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Solve the equation in $\mathbb{R}$:
\[ \sqrt{1 - x} = 2x^2 - 1 + 2x\sqrt{1 - x^2} \] |
If you form a word of length 4 by randomly selecting letters from the multiset {t: 2, r: 2, s: 1, j: 2, g: 2}, what is the expected number of times the pattern 't.*s' appears? Round to 2 decimal places.
Present the answer in LaTex format: \boxed{Your answer} |
A bucket containing a certain amount of water is hung from a spring scale, which reads 35 kg. When 7 kg of water are poured out, the scale reads 27 kg. What is the weight of the empty bucket? |
A deck of 52 cards is shuffled and two cards are drawn at random. What is the probability that the product of the ranks of the two cards is a power of 2? Express your answer as a common fraction. |
C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let Bᵅ be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest ... |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given that $a=1$, $b=2$, and $\cos B= \frac {1}{4}$, find the value of $c=$\_\_\_\_\_\_; and the area of $\triangle ABC$, $S=$\_\_\_\_\_\_. |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length 2 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $30^{\circ}$, and it is kno... |
What is the period of $y = \cos (x + \pi)$? |
N5. Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number
$$
\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}
$$
is an integer. |
8. \( a \frac{\partial u}{\partial t} + b \frac{\partial u}{\partial x} = u \), \( a, b \neq 0 \). |
Provide an example of a linear algebraic group over the field of complex numbers that has exactly two rational points. |
State the boundary-value problem for the longitudinal vibrations of a flexible rod 0 ≤ x ≤ l of variable cross-section S(x), if the ends of the rod are rigidly fixed, the volume density equals ρ(x), the modulus of elasticity equals E(x), and the vibrations are produced by the initial longitudinal displacements and velo... |
Evaluate: \( \int \frac{\sec 3x}{\sec x} \, dx \) |
There are two different numbers A and B. Find B from the two-digit addition formula AB+25=B3. |
Given two systems of relays—a series system and a parallel system—where each relay operates independently and has a probability of 0.9 of closing properly when activated, compare the probability of current flowing from point \( a \) to point \( b \) in the series system with the probability of current flowing in the pa... |
12. $y' = -y(5 - y)$ |
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. |
Find the smallest number of points that must be randomly selected from the set of vertices of a regular $1000$-gon in order that, with probability greater than $\frac{1}{2}$, some three of the selected vertices form an acute triangle. |
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt{\sqrt[3]{\frac{1}{512}}}}$$ |
Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set $\{0,1,\dots,2010\}$. Call such a polynomial [i]good[/i] if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20,... |
Given the sequence \(\{a_n\}\):
\[ a_n = 2^n + 3^n + 6^n + 1 \text{ for } n \in \mathbb{Z}_+ \]
Is there an integer \( k \geq 2 \) such that \( k \) is coprime with all numbers in the sequence \(\{a_n\}\)? If it exists, find the smallest integer \( k \); if it does not exist, explain why. |
Find the value of \( x \) between 0 and 180 such that
\[\tan (150^\circ - x^\circ) = \frac{\sin 150^\circ - \sin x^\circ}{\cos 150^\circ - \cos x^\circ}.\]
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
In each of Problems 9 through 14 find the general solution of the given system of equations.\n\n10. $\mathbf{x}' = \begin{pmatrix} 2 & 2 + i \\ -1 & -1 - i \end{pmatrix} \mathbf{x}$ |
Find the greatest integer \( k \) such that there exists a \( k \)-element subset of the set \(\{1, 2, 3, \ldots, 3^8\}\) in which no three elements form an arithmetic progression. |
Given propositions $p$ and $q$:
- Proposition $p$: The equation $x^{2}+mx+1=0$ has two distinct negative roots.
- Proposition $q$: The equation $4x^{2}+4(m-2)x+1=0$ has no real roots.
Find the range of real number $m$ that makes both propositions true. |
For \(0 < x, y < 1\), find the minimum value of the expression \((x+y+xy)( x^{y-1}+y^{x-1}) - \left(\frac{1+x}{1-x}\right)^{1-x} - \left(\frac{1+y}{1-y}\right)^{1-y}\). |
Let $a$ be a positive integer such that the fifth term of the geometric sequence $\left( \frac{1}{2} \right)^n$ is greater than $a$ and the seventh term is less than $2a$. What is the largest possible value of $a$? |
Given that the line $x+y+a=0$ intersects the circle $x^2+y^2=1$ at two distinct points $A$ and $B$, and $O$ is the origin of the coordinate system, and $$|\overrightarrow{OA} + \overrightarrow{OB}| \geq |\overrightarrow{AB}|$$, then the range of the real number $a$ is
Return your final response as 'Final Answer: \boxe... |
In a math contest, each student has to answer 20 problems, with each problem having 4 choices. If a student guesses randomly, what is the probability that the student answers exactly 3 problems correctly? Express your answer as a common fraction. |
Determine the range of the function
$$g(x)=\arcsin(\cos(x))$$
where $g$ is restricted to the domain $x \in [0, 2\pi].$ |
Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\... |
A rectangle can be divided by parallel lines to its sides into 200 congruent squares and also into 288 congruent squares. Find the smallest possible number of congruent squares into which the rectangle can be divided, given that this number is greater than 288. |
Compute the value of the sum \[\sum_{r = 1}^{11}\arcsin \frac{\cos r^\circ}{\sin 8r^\circ}.\] |
What is the leading asymptotic behavior of the solution to the Thomas-Fermi equation \( y'' = y^{3/2} x^{-1/2} \) as \( x \to \infty \), given the boundary conditions \( y(0) = 1 \) and \( y(\infty) = 0 \)? |
Given the function $f(x) = \frac{x^2 - 3x + 2}{x^2 + 2x - 1}$, determine the values of $x$ for which $f(x)$ intersects its vertical asymptote(s). |
Let \( n \geq 2 \) be an integer. Lucia chooses \( n \) real numbers \( x_1, x_2, \ldots, x_n \) such that \( \left| x_i - x_j \right| \geq 1 \) for all \( i \neq j \). She writes these numbers in an \( n \times n \) grid such that no number is repeated in the same row or column. For each cell, she calculates the absol... |
Show: \((U \times B) \setminus (A \times B) = A^c \times B\). |
Given 49 students who each solve a set of 3 problems, with each problem scored from 0 to 7, find the minimum number of students such that there are at least two students A and B where A scores at least as many points as B on each problem. |
Find the number of ways to arrange the numbers \(\{1,2,...,n\}\) in a row such that for every number \(x\) (except for the very left one), at least one of the numbers \(x\pm 1\) is to the left of \(x\). |
Let \( K \) be the simplicial complex consisting of the triangular faces, edges, and vertices of an octahedron in \( \mathbb{R}^3 \) with vertices \( P_1 = (0,0,1) \), \( P_2 = (1,0,0) \), \( P_3 = (0,1,0) \), \( P_4 = (-1,0,0) \), \( P_5 = (0,-1,0) \), and \( P_6 = (0,0,-1) \). Calculate the homology groups \( H_0(K) ... |
A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the st... |
A princess must choose from among \( r \) suitors who are presented to her one by one in random order. At each stage, she must either accept the current suitor or reject him and proceed to the next. What strategy will maximize her chance of stopping with the best suitor of all? |
Find the minimum value of the function \( f(x) = \sum_{n=1}^{20} |nx - 3| \). |
A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$, where $m$ and... |
Which shape of the pair has the greater volume, a cone that is 8 in tall and has a base diameter of 4 in, or a sphere that has a diameter of 8 in? |
The product of three consecutive odd integers is 945. If the middle integer is decreased by 2, what is the sum of the three resulting integers? |
Find the complementary and particular solutions, and thereby write the general solutions for the following differential equations:
(a) \(\frac{d^2y}{dx^2} + y = x \sin(x)\)
(b) \(\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = x e^x\)
(c) \(x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = x\) |
Calculate: $(x^{3}y^{2})^{2} \cdot \frac{x}{{y}^{3}} = \_\_\_\_\_\_$. |
Example 1 (Problem from the 35th IMO) Find all ordered pairs of positive integers $(m, n)$ such that $\frac{n^{3}+1}{m n-1}$ is an integer.
Find all ordered pairs of positive integers $(m, n)$ such that $\frac{n^{3}+1}{m n-1}$ is an integer. |
A triangle \( \triangle ABC \) is inscribed in a circle; through the vertex \( A \) a tangent line is drawn to intersect the extension of the side \( BC \) at the point \( D \). From the vertices \( B \) and \( C \) perpendiculars are dropped to the tangent line, the shorter of these perpendiculars being equal to 6 cm.... |
(Determine the polynomials $P(x, y, z, w)$ and $Q(x, y, z, w)$ satisfying:
$$
(x y+z+w)^{2}-\left(x^{2}-2 z\right)\left(y^{2}-2 w\right)=(P(x, y, z, w))^{2}-\left(x^{2}-2 z\right)(Q(x, y, z, w))^{2}
$$ |
Suppose $x$ and $y$ are real numbers such that $\frac{x-1}{x^2+4x+1}=y$. Find the smallest value of $y$. |
Show the following properties of $<_{\sigma}$ where $\sigma$ is a sign function on a commutative ring $R$ with unit 1. For all $a, b, c, d \in R$:
(a) $a<_{\sigma} b$ implies $a+c<_{\sigma} b+c$.
(b) $a<_{\sigma} b$ and $\sigma(c)>0$ implies $a c<_{\sigma} b c$.
(c) If $b^{-1}$ and $a^{-1}$ exist, then $a<_{\sigm... |
\( f(x) = 3x^2 + x - 2 \) at \( (3, 28) \) |
the diameter of a cylindrical tin is 14 cm and height is 2 cm . find the volume of the cylinder ? |
Let \( f(x) \) be an arbitrary function, differentiable infinitely many times. The \( n \)th derivative of \( f(e^x) \) has the form
\[
\frac{d^n}{dx^n} f(e^x) = \sum_{k=0}^{n} a_{kn} e^{kx} f^{(k)}(e^x).
\]
From the coefficients \( a_{kn} \), compose the sequence of polynomials
\[
P_n(x) = \sum_{k=0}^{n} a_{kn} x^k... |
There are 40 identical gas cylinders with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the connection, the gas pressure in the connected cylinders becomes equal to the arithmetic ... |
In a right triangle with sides 3, 4, and 5 (where 5 is the hypotenuse), find the tangent of angles A and B, where angle A is opposite the side of length 3 and angle B is opposite the side of length 4. |
What is the intersection point of the line \( y = 3x - 4 \) and the line perpendicular to it that passes through the point \( (3, 2) \)? Additionally, find where this perpendicular line intersects with another line parallel to \( y = 3x - 4 \) that passes through \( (0, -3) \).
Return your final response as 'Final Ans... |
What is the domain of $y = \tan x - 3$? Express your answer using interval notation. |
Consider a group of $2n+2$ people. Determine the minimum number of pairs $(A, B)$, where $A$ and $B$ are individuals in the group, such that there are exactly $n$ of the remaining $2n$ people who either know both $A$ and $B$, or do not know either $A$ or $B$. Find this minimum number for $n=5$. |
Example 4. Integrate the equation $3 y^{\prime \prime}=y^{-\frac{5}{3}}$. |
Given a triangle with sides \(a = 12\), \(c = 17\), and angle \(\alpha = 35^\circ\), determine the number of possible triangles and solve them if they exist. Round all answers to the nearest tenth. |
Consider the makespan minimization problem where the goal is to minimize the $L_2$ norm (or equivalently, the sum of the squares) of the machine loads. Suppose there are only a constant number of distinct job lengths, and each job length is at least a constant fraction of the average machine load (i.e., $\sum p_j / m$,... |
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that, for every real number \( x \):
\[ f\left(x^{3}+x\right) \leq x \leq (f(x))^{3}+f(x) \] |
A retired schoolteacher has a beautifully maintained garden where she plants flowers in neat rows. She plants her flowers in such a way that the number of flowers in each row forms an arithmetic sequence. On a particular day, she notices that the total number of flowers in the first 10 rows is 340. She also realizes th... |
Let \( a \) be a real number. Let \( a_{n+1} = 2|a_n| - 1 \) for \( n \ge 1 \) and \( a_1 = a \). Find the general term of the sequence \( \{a_n\} \). |
There are 3N participants in AtCoder Group Contest. The strength of the i-th participant is represented by an integer a_i. They will form N teams, each consisting of three participants. No participant may belong to multiple teams.
The strength of a team is defined as the second largest strength among its members. For ... |
Write each of the following ratios in lower forms and, if you can, use words.
a) $\frac{150\text{ metres}}{240\text{ metres}}$
b) $\frac{4000\text{ pages}}{5000\text{ pages}}$
c) $\frac{500\text{ dots}}{800\text{ dots}}$ |
Determine the integers $(x,y)$ where $18x + 27y = 21$ and $27x + 18y = 69$. |
Given the set \( T := [0, 12) \subseteq \mathbb{R} \), show how to express \( T \) as the coequalizer of a pair of arrows \( \mathbb{R} \rightrightarrows \mathbb{R} \). |
Let the function \( f(t) \) be continuously differentiable and positive for \( 0 < t < 1 \), and also let \( \int_0^1 f(t) \, dt \) exist. The entire function defined by the integral\n\n\[ \int_0^1 f(t) e^{xt} \, dt = F(z) \]\n\nhas no zeros\n\n- in the half-plane \(\Re z \geq 0\), if \( f'(t) > 0 \),\n- in the half-pl... |
In a random sample of size \(n = 50\) of random vectors \((X, Y)\) from a bivariate normal distribution, \(-\infty < \mu, \eta < \infty, 0 < \sigma_1, \sigma_2 < \infty, -1 < \rho < 1\), the MLE of \(\rho\) is \(\hat{\rho} = 0.85\). Apply the variance stabilizing transformation to determine asymptotic confidence limits... |
Determine the values of \( \alpha \) for which the generalized Gaussian probability density \( p_{\alpha}(t) \propto \exp (- |t|^\alpha) \) is infinitely divisible, given that \( 0<\alpha<\infty \). |
A cylindrical container has a height of 10 cm and a radius of 4 cm. A spherical marble is dropped into the container and rolls around until it comes to rest. If the volume of the marble is equal to the volume of the water displaced by the marble, and the water level rises by 2 cm, find the radius of the marble. |
Find the Riemann function for the operator\n\n\[L(u) = \frac{\partial^2 u}{\partial t^2} - a^2 \frac{\partial^2 u}{\partial x^2}, \quad a = \text{const.},\]\n\nand using it solve the boundary-value problem\n\n\[\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2} + f(x, t), \quad -\infty < x < +\in... |
Let $a$ and $b$ be real numbers. Find the maximum value of $a \cos 2\theta + b \sin 2\theta$ in terms of $a$ and $b$. |
For natural numbers \( n \) and \( k \), consider monic polynomials \( \{P_1(x), \ldots, P_k(x)\} \) of degree \( n \) with integer coefficients such that no two of them have a common factor and for every subset of \( \{P_1(x), \ldots, P_k(x)\} \), the sum of the elements of the subset has all its roots real. Calculate... |
Let $A=(a,b)$ be a point on the line $x-y=1$. Find the coordinates of the point $B$ that is symmetric to $A$ with respect to the line $y=2x$.
[The end of the response] |
There are two piles of stones: 1703 stones in one pile and 2022 in the other. Sasha and Olya take turns playing the game, with Sasha starting first. Assume that before the player's turn, the piles contain $a$ and $b$ stones, where $a \geq b$. On their turn, the player is allowed to take any number of stones from the pi... |
Let $f(x) = \frac{x^2}{(x-3)^2}.$ Determine the range of $f(x)$ for $x \in (3, \infty).$ |
Let's determine the equation of the tangents drawn from the point $P\left(\frac{3}{5} ; \frac{14}{5}\right)$ to the ellipse $4 x^{2}+9 y^{2}=36$.
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
---
Determine the equati... |
3. Find all positive integers that are 13 times the sum of their digits. |
For the equilateral triangle with base \(y = 0\), \(0 < x < 1\), the lowest eigenvalue is \(\lambda_1^2 = 16\pi^2/3\) (from Exercise 21). For one-sixth of a circular disk, the eigenvalue problem is\n\n\[\n\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\pa... |
Simplify $\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)^{-3}$. |
Return your final response within \boxed{}. Find all functions \( f: \mathbf{Q} \rightarrow \mathbf{Z} \) such that for any \( x \in \mathbf{Q}, a \in \mathbf{Z}, b \in \mathbf{Z}_{+} \), the following holds:
\[
f\left(\frac{f(x)+a}{b}\right) = f\left(\frac{x+a}{b}\right).
\] |
Return your final response within \boxed{}. Given \( a \geq b \geq c \geq d > 0 \) and \( a^{2} + b^{2} + c^{2} + d^{2} = \frac{(a + b + c + d)^{2}}{3} \), find the maximum value of \(\frac{a + c}{b + d}\). |
Madison and Gillian want to buy burgers for the students in their class. The teacher Milly gave them a discount of $0.50 on every burger. If they end up spending 24 dollars for 36 burgers, what was the original price of the burger? |
Find an asymptotic approximation for the gamma function \( \Gamma(n + 1) \) when \( n \) is large and positive, but not necessarily an integer. |
The equation of the tangent line to the function $f(x) = (x + 1)\ln x - 4(x - 1)$ at the point $(1, f(1))$ is ______. |
Solve for x: (x+2)(3x-1) = (x-1)(x+3)
Note: This new problem requires a different approach than the original problem, as it involves factoring and solving a quadratic equation. |
Given two squares with sides parallel to each other, determine the set of points \( M \) such that for any point \( P \) in the first square, there exists a point \( Q \) in the second square such that the triangle \( MPQ \) is equilateral. Let the side lengths of the first and second squares be \( a \) and \( b \) res... |
Let $f(x) = a^{|x|}$ ($a > 0$ and $a \neq 1$), determine the relationship between $f(a+1)$ and $f(2)$.
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
Let \(a\), \(b\), \(c\), and \(d\) be elements of the set \(\{1, 2, 3, \ldots, 2014, 2015\}\) such that \(a < b < c < d\), \(a + b\) is a divisor of \(c + d\), and \(a + c\) is a divisor of \(b + d\). Determine the maximum value that \(a\) can take. |
2B. Given are the lines $y=\frac{3}{4} x+6, y=\frac{3}{4} x+3$ and the point $T(7,24)$. Determine the equation of the line that contains the given point, and the segment with endpoints obtained by the intersection of this line with the given lines, has a length of 4. |
A quadrilateral has side lengths which are powers of 2, and the length of its perimeter is also a power of 2. What is its largest possible perimeter? |
Given a compact oriented \( n \)-dimensional manifold \( M \) triangulated into simplices, construct a vector field \( X \) on \( M \) such that the index of \( X \) satisfies:
\[
\text{Index}(X) = \sum_{j=0}^{n} (-1)^j \nu_j,
\]
where \( \nu_j \) is the number of \( j \)-simplices in the triangulation (e.g., \( \nu_0 ... |
9. Let $A B C D$ be a trapezoid with $A B \| C D$ and $A D=B D$. Let $M$ be the midpoint of $A B$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle B C D$ and the diagonal $A C$. Suppose that $B C=27, C D=25$, and $A P=10$. If $M P=\frac{a}{b}$ for relatively prime positive integers $a$ a... |
Given that the focus of an ellipse is on the x-axis, the length of the minor axis is 16, and the eccentricity is $$\frac {3}{5}$$, determine the standard equation of the ellipse. |
Provide a natural deduction (ND) derivation to show the following entailment: \( A \rightarrow \sim B \vdash_{ND} B \rightarrow \sim A \). |
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