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Return your final response within \boxed{}. For which \( n \) can we find a cyclic shift \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) (i.e., \( i, i+1, i+2, \ldots, n, 1, 2, \ldots, i-1 \) for some \( i \)) and a permutation \( b_1, b_2, \ldots, b_n \) of \( 1, 2, 3, \ldots, n \) such that \( 1 + a_1 + b_1 =... |
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer ... |
Mr. Bodhi is transporting some animals using a yacht across a river. He has 20 cows, 15 foxes, and three times as many zebras as foxes. To balance the yacht to ensure a smooth sail across the river, the total number of animals in the yacht needs to be 100. If he decides to add sheep to the yacht to make the yacht sail-... |
Consider the number
\[\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots + \frac{1}{1000!}.\]If this expression were extended to include the denominator $2009!,$ would it be the smallest integer greater than this expression? |
Let $R$ be a reflection over a line. What single point is invariant under $R$? |
Let $a$, $b$, $c$ be the positive integers such that $a^2+b^2+c^2 = 2abc$.
Find the remainder when $a+b+c$ is divided by $4$. |
Julia played tag with 15 kids on Monday, 18 kids on Tuesday, and some kids on Wednesday. She played with 33 kids on Monday and Tuesday. How many kids did she play with on Wednesday? |
A circular fountain has a diameter of 20 feet. A path that is 3 feet wide is constructed around the fountain. What is the area of the path? |
Given the function $f(x)=\sin \left( {\omega x+\frac{\pi }{6} } \right)(\omega > 0)$, the distance between two adjacent symmetry axes of the function's graph is $\frac{\pi }{2}$, and the graph is symmetrical about the point $(x_{0},0)$, where $x_{0} \in \left[ 0,\frac{\pi }{2} \right]$. Find the value of $x_{0}$. |
For the complex number $$\frac {2i}{1-i}=a+bi$$ (where $i$ is the imaginary unit, and $a, b\in \mathbb{R}$), find the values of $a$ and $b$. |
Find the image of the half-plane $\text{Re}(z) + \text{Im}(z) > 0$ under the conformal mapping $w = \frac{z-1}{z+i}$. Determine whether the image satisfies $|w| > 1$ or $|w| < 1$. |
In the arithmetic sequence $\{a\_n\}$, the arithmetic mean of $a\_2$ and $a\_6$ is $5 \sqrt {3}$, and the arithmetic mean of $a\_3$ and $a\_7$ is $7 \sqrt {3}$. Find the value of $a\_4$. |
Define the relation "less than" as a subset of a Cartesian product. |
Determine the complexity of the recurrence relation \( T(n) = T(n - \sqrt{n}) + n \). |
Suppose that for any function $f(x)$ defined on an interval $A$, and for all $a, b, c \in A$, $f(a), f(b), f(c)$ represent the lengths of the sides of a triangle. In this case, we call $f(x)$ a "triangle function." Given that the function $f(x) = x \ln x + m$ is a "triangle function" on the interval $\left[ \frac{1}{e^... |
Minimize the function \( \| (2x - y, x + y)^T \|^2 - 6x \) over all \( x, y \), where the norm is defined as:
(a) The weighted norm \( \| (x, y)^T \| = \sqrt{2x^2 + 3y^2} \);
(b) The norm based on the matrix \( \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix} \);
(c) The norm based on the matrix \( \begin{pmatrix} ... |
Return your final response within \boxed{}. Let $a_1$ , $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$ .
Call a nonempty contiguous subsequence of this sequence *good* if the product of the integers in it leaves a remainder of $1$ when divided by $101$ .
In other words, it is a pair of in... |
Three listers are planing to catch fish in increasing order of birth order. Calvin was the first, Andy was the second, and Phoebe was the third. If Andy catches four inside his bag, Calvin catches twice as many as Andy, and Phoebe catches 3 more inside her bag. How many total fish does Phoebe catch? |
Determine $p(x)$ such that \[4x^4 + 2x^3 - 6x + 4 + p(x) = 2x^4 + 5x^2 - 8x + 6.\] |
Find the function $f(n)$ defined for all integers $n$ such that $f(x) + f(y) = f(x + y) - 2xy - 1$ for all integers $x$ and $y$, and $f(1) = 1$. Determine $f(n)$. |
Suppose that you invest $13,000 at 9% interest compounded continuously. How much money will be in your account in 8 years? |
If $a$ represents the number of cars in Albert's garage and $b$ represents the number of bicycles in that garage, what is the total number of wheels in the garage? |
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$ . Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$ , define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way:
(a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$ .
(b) ... |
Graph the circle given by the equation \( x^2 - 6x + y^2 - 8y = 0 \). To do this, first rewrite the equation in the standard form of a circle by completing the square for both \( x \) and \( y \), then identify the center and radius, and finally sketch the graph. |
Suppose we have a ring of $n$ nodes, each of which holds a bit. At each step, we pick a node uniformly at random and have it copy the bit of its left neighbor. Suppose $n$ is even and we start with a state where positions 1 through $n/2$ are all 0 and positions $n/2 + 1$ through $n$ are all 1. What is the expected numb... |
Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + 3 $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + X $. The distance between $A$ and $B$ is 5. What is the value of unknown variable X? |
In triangle \( K M N \), the altitude \( N A \), the angle bisector \( N B \), and the median \( N C \) divide the angle \( K N M \) into four equal parts. Find the lengths of the altitude \( N A \), the angle bisector \( N B \), and the median \( N C \), if the circumradius of the triangle \( K M N \) is given as \( R... |
Given a geometric sequence $\{a_{n}\}$ that satisfies $2a_{2}+a_{3}=a_{4}$ and $(a_{2}+1)(a_{3}+1)=a_{5}-1$, then the value of $a_{1}$ cannot be:
A: $-2$
B: $\frac{1}{4}$
C: $1$
D: $2$
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the sol... |
Form the duals of the following Boolean equations:
(a) \( p \land (p' \lor q) = p \land q \)
(b) \( (p \lor q) \land (p' \land q') = (p \land (p' \land q')) \lor (p' \land (q' \land q)) \) |
Find the Laplace transform of \( f(t) = t \sin(bt) \) using the properties of Laplace transforms. |
Find the inverse matrix \( A^{-1} \) for the matrix \( A = \begin{bmatrix} 20 & 5 \\ 6 & 2 \end{bmatrix} \). |
Given that the vertex of angle \\(θ\\) is at the origin of coordinates, its initial side coincides with the positive half-axis of \\(x\\), and its terminal side is on the line \\(4x+3y=0\\), then \\( \dfrac{\cos \left( \left. \dfrac{π}{2}+θ \right. \right)-\sin (-π-θ)}{\cos \left( \left. \dfrac{11π}{2}-θ \right. \right... |
The area under the curve of the function $f(x)=\frac{1}{x^2+1}$ between $x=0$ and $x=k$ is equal to $\frac{\pi}{2}$. Determine $k$. |
John wants to earn $45 in two weeks by selling crafts in a local league. In the past two months, he had earnings that were seven times less than what he wants to earn now. How much did John make on average in a week for the last two months? |
The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ represent the edges of a parallelepiped. Find the area of the triangle formed by the vectors $\mathbf{b} \times \mathbf{c}$ and ... |
Simplify the following expression: $3[2x+(5(3x-4))]$. |
Determine the points on the paraboloid $z = x^2 + 2y^2$ where the tangent plane is parallel to the plane $2x + 4y - z = 5$. Provide the coordinates of these points. |
Calculate the principal curvatures of the helicoid parametrized by \(\sigma(u, v) = (\cos v \sinh u, \sin v \sinh u, v)\) and the catenoid parametrized by \(\sigma(u, v) = (\cosh u \cos v, \cosh u \sin v, u)\). |
$8[x]$ represents the greatest integer not exceeding the real number $x$, then $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+$ $\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2012\right]=$ $\qquad$ . |
There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$? |
Consider the curve \( C: Y^2 = 2 - 2pX^4 \), where \( p \equiv 1 \mod 8 \). What can be said about the existence of points on \( C \) over the fields \( \mathbb{Q} \), \( \mathbb{R} \), and \( \mathbb{Q}_p \) for all primes \( p \)? |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. |
Gina is 2 years older than Jim. If Jim is 18, answer these questions: What is the sum of Jim's and Gina's ages? What is the difference, in years, between the sum Jim's age in 4 years and Gina's age 5 years ago? |
A person ordered a fence with a total length of 297 meters. The fence should consist of 16 sections, each containing an integer number of meters. Eight sections must have the maximum length, while the others should be shorter by 1, 2, or 3 meters. How should this order be fulfilled? Suppose that the 8 sections of maxim... |
A factory produces 800 suits per decade (10 years) of operation. If $50\%$ of that amount are JCC tank tops, $20\%$ are open shirts and $30\%$ are closed shirts, how many suits are each category if they continue snaking at the current rate of production rate? |
Let \( X = [0, 1]^n \) be the \( n \)-dimensional torus (\( n \in \mathbb{N} \)), and let \( S \) be the shift \( x \mapsto y = S(x) \) defined by the formula \( y_k = (x_k + \alpha_k) \mod 1 \), \( k = 1, \ldots, n \). For what values of \( \alpha_1, \ldots, \alpha_n \) is the shift ergodic? |
Let $f(x)$ be a real-valued function defined on the interval $(-5, 5)$ such that $$e^{-x}f(x) = 2 + \int\limits_0^x (t^4 + 1)^{1/2} \ dt$$ for all $x \in (-5, 5)$. Find the value of $(f^{-1})^{'}(2)$. \(\) |
Axiomatize the theory \( T \) of discrete linear orderings with first but no last element in which every element other than the first has an immediate predecessor and an immediate successor, using the language \( L(=, \leq) \). |
Compute the dot product of two vectors $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 7 \\ -2 \end{pmatrix}$, but this time, one of the vectors is expressed in polar coordinates as $\begin{pmatrix} 5 \text{cis} 30^\circ \end{pmatrix}$. Can you find the dot product using this new representation? |
Given the sequence defined by \( x_0 = 10^{-3} \) and \( x_{n+1} = x_n + x_n^2 (\ln x_n)^{\frac{1}{3}} \), find the equivalent of \( x_n \) as \( n \to \infty \). |
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$ , there is a circle passing through all four or a line passing through some three.
*Carl Lian.* |
Find all triangles such that their side lengths, area, and angles (in degrees) are all rational. |
Solve the equation \( f^{-1}(g(x))=h(x) \), where \( f^{-1}(x) \) is the inverse function of \( f(x) \), given that \( f(x)=x^{3}+2x^{2}+3x+1 \), \( g(x)=x^{4}-x^{3}+4x^{2}+8x+8 \), \( h(x)=x+1 \). |
A certain point in the plane is translated 4 units to the right, then rotated 90 degrees about the origin, translated 5 units up, rotated 90 degrees about the origin, and finally translated 3 units to the left. Find the original position of the point if it ends up at the coordinates (7, -3). |
Expand the function \( f(z) = \sin z \) into a Taylor series around the point \( z = 3 \) (in terms of powers of \( z-3 \) ), using known expansions. Specify the region where the resulting series is valid. |
Given that the coordinates of a point on the terminal side of angle $α$ are $(\sin \frac{2π}{3},\cos \frac{2π}{3})$, determine the smallest positive value of angle $α$. |
Given real numbers \(a, b, c\) satisfying the equations \(a + b + c = 0\) and \(abc = 8\), determine the value of \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\):
(A) Positive
(B) Zero
(C) Negative
(D) Cannot determine if positive or negative. |
Return your final response within \boxed{}. Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$ for some polynomials $f(x)$ and $g(x)$ then find the smallest po... |
The range of the function $f(x) = \frac{3x+1}{2-x}$ is \_\_\_\_\_\_. |
What is the area, in square feet, of an isosceles trapezoid with a base of 8 feet, a smaller base of 3 feet, and legs of 5 feet? |
Find all positive real values of $x$ such that
\[\frac{\sqrt{1-2x^2}}{\sqrt{1+x^2}} + \frac{\sqrt{1+2x^2}}{\sqrt{1-x^2}} = 2.\] |
Timmy wants to buy four bottles of water and four bags of chips for an upcoming road trip. Each bottle of water costs 50 cents and each bag of chips cost $2. How much will he spend on groceries? |
Let \( f \) be a real-valued function on \( (0,1) \) that is differentiable almost everywhere with respect to Lebesgue measure on \( (0,1) \) and \( |f'| \leq M \) wherever \( f' \) is defined. Suppose \( f \) is unbounded. Calculate the infimum of all possible values of \( M \) for which such an unbounded function \( ... |
Return your final response within \boxed{}. Assume that \( z = a + b i \) is a solution to the polynomial equation
\[
c_{4} z^{4} + i c_{3} z^{3} + c_{2} z^{2} + i c_{1} z + c_{0} = 0
\]
where \( c_{0}, c_{1}, c_{2}, c_{3}, a \) and \( b \) are real constants, and \( i^{2} = -1 \). Which of the following is also a solu... |
Matthew gets between 500ml and 550ml of water in the mountains. He drinks half of that. What range of water does he consume? |
Solve the system of equations in $\mathbb{R}^2$:
\[
\left\{
\begin{array}{l}
\sqrt{\sin^2 x + \frac{1}{\sin^2 x}} + \sqrt{\cos^2 y + \frac{1}{\cos^2 y}} = \sqrt{\frac{20y}{x+y}} \\
\sqrt{\sin^2 y + \frac{1}{\sin^2 y}} + \sqrt{\cos^2 x + \frac{1}{\cos^2 x}} = \sqrt{\frac{20x}{x+y}}
\end{array}
\right.
\] |
Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, ... |
Let \[f(x) = \left\{ \begin{aligned} x^2-4 & \quad \text{ if } x < 2 \\ x^2-4 & \quad \text{ if } x \ge 2 \end{aligned} \right.\]Find all solutions to the equation $f(x) = 0.$ |
The solution set of the inequality $\frac{x-3}{x}≥0$ with respect to $x$ is ______. |
Compute
\[(3 \cos 50^\circ - i \sin 50^\circ)^{12}.\] |
What is the Gibbs energy change, ΔG*, with nucleation radius, R*, expressed in terms of the surface tension of the liquid (σ), the molecular volume (v), the Boltzmann constant (k), the temperature (T), and the supersaturation ratio (S)? |
If the curve $$y= \sqrt {1-x^{2}}$$ and the line $y=k(x-1)+1$ have two common points, then the range of the real number $k$ is \_\_\_\_\_\_.
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of the solution. |
On the faces \(BCD, ACD, ABD,\) and \(ABC\) of a tetrahedron \(ABCD\), points \(A_1, B_1, C_1,\) and \(D_1\) are marked, respectively. It is known that the lines \(AA_1, BB_1, CC_1,\) and \(DD_1\) intersect at point \(P\) with \(\frac{AP}{A_1P} = \frac{BP}{B_1P} = \frac{CP}{C_1P} = \frac{DP}{D_1P} = r\). Find all possi... |
In a mathematics competition, two students are tasked with solving problems involving trigonometric identities. Student A is given that $\cos \theta = \frac{1}{3}$ and is asked to find $\cos 5\theta$. Meanwhile, Student B is given that $\cos \theta = \frac{1}{4}$ and must calculate $\cos 3\theta$. After they finish, th... |
An equilateral triangle is inscribed in a circle. The area of the circle is $49\pi$ square units. How many square units are in the area of the triangle? Express your answer in simplest radical form. |
What is the largest possible volume of a rectangular prism with a fixed surface area of 300 square units? |
Evaluate the definite integral \( \int_0^{\pi/2} \frac{1}{1+\cot^\beta{x}}\,\mathrm{d}x \), where \( \beta \) is a positive constant. |
Find the least positive integer $M$ such that $M^{49} \equiv 21 \pmod{209}$. \( \text{(Note: 209 = 11 * 19)} \) |
Let $G$ be a non-abelian group of order $36$ and $H$ be a subgroup of $G$ with order 4. Determine whether $Z(H) = Z(G)$. |
On the number line, there are ____ points that are 3 units away from the point representing $-2$. The numbers represented by these points are ____. |
Let \(ABC\) be a triangle with orthocenter \(H\) and incenter \(I\). \(M\) is the midpoint of \(BC\). \(N\) is the symmetric point of \(I\) through \(M\). \(D\), \(E\), and \(F\) are the projections of \(N\) on lines \(BC\), \(CH\), and \(HB\), respectively. Let \(O\) be the circumcenter of triangle \(DEF\). If the cir... |
**5/1/32.** Find all pairs of rational numbers $(a, b)$ such that $0 < a < b$ and $a^a = b^b$ . |
## Task 8
Draw a rectangle and a square!
Write down how these two quadrilaterals differ from each other! |
Solve the equation $x^2 +160x + 25 = 0$. |
Find the smallest positive integer $a$ that satisfies the following condition: there exists a positive odd number $n$, such that $203^{n} + a \cdot 198^{n}$ is a multiple of 2005. |
Factor out the greatest common monomial factor.
$$2x+14y$$ |
A parallelogram has sides of 15 cm and 7 cm, and the angle between the sides is $90^\circ$. A semicircle of diameter 14 cm is cut off from the parallelogram. What is the area of the remaining region? |
How many natural numbers \( \mathrm{N} \) greater than 900 exist such that among the numbers \( 3\mathrm{N}, \mathrm{N} - 900, \mathrm{N} + 15, 2\mathrm{N} \) exactly two are four-digit numbers?
Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or mathematical expression of t... |
It is known that the sequence is given by $a_n = \frac{n - \sqrt{98}}{n - \sqrt{99}}$. Determine the largest and smallest terms among the first 30 terms of this sequence. |
Verify the infinite speed of propagation associated with diffusion-like evolution by showing that (a) for each \( t > 0 \) and all \( x, \, u(x, t) \) depends on all the initial data \( F(x), -\infty < x < \infty \); (b) for the particular data\n\n\[\nF(x) = \n\begin{cases} \n1 & |x| < \epsilon \\\n0 & |x| \geq \epsilo... |
Let \( k, m, n \in \mathbb{N} \) and \( k \leq m \). Define the sequence \( a_n(m, k) \) as follows:
\[ a_n(m, k) = k^n \sum_{i_1=k}^{m} \frac{1}{i_1} \sum_{i_2=i_1}^{m} \frac{1}{i_2} \cdots \sum_{i_n=i_{n-1}}^{m} \frac{1}{i_n} \]
Find the limit:
\[ \lim_{n \to \infty} a_n(m, k) = ? \] |
Given a line $m$ intersects with plane $\alpha$ but is not perpendicular to it, which of the following statements is correct? ( )
A: There is exactly one line within plane $\alpha$ that is perpendicular to line $m$.
B: There is exactly one plane through line $m$ that is perpendicular to plane $\alpha$.
C: A line perpen... |
Return your final response within \boxed{}. A group of adventurers is displaying their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. Additionally, the following conditions are known:
- If an adventurer has sapphires, then they ha... |
Suppose $f(x) = 2^x$, and $g(x)$ is a polynomial such that $f(g(x)) = x^2 + x + 1$. Determine all possible polynomials $g(x)$. |
Consider a rational function of the form:
$$f(x) = \frac{1}{(x^2-4)(x^2-9)}$$
Can you express $f(x)$ as a sum of two rational functions with quadratic denominators and find the coefficients of the resulting decomposition? |
How many different strings of length \( n \) can be made using the characters A, B, and C, such that the number of A's, B's, and C's all have the same parity as \( n \) (i.e., all odd or all even)? |
April 05, 1978; James E. West (Ithaca, NY)\nLet \( E \overset{p}{\rightarrow} B \) be a Hurewicz fibration with each fiber homeomorphic to a given Hilbert cube manifold \( F \).\nIf \( E \) and \( B \) are also Hilbert cube manifolds, under what conditions is \( p \) a locally trivial bundle? |
The equilateral triangles $ABF$ and $CAG$ are constructed exterior to a right-angled triangle $ABC$ with $\angle C = 90^\circ$. Let $M$ be the midpoint of $BC$. Given that $MF = 11$ and $MG = 7$, find the length of $BC$. |
Return your final response within \boxed{}. Let \( n \) be a fixed positive integer, and choose \( n \) positive integers \( a_{1}, \ldots, a_{n} \). Given a permutation \( \pi \) on the first \( n \) positive integers, let \( S_{\pi}=\left\{i \left\lvert\, \frac{a_{i}}{\pi(i)}\right.\right. \text{ is an integer} \} \)... |
Return your final response within \boxed{}. Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$ . Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$ . Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$ . If the ra... |
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