end{pmatrix}= \begin{pmatrix} -b \\ -a \end{pmatrix}.$ Find the matrix $M$ such that $f(M \mathbf{v}) = \mathbf{v}$ for all vector $\mathbf{v}.$ |
Find the tensor product \( M \otimes_A N \) of the left \( A \)-module \( M = \mathbf{k} \oplus \mathbf{k}^2 \) and the right \( A \)-module \( N = \mathbf{k} \oplus \mathbf{k} \), where the \( A \)-action is given by:
\[
\begin{bmatrix}\lambda&0\\(u_1,u_2)&\mu\end{bmatrix}\begin{pmatrix}m_1\\(m_2,m_3)\end{pmatrix}=\be... |
In a company, there are $n$ married couples together, and it turns out that the people present form $n$ sibling pairs. In how many ways is this possible for the cases where $n \leq 5$? (Siblings, of course, cannot be married to each other.) |
Let \(a\) and \(b\) be such that \(a^2 + b^2 = 1\). Find the minimum and maximum values of the expression \(S = 2a^2 + ab + b^2\). |
Find the maximal value of \( x \in \mathbb{Z} \) such that \( 4^{100} + 4^{27} + 4^x \) is a perfect square. |
Let $A_n$ be the $n$-th term of the sequence defined recursively by $A_1 = 1, A_2 = \frac{1}{2}, A_n = \frac{A_{n-1} + A_{n-2}}{2}$. Let $B_n$ be the $n$-th term of the sequence defined recursively by $B_1 = 1, B_2 = \frac{1}{3}, B_n = \frac{B_{n-1} + B_{n-2}}{3}$. Determine $\lim_{n \to \infty} \frac{A_n}{B_n}$. |
Find all integers \(a, b, c\) such that for any real numbers \(u, v\) with \(0 \leq u < v \leq 1\), there exists a positive integer \(n\) making \(a n^2 + b n + c \geq 0\),
and \(\{\sqrt{a n^2 + b n + c}\} \in (u, v)\).
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or ma... |
Solve the inequality $|x-1|+|x-2|>5$. |
Calculate the expected number of coin tosses required to obtain the sequence H-T-H-T-T-H with a fair coin. |
3. The real numbers $x_{1}, x_{2}$, and $x_{3}$ are the three roots of the equation $2 x^{3}-(p-4) x^{2}-(2 p-1) x-p+8=0$, considering their possible multiplicities. Find the smallest value of the expression $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ under these conditions. For which $p$ is it realized? |
Let \( f \) be holomorphic on a region containing the closed disc \( \bar{D}(P, r) \) and suppose \( |f| \leq M \) on \( \bar{D}(P, r) \). Using the Cauchy estimates, what is the bound on \( \left|\frac{\partial^{k}}{\partial z^{k}} f(P)\right| \)? |
Solve the quadratic equation $2x^2 - 5x + 18 = 3x + 55$ and find the positive difference between the solutions. |
In how many ways can the letters in the word PHILLIP be arranged so that neither the string PHI nor the string ILL appears? |
Find the real numbers $\gamma$ for which the inequality
$$\sum_{1\leq i<j<k\leq 5}{\frac {a_ia_ja_k}{(s-a_i)(s-a_j)(s-a_k)}}-\frac{80}{27}\geq\gamma\left(\sum_{i=1}^{5}{\frac {a_i}{s-a_i}}-\frac{10}3\right)$$
is always true, where $a_1, a_2, \ldots, a_5$ are the side lengths of a non-degenerate pentagon with semi-perim... |
Establish the validity of the Schwarz–Christoffel transformation, showing that the mapping function obtained from
\[
\frac{dw}{dz} = A(z - x_1)^{\alpha_1/\pi - 1}(z - x_2)^{\alpha_2/\pi - 1} \cdots (z - x_n)^{\alpha_n/\pi - 1}
\]
maps the real axis of the \( z \)-plane onto a given polygon in the \( w \)-plane. |
In triangle \(ABC\), the median \(AM\) is drawn. Circle \(\omega\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), point \(F\) is chosen such that \(\angle BFE = 72^{\circ}\). ... |
Find the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[ (x+y)f(x+y) = f(y)(2x+y) - xf(-x) \] |
On average, the bottle-nosed dolphin comes up for air once every 3 minutes; the beluga whale, a close relative, comes up for air on average once every 6 minutes. The number of times a bottle-nosed dolphin would come up for air in a 24-hour period is approximately what percent greater than the number of times a beluga w... |
16. $[\mathbf{9}]$ Solve for $x$ :
$$
x\lfloor x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor\rfloor=122 \text {. }
$$ |
Find the smallest positive integer \( n \) such that \( 3^{2n} - 1 \) is divisible by \( 2^{2010} \). |
by selling 15 pencils for a rupee a man loses 60 % . how many for a rupee should he sell in order to gain 60 % ? |
To have a nontrivial nullspace, the determinant must equal zero.\n\n0 = \begin{vmatrix} -1 & x \\ -x & 4 \end{vmatrix}\n\n0 = -4 + x^2\n\nx^2 = 4\n\nx = \pm 2 |
Solve the following system of equations:
$$
\begin{aligned}
& x+y=a \ldots \\
& \frac{1}{x}+\frac{1}{y}=\frac{1}{b} \ldots
\end{aligned}
$$
Determine for what values of $a$ and $b$ the roots are real. What relationship must hold between $a$ and $b$ for one of the roots to be three times the other? |
11. As shown in the figure, in the Cartesian coordinate system $x O y$, $F$ is a moving point on the positive half-axis of the $x$-axis. A parabola $C$ is constructed with $F$ as the focus and $O$ as the vertex. Let $P$ be a point on $C$ in the first quadrant, and $Q$ be a point on the negative half-axis of the $x$-axi... |
Consider the group $\mathrm{SL}_{n}(\mathbb{Z})$. Let $\phi$ be an injective endomorphism of $\mathrm{SL}_{n}(\mathbb{Z})$. Define $f(n)$ to be the smallest positive integer such that $\phi$ restricted to the subgroup of matrices in $\mathrm{SL}_{n}(\mathbb{Z})$ with entries in $f(n)\mathbb{Z}$ is surjective onto the s... |
Determine all monic polynomials with integer coefficients \( f \) that satisfy the following condition: there exists a strictly positive integer \( N \) such that for every prime number \( p \) for which \( f(p) \) is a strictly positive integer, \( p \) divides \( 2(f(p)!)+1 \).
Return your final response as 'Final A... |
Given $3^{2k}$ indistinguishable coins, one of which is counterfeit and slightly lighter, and three two-pan balances where one is faulty, devise a strategy to identify the counterfeit coin using at most $3k + 1$ weighings. The faulty scale provides random readings, and the accurate scales are unknown. Describe your met... |
Calculate \( \lim_{x \to 1^{-}} \frac{\arccos{x}}{\sqrt{1-x}} \) without using L'Hôpital's rule. |
Solve the recurrence relation given by
\[
\begin{cases}
a_1 = 1 \\
a_{n} = a_{n-1} + 1 + \sum_{k=1}^{n-1} a_k, \quad n > 1
\end{cases}
\] |
Return your final response within \boxed{}. Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$ , we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$ . |
A total of \( n \) balls are sequentially and randomly chosen, without replacement, from an urn containing \( r \) red and \( b \) blue balls (\( n \leq r + b \)). Given that exactly \( k \) of the \( n \) balls are blue, what is the conditional probability that the first ball chosen is blue? |
Evaluate the integral \(\int_{0}^{+\infty}\arctan x\ln\left(1+\frac{1}{x^2}\right)\,dx\) and express your answer in terms of a well-known mathematical constant. What is the numerical value of this constant? |
9 years ago i was 5 times as old as my eldest son . today i am 3 times his age . how old am i now ? |
As $x$ ranges over all real numbers, find the range of
\[f(x) = \cos^4 x + \sin^2 x.\]Enter your answer using interval notation. |
Find the largest real $\alpha$ such that any reduced polynomial of degree 6 with integer coefficients that satisfies the inequality $f(x) > 0$ for all real $x$ also satisfies the inequality $f(x) \ge \alpha$ for all real $x$. |
1. Let the circumcenter of $\triangle A B C$ be $O$, and
$$
3 \overrightarrow{O A}+4 \overrightarrow{O B}+5 \overrightarrow{O C}=\mathbf{0} \text {. }
$$
Then the size of $\angle C$ is $\qquad$ |
Given the set of positive integers \(\mathbf{N}^{*}\) and a function \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) such that for any positive integers \( s \) and \( t \), the following holds:
\[
f\left(t^{2} f(s)\right) = s(f(t))^{2},
\]
determine the minimum possible value of \( f(1998) \) among all such functi... |
The \(23^\text{rd}\) day of a particular month is a Wednesday. On what day of the week was the first day of that month? |
Given that one focus of the ellipse $x^2 + ky^2 = 3k$ (where $k>0$) coincides with the focus of the parabola $y^2 = 12x$, find the eccentricity of the ellipse. |
Convert the point \( (\rho,\theta,\phi) = \left( 5, \frac{\pi}{4}, \frac{\pi}{3} \right) \) in spherical coordinates to rectangular coordinates. |
Let $C$ be a regular curve on the unit sphere, and let $S$ be the surface obtained by taking all lines through the origin passing through points on $C$. Using a convenient parameterization, calculate the coefficient $E$ of the first fundamental form of $S$ at a point where the curve $C$ has a curvature of $\kappa$. Exp... |
3. Let $p>3$ be a prime number. For an arbitrary set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{Z}$, let
$$
S_{a}=\left\{x \in\{0,1, \ldots, p-1\} \mid(\exists s \in S) \quad x \equiv_{p} a \cdot s\right\}
$$
(a) How many sets $S \subseteq\{1,2, \ldots, p-1\}$ are there such that the sequence $S_{1}, S_{2}, \ldots, ... |
Determine the value of $a$ such that the expression $a^3 + a^3 + a^3$ is equal to the square root of $a^2 + a^2 + a^2$. |
Given a coframe \(\theta^i\) with connection \(\omega^i_j\) in \(\mathbb{R}^n\), the curvature form is defined as \(\Omega^i_j = d\omega^i_j + \omega^i_k \wedge \omega^k_j\). What is the transformation law for the curvature form \(\Omega\) under a change of basis \(\bar{e} = eB\)? |
Sketch \( f(x) = \frac{x^2 + 1}{x + 1} \). Find all points where \( f'(x) = 0 \) and describe the behavior when \( x \) or \( f(x) \) is large. |
Let \( a > 0 \). Three straight lines pass through the points \((0, -a, a)\), \((a, 0, -a)\), and \((-a, a, 0)\), parallel to the \(x\)-, \(y\)-, and \(z\)-axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described... |
Simplify the expression $\frac{3(x+2y)+x}{x}$. |
(Iran 2011) Let \( k \geq 7 \) be an integer. Find the number of pairs \( (x, y) \) such that \( 0 \leq x, y < 2^k \) and\n\n\[\n73^{73^x} \equiv 9^y \pmod{2^k}.\n\] |
In a $4n \times 4n$ (where $n \in \mathbf{N}_{+}$) table of real numbers, each number's absolute value does not exceed 1, and the sum of all numbers is 0. It is known that the absolute value of the sum of the numbers in each row and the absolute value of the sum of the numbers in each column is at least $c$. Find the m... |
Consider the inequality \(\max_{0 \leq x \leq 2} \prod_{j=1}^n |x - a_j| \leq C^n \max_{0 \leq x \leq 1} \prod_{j=1}^n |x - a_j|\) for any positive integer \( n \) and any real numbers \( a_1, a_2, \ldots, a_n \). Find the smallest possible value of \( C \) that satisfies this inequality for all \( n \) and all \( a_1,... |
Calculate the Lebesgue measure integral \( \int_A y^{-1/2}\sin x \;dm^2 \) over the set \( A=\{(x,y)\in[0,1]^2:x+y\notin \mathbb Q \text{ and } xy\notin \mathbb Q\} \) in \( \mathbb{R}^2 \). |
Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$. |
Find all conformal mappings from the set $\{a_1 < |z| < a_2\}$ to the set $\{b_1 < |z| < b_2\}$ where $a_1, a_2, b_1, b_2$ are positive. |
Consider the equation \( \lambda = e^{-z} + z \) where \( \lambda > 1 \) and \( z \) is a complex number with \( \text{Re}(z) > 0 \). Let \( z_0 \) be the unique solution in this set. Calculate the value of \( \text{Re}(z_0) + \text{Im}(z_0) \) when \( \lambda = 2 \). |
Given the points \( M = \left(\frac{a}{2}, \frac{b}{2}\right) \) and \( N = \left(\frac{c}{2}, \frac{d}{2}\right) \), express the vector \( \mathbf{MN} \) in terms of the vector \( \mathbf{BC} \), where \( B = (a, b) \) and \( C = (c, d) \). |
Return your final response within \boxed{}. Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotie... |
Bucket A, Bucket B, and Bucket C each contain some oil. First, $\frac{1}{3}$ of the oil in Bucket A is poured into Bucket B. Then, $\frac{1}{5}$ of the oil in Bucket B is poured into Bucket C. At this point, the amount of oil in each of the three buckets is the same. If Bucket C originally had 48 kilograms of oil, dete... |
If the inequality $|x+3| - |x+1| \leq 3a - a^2$ holds true for all real numbers $x$, then the range of values for the real number $a$ is ____. |
Find the image of the Rokovsky function \(\varphi(z) = \frac{1}{2}\left(z + \frac{1}{z}\right)\) on the set \(\{z \in \mathbb{C} : |z| > 1\}\). |
(USA 2) The incenter of a triangle is the midpoint of the line segment of length 4 joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle. |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{5 / 2} \frac{x^{2} \cdot d x}{\sqrt{25-x^{2}}}
$$ |
Lisa is making a rectangular cardboard box with square bases for a school project. She decides that the height of the box should be 5 units greater than the side of the square bases. Determine the minimum height of the box if she requires the surface area to be at least 150 square units. |
Consider a multistage search allocation game where the searcher and the target play a zero-sum game. At stage \( n \), the searcher distributes searching resource \( \varphi_i \) in cell \( i \) with cost \( c_i \), and the target moves to cell \( i \) with probability \( p_i \). The nondetection probability in cell \(... |
Find all polynomials \( p(x) \) with real coefficients such that \( p(x^2) = p(x) p(x+1) \). |
Given a sequence $\{a_n\}$ satisfies $a_1=3$ and $a_{n+1}=4a_n+3$ ($n\in\mathbb{N}^*$), then the general formula for the sequence $\{a_n\}$ is ______. |
The equation of a parabola is given by $y = -x^2 + ax + b$ and its solution to the inequality $y \le 0$ is $(-\infty, -1] \cup [7, \infty)$. Determine the vertex of this parabola. |
Derive the contravariant components of the acceleration vector \( \gamma^i \) for a moving point in a Riemannian space, where the point's trajectory is parameterized by a scalar parameter \( t \). |
Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x} \cdot f(x) = 2 + \int_{0}^{x} \sqrt{t^4+1} \, dt$ for $x \in (-1,1)$. If $f^{-1}$ is the inverse of $f$, find the value of $(f^{-1})'(2)$. \( \boxed{} \) |
Given \( m \leq p \log(p) + q \log(q) \leq M \) and \( p, q > 0 \) with \( p + q = 1 \), find the values of \( M \) and \( m \). |
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define $$f(A)=\sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.$$If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the... |
Solve the differential equation: x'' + (x^2 - 1)x' + x = 0 |
Using the Cauchy-Schwarz inequality, determine the minimum value of \( \int_0^1 f^2(x) \, \mathrm{d}x \) for a non-negative Riemann integrable function \( f(x) \) satisfying \( \int_0^1 f(x) \, \mathrm{d}x = 1 \) and \( \int_0^1 x f(x) \, \mathrm{d}x = \frac{1}{6} \). |
A settlement is built in the shape of a 3-block by 3-block square (each block is a square with side $b$, for a total of 9 blocks). What is the shortest path the paver needs to take to pave all the streets if he starts and ends his journey at the corner point $A$? (The sides of the square are also streets). |
Determine the cardinality of the set of all circles in the plane whose centers have rational coordinates and whose radii are the square roots of prime numbers. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.