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Consider a function $g(x) = \sqrt{ax+b}$, where $a$ and $b$ are nonzero constants. Find all values of $x$ for which the derivative of $g(x)$ is equal to $1$. Express your answer in terms of $a$ and/or $b$.
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that...
Determine the minimum number of integers that must be removed from the set of integers between 1 and 50,000, inclusive, so that the product of the remaining integers ends with '2225'.
For every polynomial of degree 45 with coefficients $1,2,3, \ldots, 46$ (in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers? Alexey Glebov
Find the number of solutions to the equation $x_1 + x_2 + \cdots + x_n = k$ in nonnegative integers.
The solution of the equation $2^{x}+3x=k$ is within the interval $[1,2)$. Then, the range of $k$ is \_\_\_\_\_\_\_\_.
Given a convex quadrilateral $ABCD$ where $\angle BAC = 20^\circ$, $\angle CAD = 60^\circ$, $\angle ADB = 50^\circ$, and $\angle BDC = 10^\circ$. Find $\angle ACB$.
Given a hyperbola $C$ with one of its foci on the line $l: 4x-3y+20=0$, and one of its asymptotes is parallel to $l$, and the foci of the hyperbola $C$ are on the $x$-axis, then the standard equation of the hyperbola $C$ is \_\_\_\_\_\_; the eccentricity is \_\_\_\_\_\_.
In the tetrahedron $ P-ABC $, $\angle APB = \angle BPC = \angle CPA = 90^{\circ}$, and the sum of all edge lengths is $ S $. What is the maximum volume of this tetrahedron?
If $a$ and $b$ are positive integers such that $(a^b)^a > (b^a)^b$, then find all possible values of $b-a$.
Given that $ p $ is a prime and $ a, b $ are positive integers such that $ p \mid a^3 + b^3 - ab(a + b) $, find the smallest possible value of $ p $ for which $ p^3 \mid (a + b)^p - (a^p + b^p) $ holds true for all such $ a $ and $ b $.
In an acute triangle $ABC$, point $D$ is on $BC$ such that $AD \perp BC$. Point $E$ is on $AC$ such that $\angle ABE = \angle EBC$. The intersection of $AD$ and $BE$ is $P$. Given that $AD = DC$ and $\sin \angle PBD = \frac{1}{\sqrt{5}}$, find the value of $\frac{PC}{BE}$.
The negation of the proposition "There exists $x\_0 > 1$ such that $x\_0^2 - x\_0 + 2016 > 0$" is $\_\_\_\_\_\_\_\_\_\_$.
What is the range of the function $y = \frac{x^2 - 5x + 6}{x-3}$? (Express your answer using interval notation.)
What is the smallest number of queens required to attack all of the squares of a $6$ by $6$ chessboard?
Exercise 3. 42 students are lined up. Paul gives each student a certain positive number of pebbles. It is assumed that each student has strictly more pebbles than their right neighbor (except for the student at the far right of the line). How many pebbles did Paul distribute in total, at a minimum?
Find all continuous functions $ f: \mathbb{R} \to \mathbb{R} $ such that \[ f\left(2x - \frac{f(x)}{3}\right) = 3x \] for all $ x \in \mathbb{R} $.
200 ml of saltwater with 5% salt concentration is mixed with 300 ml of saltwater with 8% salt concentration. What is the salt concentration of the mixture?
Given the second-order differential equation $ u''(t) + \omega^2 u(t) = 0 $, convert it into a system of first-order differential equations and apply the Forward Euler method to discretize the system.
The cycle time for trucks hauling concrete to a highway construction site is uniformly distributed over the interval (50, 70) minutes. Find the mean and variance of the cycle times for the trucks.
44. Use the method of inversion to deduce the Green's function for the Dirichlet problem in the case where $ D $ is a sphere of radius $ a $ with its center at the origin $ O $, assuming that the expression for the Green's function of a half-space is known.
Sketch the graph of the function $ f(x) = 2 + \frac{x}{x^2 - 1} $, $ x \neq \pm 1 $. Identify and label all important features, including vertical and horizontal asymptotes, axial intercepts, and any stationary or inflection points.
Classify the splittings of the short exact sequence $1 \rightarrow A \stackrel{\iota}{\rightarrow} G \stackrel{\pi}{\rightarrow} Q \rightarrow 1$, where $A$ is an abelian group, up to $A$-conjugacy.
Solve the equation $ 4x^2 = 81 $ for $ x $.
Evaluate the natural logarithm of $-e$ in the complex plane, expressing the result in the form $x + iy$.
Let $ a \in \mathbb{N} $. Consider the sequence $ a_n $ defined by $ a_0 = 0 $, $ a_1 = 1 $, and $ a_{n+1} = 2a_n + (a-1)a_{n-1} $. For a given prime $ q $, find the minimal $ a $ such that for all primes $ p \leq q $, $ p \mid a_p $, and for all primes $ p > q $, $ p \nmid a_p $.
The monotonic decreasing interval of the function $y=\log_{\frac{1}{2}}\|x-3\|$ is ______.
Let $ G $ be a finite group with irreducible characters $ \chi_i \in \text{Irr}(G) $ for $ i = 1, \ldots, h $. Define $ p_i = \frac{\chi_i(1)}{\|G\|} \sum_{s \in G} \chi_i(s^{-1}) s $. Calculate the value of $ \sum_{i=1}^h \text{Tr}(p_i^2) $ where $ \text{Tr} $ denotes the trace in the group algebra \( \mathb...
Determine the Galois group of the splitting field of the polynomial $ X^5 - 4X + 6 $ over $ \mathbb{Q} $.
Formulate the correct statement of the Künneth formula for the multiplicative operation (wedge product) on the de Rham cohomology of the product manifold $ M \times N $. Include the relationship between the wedge products on $ M $, $ N $, and $ M \times N $.
Tom dances 4 times a week for 2 hours at a time and continues this routine for 10 years. Meanwhile, James is training for the Olympics, dedicating 4 hours to training twice a day for all but 2 days each week. After a year of their respective activities, who trained more hours in total?
Evaluate the Riemann sums $ L_{4} $ (left endpoint) and $ R_{4} $ (right endpoint) for the function $ y = \ln(x^{2} + 1) $ over the interval $[0, e]$. Compare your answer with the exact answer if possible, or use a calculator to determine the answer.
In triangle $ABC$, let $O$ be the center of the circumscribed circle. Let $A', B', C'$ be the reflections of $A, B, C$ with respect to the opposite sides $BC, CA, AB$ respectively. Let $A_1, B_1, C_1$ be the intersection points of the lines $OA'$ and $BC$, $OB'$ and $AC$, $OC'$ and $AB$ respecti...
Consider a closed convex polygonal region $K$ and a point $X$ in the plane of $K$. Suppose a sequence of reflections in the sides of $K$ is performed on $X$. Let $N$ be the minimum number of reflections required such that the image of $X$ after these reflections lies inside $K$. If $K$ is a regular hexagon with side le...
Two positive integers are added together to get a two-digit number with the same digits. When these two numbers are multiplied, the result is a three-digit number with the same digits. What are the original two numbers?
In a class of 26 students, there are 12 boys. The number of female students who are taller than the average female student in the class is two less than the number of female students shorter than the average female student in the class. How many female students in the class are taller than the average female student?
Find the number of integers $n$ such that $2000 \leq n \leq 2010$ and $7$ divides $\left\lfloor \frac{\sqrt{2}}{4}(1+\sqrt{2})^n \right\rfloor$. Provide your answer as a single integer.
The polynomial $x^4+6x^3-20x^2-12x+9$ can be factored into the product of two quadratic polynomials, each of which has integer coefficients. If $p$ and $q$ are the constant terms of these two polynomials, and $r$ is the leading coefficient of one of the polynomials, then what is $p+q+r$?
10.35 Two people start walking towards each other from points $M$ and $N$, which are 72 miles apart. The first person walks at a speed of 4 miles per hour, while the second person walks 2 miles in the first hour, $2 \frac{1}{2}$ miles in the second hour, 3 miles in the third hour, and so on (forming an arithmetic seque...
Calculate the variance of the stochastic integral: $$X = \int_0^2 2t \, dW(t) + \int_4^6 W(t) \, dW(t)$$
Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that \[\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\mbox{.}\] Determine the set of elements of $(a_n)$.
Find another 4-digit number which is a (non-trivial) multiple of its reverse.
Determine $\theta$ such that for all roots $z$ of the equation \[z^6 - z^4 + z^2 - z + 1 = 0,\] the maximum imaginary part of a root can be expressed as $\sin \theta$, where $-90^\circ \leq \theta \leq 90^\circ$.
Find all polynomials $f \in \mathbb{R}[x]$ such that \[ f^2(x) = f(x^2) \quad \forall x \in \mathbb{R}. \]
Find all real functions $ f, g, h: \mathbf{R} \rightarrow \mathbf{R} $ such that for any real numbers $ x $ and $ y $, the following inequalities hold: $$ (x - y)f(x) + h(x) - xy + y^2 \leq h(y) \leq (x - y)g(x) + h(x) - xy + y^2. $$
Using the example of a telephone switchboard where calls arrive randomly in time, derive the Poisson distribution directly (without using the binomial distribution limit). Assume the probability of receiving a call in a small time interval $ \Delta t $ is $ \lambda \Delta t $, and derive the probability \( P_x(t) \...
Problem: Find the remainder when $x^6 + 7x^4 - 13x^3 + 17x^2 - 8x + 3$ is divided by $x^2 - 2x - 1$.
Find the range of the function \[f(x) = \frac{x^4 + x^2 + 1}{x^2 + x + 1}.\]
Consider a convex polygon $A_{1}A_{2}\ldots A_{n}$ and a point $M$ inside it. The lines $A_{i}M$ intersect the perimeter of the polygon a second time at points $B_{i}$. The polygon is called balanced if each side of the polygon contains exactly one of the points $B_{i}$ (strictly inside). Find all balanced po...
Let $ S $ be a regular surface without umbilical points. Suppose $ S $ is a minimal surface. Calculate the value of $ \lambda(p) $ at a point $ p \in S $ given that $ \langle dN_p(w_1), dN_p(w_2) \rangle_{N(p)} = \lambda(p) \langle w_1, w_2 \rangle_p $ for all $ w_1, w_2 \in T_p(S) $. Express your answer in...
If 10 people are to sit on a bench with 1000 seats, how many different ways are there for them to sit such that each person occupies a seat?
In a regional chess competition, each of the 5 participating teams consists of 8 players. Each team selects 3 players to represent them in a tournament, except the defending champion team, which selects 4 players. If each team takes turns hosting the tournament, how many different 16-player tournament teams are possibl...
Positive integers $a,b,c$ satisfy $b\leq c$ and $$\frac{(ab-1)(ac-1)}{bc}=2023.$$ Find all positive values of $c$.
Let $ M \subset \{1, 2, 3, \ldots, 2007\} $ be a set with the following property: Among every three numbers from $ M $, you can always choose two such that one is divisible by the other. What is the maximum number of elements $ M $ can contain?
The area under the curve $\sin 3x$ is revolved about the $x$-axis over an $2 \pi/3$ interval. What volume does it create when revolved about the $x$-axis?
What is the relationship between the infinite cardinal numbers $\aleph_0$ (the cardinality of $\mathbb{N}$) and $c$ (the cardinality of $\mathbb{R}$)?
For $ n \geq 2 $, suppose $ n $ points are chosen uniformly and independently at random on a circle of circumference 1. Calculate, as a function of $ n $, the expected minimum distance along the circumference between two points.
The number 890 has the following property: by changing any of its digits by 1 (either increasing or decreasing), one can obtain a number that is a multiple of 11. Find the smallest three-digit number with the same property.
Suppose $ A $ is an $ n \times n $ matrix with $ n $ distinct eigenvalues. What is the relationship between the minimal polynomial and the characteristic polynomial of $ A $?
If $\sin 2x = \frac{3}{5},$ then find $\cos (x + \pi).$
Compute the quadratic variation $[X,X]_t$ of the zero-mean Ornstein-Uhlenbeck process $(X_t)_{t\geq 0}$ given by the representation: \[ X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \]
Sketch a graph of the polar equation $ r = 3 + \sec(\theta) $. What type of curve does this equation represent, and what are its key features?
Find the weak formulation of the problem\[\begin{cases}\Delta^2 u = f & \text{in } \Omega, \\u = 0 & \text{on } \partial \Omega, \\\frac{\partial u}{\partial n} = 0 & \text{on } \partial \Omega,\end{cases}\]$\Omega \subset \mathbb{R}^2$ being a bounded open set with regular boundary $\partial \Omega$, \(\Delta^2 = ...
Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{S_1(n - 1)}{n^2}$, where $S_1(n - 1)$ is the harmonic number defined as $S_1(n - 1) = \sum_{j=1}^{n-1} \frac{1}{j}$.
A line $l$ passing through the right focus $F$ of the hyperbola $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) intersects $C$ at only one point. The equation of $l$ is $y = \sqrt{3}x - 4\sqrt{3}$. Find the focal length and the eccentricity of $C$. Return your final response as 'Final Answer: \boxed{...
Natural numbers $ a, b, c $ are chosen such that $ a < b < c $. It is also known that the system of equations $ 2x + y = 2025 $ and $ y = \|x - a\| + \|x - b\| + \|x - c\| $ has exactly one solution. Find the minimum possible value of $ c $.
Find the tangent space at any tangent point on $\mathbb{P}^{2}$, the projective plane.
Simplify the fraction $7-\dfrac{30+\sqrt29-13}{5}$.
In a scalene triangle $ ABC $, one of the angles is equal to the difference of the other two, and one of the angles is twice another angle. The angle bisectors of $ \angle A $, $ \angle B $, and $ \angle C $ intersect the circumcircle of the triangle at points $ L $, $ O $, and $ M $ respectively. Find th...
A box contains 24 identical-looking coins, and each coin has a different number of grams of gold in it. The total weight of gold in the coins is 432 grams. If the weight of gold in each coin is at most 20 grams and is a multiple of 3, what is the minimum number of coins that need to be taken out of the box to guarantee...
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $9:8$. Find the minimum possible value of their common perimeter. Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number or m...
Give an example of a function $ f \in L_2(\mathbb{R}^d) $ which is not in $ L_1(\mathbb{R}^d) $, but such that $ \hat{f} \in L_1(\mathbb{R}^d) $. Under what circumstances can this happen?
Given that the circumference of a sector is 6 cm, and its area is 2 cm2, then the radian measure of the central angle of the sector is.
Evaluate: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx \]
Sketch the phase diagram for the quadratic system $ \dot{x} = 2xy, \, \dot{y} = y^2 - x^2 $.
Let $x$ and $y$ be two vectors such that $x \times y = \begin{pmatrix} 4 \\ 1 \\ -3 \end{pmatrix}$ and $x \cdot y = 3$. Find the angle between $x$ and $y$.
Given a point in cylindrical coordinates $(r,\theta,z) = \left( 5, \frac{3\pi}{4}, 2 \right)$, find the rectangular coordinates of the point that is symmetric to it with respect to the $xy$-plane.
Find all functions $ f: (1, \infty) \to \mathbb{R} $ satisfying \[ f(x) - f(y) = (y - x) f(xy) \] for all $ x, y > 1 $.
Given positive real numbers $a$, $b$, and $c$ such that $21ab + 2bc + 8ca \leq 12$, find the least possible value of the expression $\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$.
Given two strictly positive functions $ f $ and $ g $ such that $ f(x) \sim g(x) $ as $ x \to +\infty $ (i.e., $ \lim_{x \to +\infty} \frac{f(x)}{g(x)} = 1 $), what condition must be satisfied to ensure that $ \log f(x) \sim \log g(x) $ as $ x \to +\infty $?
Return your final response within \boxed{}. Let $ A \cong \{0, 1, 2, \cdots, 29\} $ satisfy: for any integer $ k $ and any numbers $ a, b $ in $ A $ ( $ a $ and $ b $ can be the same), $ a + b + 30k $ is not the product of two consecutive integers. Determine all sets $ A $ with the maximum number of ele...
Evaluate the limit of the integral $\int_{0}^{a} \left(\frac{x^{2}}{2n} - 2 \pi \left\lfloor \frac{x^{2}}{4n \pi} \right\rfloor \right) \left(\frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^{2}} \right) dx$ as $a \to \infty$, where $n, \mu, \sigma \in \mathbb{R}^{+}$.
A train is moving at a speed of 8 miles per hour along a circular track with a radius of 2500 feet. Through what angle (in radians and degrees) does the train turn in one minute?
A glass with the form $ z = x^2 + y^2 $ and $ z \leq 1 $ is filled with water to a certain height $ h $. If the glass is tilted $ 45^\circ $, the water is just about to stream out. Determine $ h $.
Given a complete directed graph $ G = (V, A) $ with costs $ c_{ij} \geq 0 $ for all arcs $ (i, j) \in A $ that obey the triangle inequality, how can we obtain an $ O(\log n) $-approximation algorithm for the metric asymmetric traveling salesman problem via randomized rounding?
The average age of 20 students of a class is 20 years. Out of these, the average age of some students is 11 years and that of the other 10 students is 24 years. The age of the 20th student is 61 years. How many students have an average age of 11 years?
Evaluate the integral $ I = \frac{1}{2\pi i} \int_{\gamma} z^7 \cos \frac{1}{z^2} \, dz $ where $ \gamma $ is a positively oriented circle centered at the origin with radius $ R $.
Find the coordinates of the point where the line $3x - 2y = 5$ intersects the perpendicular bisector of the line segment joining the points $(-1, 3)$ and $(7, 1)$.
What is $a^{5 b^{-1}} \times a^{8 b^{-1}}$ simplified?
Find integers $ x $ and $ y $ that satisfy the equation: \[ 2(x + y) + xy = x^2 + y^2 \]
Evaluate the more general integral analytically: \[ \int_{0}^{+\infty} e^{i t (\omega - i \eta)} j_l(\mu t) j_m(\mu t) \, dt \] where $ j_l $ and $ j_m $ are spherical Bessel functions of the first kind of orders $ l $ and $ m $, respectively, and $\eta$ is a small positive number.
John orders food for a massive restaurant. He orders 1000 pounds of beef for a certain price per pound. He also orders twice that much chicken at $3 per pound. Everything cost $14,000. How much did John pay per pound of beef?
For real $\theta_i$ , $i = 1, 2, \dots, 2011$ , where $\theta_1 = \theta_{2012}$ , find the maximum value of the expression \[ \sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}. \] Proposed by Lewis Chen Return your final response as 'Final Answer: \boxed{<answer>}', where <answer> is the number o...
What is the algebra generated by the nilpotent matrices in $M_n(\mathbb{C})$?
Let $G$ be a finite group and $g \in G$ an element with order $o(g)$. If $r$ is a divisor of $o(g)$, what is the order of $g^r$?
A rectangular storage container with an open top is to have a volume of 10 m³. The length of its base is twice the width. Material for the base costs \$10 per square meter, and material for the sides costs \$6 per square meter. Find the cost of materials for the cheapest such container.
Find all continuous functions $ f: (0, \infty) \to (0, \infty) $ such that for all $ a, b, c, d \in (0, \infty) $, if $ \frac{a}{b} \geq \frac{c}{d} $, then $ \frac{f(a)}{f(b)} \geq \frac{f(c)}{f(d)} $.
Simplify $\dfrac{3+4i}{1+2i} \cdot \dfrac{1-2i}{1-2i}$ and express the result in the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary).
Find a monic quartic polynomial $Q(x)$ with rational coefficients such that \[Q(\sqrt[4]{2} + \sqrt[4]{3}) = 0.\]
The number $x$ is three times the number $y$, and $y$ is equal to half the number $z$. If $z$ is 6/5 times the number $w$, find the number that represents $x$ in terms of the value of $w$.

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