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In the function $y=\frac{\sqrt{3-x}}{{x}^{0}-3}$, the range of the independent variable $x$ is ____. |
Determine all ordered pairs (m, n) of positive integers for which \(\frac{n^3 + 1}{mn - 1}\) is an integer. |
An equilateral triangle is inscribed in a circle with radius $R$. What is the radius of the largest possible equilateral triangle whose base is half the base of the larger triangle? |
A cone has a base radius of \( R \) and a height \( m = 2R \). At what distance from the apex should we cut the cone with a plane parallel to the base so that the total surface areas of the resulting sections are equal? |
y^{(4)} - 4y'' + 6y' - 4y' + y = 0; y(0) = 0, y'(0) = 1, y''(0) = 0, y'''(0) = 1 |
(a) Find the distribution of temperature in a rod \(0 \leq x \leq l\) thermally insulated along the surface, if the temperature of its ends is maintained equal to zero, and the initial temperature equals an arbitrary function \( f(x) \). (b) Consider, in particular, the case where \( f(x) \equiv U_0 = \text{const.} \),... |
A moving point has coordinates $(x,y,z) = (5\sin t, 5\cos t, 3t)$ at time $t$. Find the rectangular coordinates of the point when it intersects the $xy$-plane. |
The sum of an infinite geometric series is $27$ times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio?
A new problem:
In a certain city, the number of people infected with a certain disease is increasing geometrically every week. If th... |
Find the geodesics on the cone defined by \( z = r \cot \alpha \), where \( r^2 = x^2 + y^2 \). |
$A, B, C,$ and $D$ are distinct positive integers such that the product $AB = 72$, the product $CD = 72$, and $A - B = C + D$. What is the value of $A$? |
5. Lorna, Rita, and Mirna together saved 750 kn. We know that Rita saved three times more than three quarters of Lorna's amount, and Mirna's savings are $85 \mathrm{kn}$ more than two thirds of Rita's savings. How much did each of them save?
## COUNTY COMPETITION IN MATHEMATICS |
Given point $M(2,1)$, a line $l$ passing through $M$ intersects the hyperbola $x^{2}- \dfrac {y^{2}}{2}=1$ at points $A$ and $B$. If $M$ is the midpoint of $AB$, find the equation of line $l$. |
Determine a vector equation for the line with the same x-intercept as \([x,y,z] = [3,0,0] + t[4,-4,1]\) and the same z-intercept as \([x,y,z] = [6,-2,-3] + t[3,-1,-2]\). |
Solve for $x$ in the equation
\[\frac{x^2 - 3x - 1}{x^2 + 1} = \frac{x + 1}{x - 3}.\] |
Consider the periodic function $f(t)$ with period 1, defined by $f(t) = e^{2\pi i z t}$ for $|t| < \frac{1}{2}$, where $z$ is a complex number not equal to any integer. Find its Fourier coefficients $\hat{f}(n)$. Using these coefficients and the Parseval-Bessel equality, deduce the identity:
$$
\frac{1}{\sin^2 z} = \su... |
Find a unit vector that is orthogonal to both $\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}.$ |
Let \( a \) be an arbitrary positive number and let \(\{b_n\}\) be a positive sequence converging to \( b \). Study the convergence of the series\n\[\n\sum_{n=1}^\infty \frac{n!a^n}{(a+b_1)(2a+b_2)\cdots (na + b_n)}.\n\] |
The heading of an airplane is 75°, and the airspeed is 200 mi/h. Find the groundspeed and course if there is a wind of 40 mi/h from 165°. |
Find all the extreme points of the function \( f(x, y, z) = xyz \) subject to the constraint \( x^2 + y^2 + z^2 = 3 \) using Lagrange multipliers. |
A drum consists of a stretched circular membrane of unit radius whose rim is fixed. The membrane is struck so that its initial displacement is \( F(\rho, \phi) \) and is then released. The boundary value problem for the displacement \( z(\rho, \phi, t) \) from the equilibrium position is given by:
\[
\frac{\partial^2 z... |
If the even function $f(x)$ is a decreasing function in the interval $[-5, -2]$ and its maximum value is 7, then $f(x)$ in the interval $[2, 5]$ is ( )
A: An increasing function and its minimum value is -7
B: An increasing function and its maximum value is 7
C: A decreasing function and its minimum value is -7
D: A dec... |
Determine the values of the genus $g$ for which there exists a rank 2, degree $2g-2$, holomorphic vector bundle $E \to C$ over a curve of genus $g$ such that $H^0(E) = 0$. |
A light particle oscillates at speed 1 ft/s, or 30 cm/s at rest. The position of the particle from its center can be modeled by
\[h(t) = \frac{50\sin 2\pi t}{c} + d,\]
where time, $t$, is measured in seconds, $h(t)$ is the displacement of the particle from its center in centimeters, and $c$ and $d$ are constants. Fin... |
Example 12. Calculate the circulation of the vector field given in cylindrical coordinates: $2=\rho \sin \varphi \mathrm{e}_{\rho}+\rho z \mathrm{e}_{\varphi}+\rho^{3} \mathrm{e}_{z}$, along the curve L: $\{\rho=\sin \varphi, z=0,0 \leqslant \varphi \leqslant \pi\}$ directly and using Stokes' theorem. |
In triangle $\vartriangle ABC$, extend side $BC$ beyond point $B$ to point $D$ such that $DB = AB$. Let $M$ be the midpoint of segment $AC$, and let $P$ be the point of intersection of $DM$ with the angle bisector of $\angle ABC$. If $\angle BCA = 40^\circ$, find the measure of $\angle BAP$. |
For the function $f(x)$ defined as
$$\displaystyle f(x) = \frac{x^3-4x^2-x-7}{x-5}$$find $f(1-\sqrt{3})$. |
Solve the first-order quasilinear partial differential equation (PDE) using the method of characteristics:
\[ x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = 2x^2 - 6y^2 \]
subject to the boundary condition:
\[ u(x = 1, y) = 3y^2 + 2y - 5. \] |
What is the domain of the function $f(x)$, defined as $f(x) = \frac{x + 1}{x - 7}$? (Express your answer in interval notation.) |
Let \( (G, \star) \) be a finite group, and let \( A \) and \( B \) be subsets of \( G \) such that \( |A| + |B| = |G| + 1 \). Define the function \( f : A \to G \setminus B \) by \( f(x) = x^{-1} \star a \) for some fixed \( a \in G \setminus (A \star B) \). Calculate the number of distinct elements in the image of \(... |
Given that $|\overrightarrow{OA}|=1$, $|\overrightarrow{OB}|=2$, $\angle AOB= \frac{2\pi}{3} $, and $\overrightarrow{OC}=\frac{1}{2}\overrightarrow{OA}+\frac{1}{4}\overrightarrow{OB}$, find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OC}$. |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta$, and $|\overrightarrow{a}| = 3$, $|\overrightarrow{b}| = 4$, $|\overrightarrow{a} + \overrightarrow{b}| = 5$, find the value of $\theta$. |
Let \(\mu\) be the Lebesgue measure on \([0, \infty]\). Define\n\n\[\mu_1(E) = \sum_{n=1}^{\infty} \frac{1}{n^3} \int_{E \cap [n, n+1]} x \, d\mu\]\n\n\[\mu_2(E) = \int_{E \cap [1, \infty]} \frac{1}{x^2} d\mu\]\n\nfor any Lebesgue measurable subset \( E \) of \([0, \infty]\). Is \( \mu \ll \mu_2 \) or/and \( \mu_2 \ll ... |
If \[a =
\begin{pmatrix}
18\\
7
\end{pmatrix},\]and \[b =
\begin{pmatrix}
4\\
1
\end{pmatrix},
\]then what is the projection of $a$ onto $b$? Express your answer as a column vector. |
Determine the order, degree, linearity, unknown function, and independent variable of the differential equation \( y'' - 5t^2y' = e^t + 1 \). |
What are the projective objects in the category of chain complexes \( \text{Ch}(A) \)? What about in the subcategories of bounded-below, bounded-above, and bounded complexes? Assume that \( A \) has enough projectives. |
Verify that the distance from the function \( f = x \) to the set \( Y = \{ f \in C[0,1] : \int_0^1 f = 0 \} \) is 0.5, given that \( Y \) is a closed subset of \( X = \{ f \in C[0,1] : f(0) = 0 \} \). |
Given that $a$ and $b$ are positive numbers, if the line $2ax + by - 2 = 0$ intercepts a chord of length $2\sqrt{3}$ on the circle $x^2 + y^2 = 4$, find the maximum value of $a\sqrt{1 + 2b^2}$. |
Let \(\pi\) be a group and consider the exact sequence of groups \(1 \rightarrow L \rightarrow \pi \rightarrow H \rightarrow 1\). What is the relationship between the homotopy fiber of the map \(g: K(\pi, 1) \rightarrow K(H, 1)\) inducing \(\gamma\) and this exact sequence? |
Find all pairs $(n,d)$ of positive integers such that $d \mid n^2$ and $(n-d)^2 < 2d$. |
Circle $A$ is in the interior of circle $B$. The diameter of circle $B$ is 20 cm. What is the diameter of circle $A$ for which the ratio of the shaded area to the area of circle $A$ is 5:1, given that the center of circle $A$ is 5 cm from the center of circle $B$? |
Consider the sequence \(a_n\) defined by the property that its terms \(a_n\) and \(a_{n+1}\) are the roots of the quadratic equation \[x^2 - c_nx + \frac{1}{3^n} = 0.\] The sequence starts with \(a_1 = 2\). Using Vieta's formulas, we have \(a_{n+1} = \frac{1}{3^na_n}\) and \(c_n = a_n + a_{n+1}\). The first few terms o... |
Find all values of the parameter \( a \) for which the equation
\[ 3|x+3a|+\left|x+a^{2}\right|+2x=a \]
has no solution. |
Mr. $A$ owns a home worth $15,000. He sells it to Mr. $B$ at a $20\%$ profit. Mr. $B$ then sells the house back to Mr. $A$ at a $15\%$ loss. How much does Mr. A gain or lose in the transaction?
$\mathrm{(A) \ A\ gains\ 2700\ dollars}$ $\qquad$ $\mathrm{(B) \ A\ gains\ 3000\ dollars}$ $\qquad \mathrm{(C) \ A\ loses\ 270... |
Compute the expected value \( \mathbb{E}(n^2) \) given by the expression:
\[ \mathbb{E}(n^2) = \sum \limits_{n= -N}^{N} \left( \frac{1}{2} \right)^N \frac{N!}{(\frac{1}{2}(N+n))!(\frac{1}{2}(N-n))!} n^2 \] |
Find the remainder when $x^4 + 2x^2 - 3x + 7$ is divided by $x^2 - 2x - 3.$ |
Joan found 70 seashells on the beach, and Sam gave her 27 seashells. Joan decided to keep 1/3 of the seashells for herself and share the remaining seashells equally among her 4 friends. How many seashells does each of her friends receive? |
Given sets $A=\{x| -1 \lt x \lt 2\}$, $B=\{x| -1 \lt x \lt m+1\}$, if $x\in A$ is a sufficient but not necessary condition for $x\in B$, then the range of real number $m$ is ____. |
In the complex plane, what is the distance from the origin to $z_{111}$, where the sequence of complex numbers is defined as $z_1 = 0$ and $z_{n+1} = z_n^2 + i$ for all $n \ge 1$? |
Anne's family has a toothpaste tube containing 105 grams. Each member of the family brushes their teeth three times a day using the following amounts: Anne's dad uses 3 grams, her mom uses 2 grams, Anne uses 1 gram, and her brother uses 1 gram. Calculate how many days it will take for the toothpaste to run out.
Meanw... |
Let \( a_1, a_2, \ldots, a_k \) be positive real numbers and \( m_1, m_2, \ldots, m_k \) be natural numbers. Suppose \( n = m_1 + m_2 + \cdots + m_k \). If \( a_1 = 2, a_2 = 3, a_3 = 5 \) and \( m_1 = 2, m_2 = 3, m_3 = 5 \), calculate the value of
\[
\frac{m_1}{n} a_1 + \frac{m_2}{n} a_2 + \frac{m_3}{n} a_3 - \left( ... |
The function
\[f(z) = \frac{(1 - i \sqrt{3}) z + (-4 \sqrt{3} + 24i)}{3}\]represents a rotation around some complex number $c$. Find $c$. |
A parabolic antenna is described by the equation $y^2 + 6y + 2x + 5 = 0$. Find the focus of the antenna.
Note: This problem requires a different approach as it involves finding the focus of a parabola, which is a concept in geometry and physics, rather than just finding the vertex of a parabola. |
Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ |
Derive the spectral (Källen–Lehmann) representation for the vacuum expectation value \( \langle T \{ J^\mu(x) J^\nu(y) \} \rangle_0 \), where \( J^\mu(x) \) is a complex conserved current. |
An isosceles trapezoid has bases of lengths 25 units (AB) and 13 units (CD), with the non-parallel sides (AD and BC) both measuring 13 units. How long is the diagonal AC? |
Given the function $f(x) = ax^{3} + (a-1)x^{2} - x + 2$ for $0 \leq x \leq 1$, find the range of values for the real number $a$ if the function attains its minimum value at $x = 1$.
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
Find the dimensionless temperature \( y(\xi_1, \xi_2, \zeta) \) that satisfies the partial differential equation:
\[
\tilde{L}^2 \left( \frac{\partial^2 y}{\partial \xi_1^2} + \frac{\partial^2 y}{\partial \xi_2^2} + \frac{\partial^2 y}{\partial \zeta^2} \right) = 0 \quad \text{in } D,
\]
where \( D \) is the domain def... |
Which is greater? $$21 \div 7 - 2 + 4 - 1 \times 3 \div 7$$ or $$20 \div 10 \times 4 - 3 + 2 \div 10$$? |
State a theorem about obstructions to extending a map \( f: X \to Y \) defined on the \( n \)-skeleton of a CW-complex \( X \) into a non-simple space \( Y \). Include the conditions under which the obstructions are defined and the cohomology groups where these obstructions lie. |
The areas of three of the faces of a right rectangular prism are $18 \text{ cm}^2$, $50 \text{ cm}^2$, and $75 \text{ cm}^2$. Determine the volume of the prism in cubic centimeters. |
Petya and Vasya created ten fifth-degree polynomials. Vasya then began calling out consecutive natural numbers, starting from a certain number. Petya chose one of the polynomials for each number called and substituted the number into it, writing the resulting values on the board in sequence. These values formed an arit... |
Simplify ${1\over (3-{ i\over 2})}$. |
Given a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) with real coefficients \( a_i \in \mathbb{R} \) and the condition \( |f(x)| \le 1 \) for all \( |x| \le 1 \), find the maximum value of \( |a_n| + |a_{n-1}| + \cdots + |a_1| + |a_0| \). |
30 people are arranged in six rows of five people each. Each of them is either a knight, who always tells the truth, or a liar, who always lies, and they all know who among them is a knight or a liar. A journalist asked each of them: "Is it true that there are at least 4 rows in which more than half are liars?" What is... |
Convert the number $235_{16}$ to base 12. |
\( u_{tt} = 16 u_{xx} \), \quad \( 0 < x < \pi \), \quad \( t > 0 \), \n\( u_x(0, t) = 0 \), \quad \( u_x(\pi, t) = 0 \), \quad \( t > 0 \), \n\( u(x, 0) = x^3(x-\pi)^2 \), \quad \( u_t(x, 0) = 0 \), \quad \( 0 \leq x \leq \pi \) |
Find all pairs \((f,h)\) of functions \(f,h:\mathbb{R}\rightarrow\mathbb{R}\) such that for all real \(x\) and \(y\), the equation holds:
\[ f(x^2 + yh(x)) = xh(x) + f(xy). \] |
A positive integer \( n \) makes it so that every \( n \)-element subset of the set \(\{1,2,3, \cdots, 2008\}\) contains 2 elements (not necessarily distinct) whose sum is a positive integer multiple of 2. What is the smallest value of \( n \)? |
Jeanne climbs a mountain with an elevation of 1,900 feet in 10 minutes. Later, she comes down the same path and loses 50 feet every minute. How many minutes in total did she climb the mountain? |
Suppose $f(x) = 3x^2 - 2$ and $g(f(x)) = 5x^2 + x + 2$. Find $g(11)$. |
In a discrete-time financial market model with discounted stock prices \( \{S_n, n = 0, 1, \dots, N\} \) and a risk-neutral martingale measure \( Q \), consider a contingent claim \( H \in L_2(Q) \). Find a constant \( c \) and a predictable trading strategy \( \gamma = (\gamma_n, n = 1, 2, \dots, N) \) such that the m... |
Find the solution to $x|x| = 3x - 2$ which has the smallest value. |
Suppose that an entire function \( f \) has exactly one curve \(\Gamma\) of maximum modulus (that is, \(\Gamma\) is connected, joins 0 to \(\infty\), and \( f \) has no other maximum modulus points). What can be said about the minimum rate of growth of \( M(r, f) \), if one is given information about the geometry of th... |
Determine all natural numbers \( n \) with two distinct positive divisors that are the same distance from \( \frac{n}{3} \). |
Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number? |
103 persons can repair a road in 22 days, working 8 hours a day. In how many days will 57 persons, working 10 hours a day, complete the work? |
For real numbers \(a, b,\) and \(c\), calculate the minimum value of the expression \(\sum_{\text{cyc}} \left(\frac{b}{a-b}\right)^2\). |
Two positive real numbers have arithmetic mean 4 and geometric mean 2. Enter the two numbers, separated by commas. |
If cos(2x) = 1/2, what is the measure of x in degrees for the interval 90 degrees less than or equal to x less than or equal to 270 degrees? |
When driving to a destination, one must first go three hours to the river and then drive the remaining distance to the destination. After 2 hours of traveling on the river, Malcom is stopped at a bridge for 5 minutes. When he leaves the bridge, he has 480 feet left to reach the destination. If Malcom's car is traveling... |
Find the set of numbers $k$ such that there exist infinitely many distinct positive integers $x, y$ such that $(x+i)(y+i)$ is a perfect square for each $i = 1, 2, \ldots, k$. |
Factorize $$P(x) = x^4 - 3x^3 + 5x^2 - x - 15$$ as a product of two quadratic polynomials with integer coefficients, and find the sum of squares of the coefficients of both factors. |
Let \( H \cong \mathbb{Z}_p \times \mathbb{Z}_p \) be a subgroup of a finite abelian group \( G \). Find all automorphisms in \( \text{Aut}(H) \) that can be extended to an automorphism in \( \text{Aut}(G) \). |
For which natural numbers does the equation
$$
x^{3}-y^{3}=x y+61
$$
hold? |
The exam consists of \(N \geqslant 3000\) questions. Each of the 31 students learned exactly 3000 of these questions, and any question is known by at least 29 students. Before the exam, the teacher placed all the question cards in a circle. He instructed the students to point to one of the questions and explained that... |
Given the sequence $\{a\_n\}$ that satisfies: $a\_1=1$, $a\_{n+1}+a\_n=(\frac{1}{3})^{n}$, $n∈N^{*}$, find $\lim\limits_{n→∞}a_{2n}$ $\_\_\_\_\_\_\_\_$. |
In pentagon $ABCDE$, where $AB=BC=CD=DE=2$ and $AE=1$, point $F$ is such that $ABFC$ is a cyclic quadrilateral. The segments $BF$ and $CF$ intersect the diagonal $AD$ at points $P$ and $Q$, respectively. If $PF \parallel AE$ and $FQ \parallel AE$, find $BF + CF$. |
Find all \( a > 0 \) such that the improper integral \(\int_0^\infty \frac{1}{1 + x^a (\sin x)^2} \, dx\) converges. |
Black and white balls are arranged in a circle, with black balls being twice as many as white balls. It is known that among the pairs of neighboring balls, there are three times as many pairs of the same color as there are pairs of different colors. What is the minimum number of balls that could have been arranged?
Re... |
An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there? |
Using a compass and straightedge, construct a circle that passes through a given point and is tangent to a given circle at a specific point on it. |
What is the derivative of $$\tan^{-1}\left(\frac{1 - x}{x + 2}\right)$$? |
The sequence of complex numbers $z_n$ satisfies
\[z_{n+1} - 3z_n + z_{n-1} = 2i^n\]for all integers $n.$ Find $z_5$ assuming that $z_0 = 1$ and $z_1 = 1 + i.$ |
1. Find all natural numbers $n \geq 10$ with non-zero digits, for which the following holds: if we remove any digit of the number $n$, we get a number that divides $n$. |
Suppose the null hypothesis, \( H_0 \), is: "Frank's rock climbing equipment is safe." The alternative hypothesis, \( H_a \), is that the equipment is not safe. What are the Type I and Type II errors in this context? |
A pumpkin weighs 2 and 9/10 of a kilogram (kg), and a watermelon weighs 41/10 kilograms (kg). Which is heavier, the pumpkin or the watermelon? |
What is the center of the circle with equation 2(x - 2)^2 + 2(y - 3)^2 = 32? |
For a sequence of integers \(a_1 < a_2 < \cdots < a_n\), a pair \((a_i, a_j)\) where \(1 \leq i < j \leq n\) is said to be balanced if the number \(\frac{a_i + a_j}{2}\) belongs to the sequence. For every natural number \(n \geq 3\), find the maximum possible number of balanced pairs in a sequence with \(n\) numbers. |
Find all quadruplets \((x, y, z, w)\) of positive real numbers that satisfy the following system:
\[
\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\
x+y+z+w= 48
\end{cases}
\] |
Solve the stochastic differential equation (SDE) \( dX = b(X,t)dt + \sigma(X,t)dB \), where \( b \) and \( \sigma \) are smooth coefficient functions. Discuss the solution under both the Itô and Stratonovich interpretations of the stochastic integral. |
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