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Determine the values of \( \lambda > 0 \) for which the set \( I_{\lambda} \), consisting of all irrational numbers \( \rho = [a_{1}, a_{2}, \ldots, a_{n}, \ldots] \in (0,1) \) with continued fraction partial quotients satisfying \( a_{n} \leq \text{const} \cdot n^{\lambda} \), has a positive Lebesgue measure. |
Let \( f(x) = x^m + a_1x^{m-1} + \cdots + a_{m-1}x + a_m \) and \( g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n \) be two polynomials with real coefficients such that for each real number \( x \), \( f(x) \) is the square of an integer if and only if \( g(x) \) is the square of an integer. Given that \( n + m > 0 ... |
An investment fund has the following cash flows (in thousands of pounds): +100 at time 0, +50 at time 1, -60 at time 2, and -125 at time 3 (where time is measured in years). Calculate the money-weighted rate of return for this investment fund. |
There are 45 students in a summer camp. The counselor has 12 liters of juice. She pours 150 milliliters of juice for every student as a refreshing drink. Unfortunately, one of the students accidentally knocked over a cup and 300ml of juice was spilled. How much juice was left over? |
Solve the equation for $x$: $\tan x - \tan (x + 7) = 1$ |
Provide an example of a bounded interpolation operator on \( L^2 \) that is not a Hilbert-Schmidt operator and is also not compact. |
Consider the series \(\sum_{k=1}^{n} \frac{1}{\sinh(2^k \pi)}\). Let \(S_n\) be the sum of this series for a finite \(n\). Find the value of \(S_5\). |
22. N3 (BUL) ${ }^{\mathrm{MO} 2}$ Determine all pairs $(a, b)$ of positive integers such that
$$
\frac{a^{2}}{2 a b^{2}-b^{3}+1}
$$
is a positive integer. |
Suppose $\{a,b,c\}$ and $\{d,e\}$ are sets. Find the Cartesian product $\{a,b,c\}\times\{d,e\}$. |
A trapezoid has two parallel sides of lengths $x$ units and $2x$ units and the height is $x\sqrt{3}$ units. The area of the trapezoid is $15\sqrt{3}$ square units. What is the value of $x$? Express your answer in simplest radical form. |
Given the ellipse $mx^{2}+4y^{2}=1$ with an eccentricity of $\frac{\sqrt{2}}{2}$, find the real number $m$ equal to \_\_\_\_\_\_. |
Compute $(3 - 7i)^2$ and write your answer in simplest form. |
A train that is 360 meters long is running at a speed of 30 km/hour. It takes approximately 60 seconds to pass a bridge of a certain length. What is the length of the bridge? |
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space \( \mathcal{X} = \{0, 1, \ldots, n\} \), where \( P = [p, 0, \ldots, 0, 1-p] \) and \( Q \) follows a binomial distribution \( Q = \binom{n}{k} q^k (1-q)^{n-k} \) for all \( k \in \{0, 1, 2, \ldots, n\} \), determine the pa... |
2xy' + y = y^2 \sqrt{x - x^2y^2}. |
A coin is tossed. Each time a head appears, 1 point is awarded; each time a tail appears, 2 points are awarded. Repeatedly toss the coin. What is the probability of obtaining exactly \( n \) points?
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression ... |
Determine the region within the domain \(D = \{(x, y) \in \mathbb{R}^2 \mid x \geq 1, y \geq 1\}\) where the function \(f(x, y) = \begin{cases} \frac{2x^2 - xy - y^2}{x^2 - y^2} & x \neq y \\ \frac{3}{2} & x = y \end{cases}\) is continuous. |
On the same side of a street, six adjacent houses will be built. The houses can be brick or wooden, but as a safety measure against fire, two wooden houses cannot be adjacent. In how many ways can the construction of these houses be planned? |
Ellen says that the closest point to $(0, 0)$ on the line whose equation is $y = 2$ is at $(0, -2)$. Frank says that the point $(2, 0)$ is on the line, and every point $(x, 2)$ would be on the line. Who is correct? |
Let \( M \) be a compact oriented \( n \)-dimensional manifold, and let \( N_1, \ldots, N_k \) be disjoint proper oriented \((n-1)\)-dimensional submanifolds of \( M \). Given a trivial tubular neighborhood \([-1,1] \times (N_1 \cup \cdots \cup N_k)\) in \( M \), how is the Thom-Pontryagin map \( \rho_{N_1, \ldots, N_k... |
The ratio of three numbers is 1 : 2 : 5 and the sum of their squares is unknown. The sum of the numbers is 12. What is the sum of their squares? |
Find all infinite sequences \( a_1, a_2, a_3, \ldots \) of positive integers such that:
a) \( a_{nm} = a_n a_m \), for all positive integers \( n, m \), and
b) there are infinitely many positive integers \( n \) such that \( \{1, 2, \ldots, n\} = \{a_1, a_2, \ldots, a_n\} \). |
Consider a war of attrition game between two grocery stores. Each month, both stores independently decide whether to stay in business or quit. If both stay, each loses \$1000 that month. If one quits, the remaining store earns \$500 profit per month thereafter. Once a store quits, it cannot re-enter. Both stores aim to... |
Given \(x_{1}, x_{2}, \cdots, x_{n}\), where each \(x_{i}\) can only be one of \(0, 1, -1\), and \(S\) represents the sum of the products of any two selected values from \(x_{1}, x_{2}, \cdots, x_{n}\):
\[ S = \sum_{1 \leq i < j \leq n} x_{i} x_{j} \]
Find the minimum value of \(S\). |
Determine the boundary of the topological space $S^1 \times I^2$. |
Point $P$ moves on the curve $y=\frac{1}{3}x^{3}-2x^{2}+3x$. Let the slope angle of the tangent line at point $P$ be $\alpha$, then the range of values for angle $\alpha$ is
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
Find all strictly monotonous functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)) = 3n$. |
Calculate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( S \) is the boundary of the solid \( E \) enclosed by the paraboloids \( z = x^2 + y^2 \) and \( z = 12 - 2x^2 - 2y^2 \), and \( \mathbf{F}(x, y, z) = \langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \rangle \). |
The circles whose equations are $x^2 + y^2 - 2x - 4y - 6 = 0$ and $x^2 + y^2 + 8x + 10y - 16 = 0$ intersect in the points $C$ and $D.$ Compute the equation of the line that passes through the centers of the circles and the midpoint of $\overline{CD}.$ |
In Section 8.2, you'll see the identity \( \cos^2 x = \frac{1}{2} + \frac{1}{2} \cos 2x \). Use this identity to graph the function \( y = \cos^2 x \) for one period. |
For the function \( f(x) = 3 - x \) defined on \( 0 < x < 3 \), investigate the dependence on \( n \) (the number of terms in the series) of the maximum error on \([0, 3]\) for:
(a) The Fourier cosine series (even extension).
(b) The Fourier sine series (odd extension). |
Find the value of an annuity with a monthly deposit of \$100 at the beginning of each month, an annual interest rate of 9% compounded monthly, and 10 years of deposits. Calculate the value right after the last deposit. |
Find the second-order derivative \( y_{xx}^{\prime \prime} \) of the function given parametrically.
$$
\left\{\begin{array}{l}
x=2(t-\sin t) \\
y=4(2+\cos t)
\end{array}\right.
$$ |
Compare the integrals \( I_{1}=\int_{0}^{1} x \, dx \) and \( I_{2}=\int_{0}^{1} x^{3} \, dx \) without calculating them. |
Solve for \(x\): \(x^4 - 1 = 0\). |
In a triangle, the length of the hypotenuse is $15$ centimeters, and one of the acute angles is $30$ degrees. What is the area of the triangle? |
Calculate the surface area of the solid formed by rotating the curve \(x^{2/3} + y^{2/3} = 1\) around the \(y\)-axis from \(y = 0\) to \(y = 1\). |
Express the following numbers in the form \(\sqrt{a} \pm \sqrt{b}\), where \(a\) and \(b\) are natural numbers:
a) \(\sqrt{3+2 \sqrt{2}}\)
b) \(\sqrt{9+4 \sqrt{5}}\)
c) \(\sqrt{7-2 \sqrt{10}}\).
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of... |
\( \frac{d^2 x}{dt^2} + 9x = 5 \sin 3t \),\n \( x(0) = 2 \), \n \( x'(0) = 0 \) |
Given that \( f(x-3) = f(x+5) \) and \( 0 \) is a root of \( f(x) \), what are the other roots of \( f(x) \)? |
For Norm-Bound Linear Differential Inclusions (NLDIs), what is the smallest upper bound on the output energy (defined as the maximum of \(\int_{0}^{\infty} z^T z \, dt\) given an initial state \(x(0)\)) provable via quadratic Lyapunov functions? State the corresponding optimization problem and its variables. |
A width of a rectangle is 3 inches shorter than the length. If the area of the rectangle is 104 square inches, what is the length of the rectangle? |
Find the analytic center of the linear matrix inequality \( F(x) \succeq 0 \), where \( F(x) = F_0 + x_1 F_1 + \cdots + x_n F_n \) is an affine function mapping \( \mathbf{R}^n \) to \( \mathbf{S}^p \), and \( F(x) \succ 0 \) defines the domain of the logarithmic barrier function. |
Simplify \( \sqrt{4 p^{8}} \). Assume \( p \) is a real number (positive or negative). |
Find the Cauchy principal value of the integral \( \int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx \) using the method of residues. |
Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 3,$ and $a \ge -\frac{1}{2},$ $b \ge -\frac{3}{2},$ and $c \ge -2.$ Find the maximum value of
\[\sqrt{4a + 2} + \sqrt{4b + 6} + \sqrt{4c + 8}.\]
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical express... |
Find the functional derivative of the differential entropy with respect to the probability density function, i.e., determine \( \frac{\delta F[p]}{\delta p(x)} \), where \( F[p] = \int_X p(x)\ln(p(x))dx \) and \( p(x) \) is a probability density function. |
Consider the polynomial $p(x) = x^4 - 2ax^2 + 3b^2x - 4c^3$ with roots $r_1, r_2, r_3, r_4$. Suppose that the sum of the products of the roots taken two at a time is equal to $2a - b + c$. Determine the value of $b$ such that the product of the absolute values of the roots is minimized. |
Given the function $f(x)=\sin (2x+φ)$ where $0 < φ < π$, the graph of the function is shifted to the left by $\frac {π}{6}$ units to obtain the graph of the function $y=g(x)$. If $y=g(x)$ is an even function, find the value of $φ$. |
Anya writes a natural number, and Boris replaces any digit with a digit that differs by 1. What is the smallest number that Anya must write so that the result is guaranteed to be divisible by 11?
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of ... |
Let $e_1 = a^2 + 3^a + a \cdot 3^{(a+1)/2}$ and $e_2 = a^2 + 3^a - a \cdot 3^{(a+1)/2}$. For $1 \le a \le 25$, determine all integral values of $a$ such that $e_1 \cdot e_2$ is a multiple of 3. |
A team of workers must carve $n$ ice sculptures for an ice town in two days. On the first day, they carved 11 sculptures, with each worker completing an equal portion of the work. On the second day, only 7 workers came to work because some were reassigned to another project. Each worker did three times the amount of wo... |
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$ . |
Find the values of \( x_1, x_2, \) and \( x_3 \) such that the following matrix equality holds:
\[
\left[\begin{array}{ccc}
x_{1}+x_{2}+2 x_{3} & 0 & 1 \\
2 & 3 & 2 x_{1}+4 x_{2}-3 x_{3} \\
4 & 3 x_{1}+6 x_{2}-5 x_{3} & 5
\end{array}\right]=\left[\begin{array}{ccc}
9 & 0 & 1 \\
2 & 3 & 1 \\
4 & 0 & 5
\end{array}\right... |
Let \( m \) be a positive integer. Let \( \mathcal{S} \) be the set of integers from \( 0 \) to \( 10^m-1 \) inclusive whose base-2 and base-5 representations end in the same \( m \) digits. Compute the remainder when the sum of the elements of \( \mathcal{S} \) is divided by \( 10^m \). |
If $\frac{1}{A} + \frac{1}{B} + \frac{1}{C} = 3$, then what is the value of $C$ in terms of $A$ and $B$? Express your answer as a common fraction. |
The Pythagorean triple $(a,b,c)$ must satisfy which equation? |
Given a 3% chance of eating Tylenol and a 1% chance of drinking beer every hour, where the effects of each last for one hour and overlapping effects cause vomiting, what is the expected amount of time before vomiting occurs? Assume that eating or drinking either item while already affected does not reset the effect tim... |
Given that the roots $\alpha$ and $\beta$ of the equation $x^2 - 3x + 1 = 0$ are also roots of the equation $x^4 - px^2 + q = 0$, find the values of $p$ and $q$. |
The base of a triangle is given as twice its height. If its area is 300 square meters, what is its height? |
Which of the following are propositions? Give the truth value of the propositions.
a. \(2 + 3 = 7\).
b. Julius Caesar was president of the United States.
c. What time is it?
d. Be quiet! |
A cylindrical vessel has a height of 20 cm and a base radius of 10 cm. Water is poured into the vessel at a rate of 2 cm^3/s. At the same time, water is leaking out of a small hole at the bottom of the vessel at a rate of 0.5 cm^3/s. Find the rate at which the water level is rising when the water level is 8 cm above th... |
Hyungjun has 324 grape-flavored candies and 541 strawberry-flavored candies, and Junghyeok has 416 orange-flavored candies and 253 cherry-flavored candies. Who has more candies, Hyungjun or Junghyeok? |
There is a pile of 1001 stones on the table. In the first step, one stone is removed, and the remaining stones are divided into two piles. In each subsequent step, one stone is removed from a pile that contains more than one stone, and that pile is then divided into two piles. Can all the piles on the table eventually ... |
Find all positive integers $a$ and $b$ such that $a / 5$ and $b / 7$ are less than 1 and satisfy the condition
$$
1<\frac{a}{5}+\frac{b}{7}<2
$$ |
Anna, Percy, and Morgan weigh a combined 370 lb. Morgan weighs 20 lb more than Percy, and Anna weighs 1.5 times as much as Percy. Write a system of equations representing this situation and solve it using the inverse of a matrix to find each person's weight. |
The set \( S = \{1, 2, \ldots, 2022\} \) is to be partitioned into \( n \) disjoint subsets \( S_1, S_2, \ldots, S_n \) such that for each \( i \in \{1, 2, \ldots, n\} \), exactly one of the following statements is true:
(a) For all \( x, y \in S_i \) with \( x \neq y, \operatorname{gcd}(x, y) > 1 \).
(b) For all \( x,... |
Given a parallelogram \( ABCD \) in which \(\angle B = 111^\circ\) and \( BC = BD \). Point \( H \) is marked on the segment \( BC \) such that \(\angle BHD = 90^\circ\). Point \( M \) is the midpoint of side \( AB \). Find the angle \( AMH \). Give your answer in degrees. |
The equation of the line obtained by translating the line $y = 3x - 3$ upwards by 5 units along the y-axis is ____. |
Each cell of a $100 \times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors needed to paint these triangles such that any two triangles sharing a common side or vertex have different colors? |
Find the Singular Value Decomposition (SVD) of the matrix and determine the rounded largest singular value:
[[2, 4, 2, -3, 6],
[2, -4, 0, -1, 1],
[4, 2, -2, -6, -4],
[3, 5, 1, 4, -1],
[1, 2, -3, -6, -6]]
Present the answer in LaTex format: \boxed{Your answer} |
How many 3-digit numbers are there such that if you create all the different 3-digit numbers from their digits and add these numbers, the sum is divisible by 111? |
If $|3x - 2| = 11$, then what is the value of $x$? |
Return your final response within \boxed{}. In the triangular pyramid \( P-ABC \), the angles between each of the three lateral faces and the base face are equal. The areas of the three lateral faces are 3, 4, and 5, and the area of the base face is 6. What is the surface area of the circumscribing sphere of the triang... |
If \( f(x) = \tan^{-1} (\operatorname{sgn}(x^2 - \lambda x + 1)) \) has exactly one point of discontinuity, then the value of \( \lambda \) can be:\n- (a) 1 \n- (b) -1 \n- (c) 2 \n- (d) -2 |
Find the largest constant \( C \) such that
\[ \left(\sum_{\text{cyc}} a_i^2 + b_i^2 - a_ia_{i+1} - b_ib_{i+1} \right) \ge \frac{C}{n} \left(\sum_{\text{cyc}} a_i(b_{i+1} - b_{i-1})\right) \]
for all integers \( n \) and corresponding reals \( a_1, \dots, a_n, b_1, \dots, b_n \). |
Given the function defined on \(\mathbf{R}^{+}\), \( f(x) =
\begin{cases}
\left|\log_3(x) - 1\right| & \text{if } 0 < x \leq 9,\\
4 - \sqrt{x} & \text{if } x > 9
\end{cases} \), where \(a\), \(b\), and \(c\) are three distinct real numbers satisfying \(f(a) = f(b) = f(c)\), find the range of values for \(a \cdot b ... |
Let $(u_{n})$ and $(v_{n})$ be two sequences defined by:
- $u_{1}=3$
- $v_{1}=2$
- $u_{n}=u_{n-1}^{2}-2v_{n-1}^{2}$
- $v_{n}=2u_{n-1}v_{n-1}$
Find the general formula for the sequences $(u_{n})$ and $(v_{n})$. |
Example 4 The equation $k \cdot 9^{x}-k \cdot 3^{r+1}+6(k-5)=0$ has a solution for $x \in[0,2]$, find the range of real number $k$.
---
The translation maintains the original text's format and line breaks as requested. |
Let $h(x) = 2x - 3$ and $j(x) = x^2 + 1$. Also denote the inverses to these functions as $h^{-1}$ and $j^{-1}$. Compute
\[j(h^{-1}(j^{-1}(h(j(h^{-1}(h(4))))))).\] |
In an acute triangle $\triangle{ABC}$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that circle $(AHO)$ is tangent to the circumcircle of $\triangle{ABC}$, $\tan A = \frac{26}{15}$, and $BC = 52$, compute the area of $\triangle{ABC}$. |
Return your final response within \boxed{}. How many four-digit numbers contain the digit 9 followed immediately by the digit 5? |
For a certain exam, a score of 58 was 2 standard deviations below the mean, a score of 98 was 3 standard deviations above the mean, and a third score was 1.5 standard deviations above the mean. What was the mean score (q) for the exam and what was the value of the third score? |
Solve in positive integers the equation \((a+1)(b+1)(c+1)=2abc\). |
Evaluate \( \lim_{x \to \infty} \frac{\gamma(x+1, x)}{\Gamma(x+1)} \), where \( \gamma(x+1, x) \) is the lower incomplete gamma function and \( \Gamma(x+1) \) is the complete gamma function. |
Let f^{right}(n) denote the largest number such that from any set of n points in the plane one can select f^{right}(n) elements such that no three of them determine a right angle. What is the largest constant c such that f^{right}(n) ≥ (c + o(1))√n? |
Example 10. Solve the equation
$$
y^{\prime \prime \prime}-2 y^{\prime \prime}+2 y^{\prime}=4 \cos x \cos 3 x+6 \sin ^{2} x
$$ |
Given $\alpha \in [0, 2\pi]$ and $\cos(\alpha) = \frac{1}{3}$, express $\alpha$ using the arccosine function. |
Let \( f: \mathbb{R} \longrightarrow \mathbb{R} \) be a function such that
\[ \left| f(x+y) + \sin x + \sin y \right| \le 2 \]
for all real numbers \( x, y \). Find the maximum possible value of \( \left| f(x) \right| - (1 + \cos x) \) for all real numbers \( x \). |
Find the equation of the line that passes through the point P(1,1) and is perpendicular to the line 2x - y = 0. Represent the answer in the general form of the equation of a line. |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2}\left(\frac{\cos x}{\cos 2}\right)^{\frac{1}{x-2}}$ |
What is the highest common divisor of 77988690522088 and 189862899395187080074?
Present the answer in LaTex format: \boxed{Your answer} |
Consider the inequality $|x^2-5x+6|<1$. What are the possible values for $x$? Express your answer in interval notation. |
Find the domain of the function
\[f(x) = \frac{x^2 - 2}{\sqrt{x + 3}}.\] |
Convert \(3.85 \, \text{in}^3\) to cubic centimeters. |
Given the equation $\frac{x+y}{x-y} = 2y$, can it be manipulated to $\frac{x+y} = 2y(x-y)$ with the restriction $x \neq y$? If so, can implicit differentiation be applied to the second equation to find $\frac{dy}{dx}$? Would the restriction $x \neq y$ need to be included in the final answer for $\frac{dy}{dx}$? |
In each of the following, calculate the shortest distance between the two given lines:\n1) \(\frac{x + 7}{3} = \frac{y + 4}{4} = \frac{z + 3}{-2}; \quad \frac{x - 21}{6} = \frac{y + 5}{-4} = \frac{z - 2}{-1};\)\n\n2) \(x = 2t - 4, \quad y = -t + 4, \quad z = -2t - 1; \)\n\[ x = 4t - 5, \quad y = -3t + 5, \quad z = -5t ... |
Five identical rectangles are arranged to form a larger rectangle PQRS, as shown. The area of PQRS is 4000. Calculate the length, x, of each of the identical rectangles. Round your answer to the nearest whole number. |
Find all positive integers \( n \) less than 1999 such that \( n^2 \) is equal to the cube of the sum of \( n \)'s digits.\n\n(1999 Iberoamerican Mathematical Olympiad) |
Approximate the value of \(\log_{5}(80)\) using the change of base formula. |
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