problem stringlengths 16 3.48k | answer stringlengths 1 858 |
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Given that $a$ and $b$ are constants, $ab\neq 0$, and the maximum value of the function $f(x)=ax^{3}+b\arcsin x+3$ is $10$, determine the minimum value of $f(x)$. | -4 |
A bus travels at an average speed of 40 miles per hour. Calculate how many minutes longer a 520-mile trip would take compared to a 480-mile trip. | 60 |
$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possibl... | 1009 |
Compute
\[
\sum_{n = 3}^{200} \frac{2}{n \sqrt{n - 2} + (n - 2) \sqrt{n}}.
\] | \frac{19}{10} |
Determine the range of the function $y=\cos ^{2}x+\sin x-1$. | \left[-2, \dfrac {1}{4}\right] |
Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 3$. Find the minimum value of
\[
\frac{1}{(a + b)^2} + \frac{1}{(a + c)^2} + \frac{1}{(b + c)^2}.
\] | \frac{3}{2} |
Given the line $x+ \sqrt {3}y-2=0$ intersects the circle $x^{2}+y^{2}=4$, determine the length of chord $AB$. | 2\sqrt{3} |
Find \( \cos \frac{7\pi}{6} \). | -\frac{\sqrt{3}}{2} |
Quadrilateral $EFGH$ is a rhombus with a perimeter of $40$ meters. The length of diagonal $\overline{EG}$ is $16$ meters. Calculate the area of rhombus $EFGH$. | 96 |
3 square meters = square decimeters
2 hectares = square meters
5000 square centimeters = square decimeters
8 square kilometers = hectares. | 800 |
Given two lines $l_{1}$: $x+ay-a=0$ and $l_{2}$: $ax-(2a-3)y-1=0$ are perpendicular to each other, the value of $a$ is ____. | a=0 \quad \text{or} \quad a=2 |
Josh buys $5$ cookies and $3$ cupcakes, paying a total of $23$ dollars, and Emily purchases $4$ cookies and $4$ cupcakes and spends $21$ dollars total; assuming cookies and cupcakes have constant prices, determine the ratio of the price of a cupcake to the price of a cookie. | \frac{13}{29} |
In triangle $XYZ$, angle $XZY$ is 60 degrees, and angle $YZX$ is 80 degrees. Let $P$ be the foot of the perpendicular from $X$ to $ZY$, $O$ the center of the circle circumscribed about triangle $XYZ$, and $Q$ the other end of the diameter which goes through $X$. Find the angle $XPQ$, in degrees. | 20^\circ |
In triangle $XYZ,$ angle bisectors $\overline{XF}$ and $\overline{YG}$ meet at $Q.$ If $XY = 9,$ $XZ = 6,$ and $YZ = 4,$ calculate $\frac{YQ}{QG}.$ | 3.25 |
Calculate the sum $(-1)^{-11} + (-1)^{-10} + (-1)^{-9} + \cdots + (-1)^9 + (-1)^{10} + (-1)^{11}$. | 0 |
The circle $C$ has its center on the $x$-axis and passes through point $A(-1,1)$ and $B(1,3)$. What is the equation of circle $C$? | \text{The equation of circle } C \text{ is } (x-2)^{2}+y^{2}=10 |
Let the function $f(x) = e^x - e^{-x}$, and $g(x) = \log(mx^2 - x + \frac{1}{4})$. If for any $x_1 \in (-\infty, 0]$, there exists $x_2 \in \mathbb{R}$ such that $f(x_1) = g(x_2)$, determine the minimum value of the real number $m$. | -\frac{1}{3} |
Consider four segments that are cut off from a circle by an inscribed quadrilateral and are situated outside this quadrilateral. Find the sum of the angles inscribed in these segments. | 180^\circ |
Hannah is interested in numbers that are divisible by 4 but dislikes any number ending in 0. How many different units digits are possible in numbers that Hannah likes? | 4 |
Given that $x$ is a positive integer less than 150, how many solutions does the congruence $x + 17 \equiv 75 \pmod{46}$ have? | 3 |
Solve the following equations using appropriate methods:<br/>
(1) $x(x+2)=-3(x+2)$;<br/>
(2) $(2y-1)^{2}-5=0$;<br/>
(3) $(x-3)(x-1)=5$;<br/>
(4) $(x-2)^{2}-7(x-2)=18$. | x_{1}=11, \, x_{2}=0 |
Let \( y \) be a positive integer such that \( 7y \equiv 1 \pmod{31} \).
What is the remainder when \( 9 + y \) is divided by \( 31 \)? | 18 |
Given the proposition P: "For all $x \in \mathbb{R}$, there exists an $m \in \mathbb{R}$ such that $4^x - 2^{x+1} + m = 0$," if the negation of P, $\neg P$, is false, then determine the range of values for the real number $m$. | m \leq 1 |
Given a line $l$ passing through the point $(1, 4)$:
$(1)$ If line $l$ is parallel to the line $l_{1}: y = 2x$, find the equation of line $l$ and the distance between $l$ and $l_{1}$;
$(2)$ If the intercepts of line $l$ on the $x$-axis and the $y$-axis are both equal to $a$, and $a \neq 0$, find the value of $a$. | 5 |
In a factory, workers produce gadgets and gizmos. Each type of product requires a constant but different amount of time per worker. In one hour, 150 workers can produce 450 gadgets and 300 gizmos. In two hours, 90 workers can produce 360 gadgets and 450 gizmos. In four hours, 75 workers can produce 300 gadgets and $n$ ... | \frac{600}{7} |
When \( x = -2 \), evaluate the value of \( (x + 1)^{3} \). | -1 |
Passing through point $A(2,-3)$ and perpendicular to the line $l:x-2y-3=0$, the equation of the line is _____. (Please express in general form) | 2x+y-1=0 |
Mr. $A$ starts with $\textdollar{15000}$ and a house valued at $\textdollar{12000}$. Mr. $B$ starts with $\textdollar{13000}$. Mr. $A$ sells the house to Mr. $B$ for $\textdollar{14000}$. After a while, Mr. $B$ sells the house back to Mr. $A$ for $\textdollar{10000}$. Calculate the final amount of cash and analyze the ... | \textdollar{4000} |
Given two curves $f(x) = \cos x, g(x) = \sqrt{3}\sin x, x \in \left( 0, \frac{\pi}{2} \right)$ intersect at point $A$. If the tangents to the curves at point $A$ intersect the $x$-axis at points $B$ and $C$ respectively, then the length of segment $BC$ is. | \frac{4\sqrt{3}}{3} |
The terms $210, b, \frac{35}{36}$ are the first, second, and third terms, respectively, of a geometric sequence. If $b$ is positive, what is the value of $b$? | 14.29 |
Given $f(n) = n^2 \cos(n\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \cdots + a_{100}$. | -100 |
When $x$ is divided by each of $5$, $6$, and $7$, remainders of $4$, $5$, and $6$ (respectively) are obtained. What is the smallest possible positive integer value of $x$? | 209 |
Given the equation of the parabola $y^2 = -8x$, determine the coordinates of its focus. | (-2,0) |
Given the function $f^{(0)}(x) = \sin x$, and define the recursive function $f^{(n)}(x) = f'[f^{(n-1)}(x)]$, calculate the value of $f^{(1)}(15^{\circ}) + f^{(2)}(15^{\circ}) + f^{(3)}(15^{\circ}) + \ldots + f^{(2017)}(15^{\circ})$. | \dfrac{\sqrt{6} + \sqrt{2}}{4} |
Given $f(x) = 2x^2 + x - k$ and $g(x) = x^3 - 3x$, if for any $x_1 \in [-1, 3]$, there always exists an $x_0 \in [-1, 3]$ such that $f(x_1) \leq g(x_0)$ holds, then the range of real number $k$ is ______. | k \geq 3 |
If the arithmetic square root of a number is $8$, calculate the cube root of this number. | 4 |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, d x}{5 + 3 \sin x}
$$ | \frac{\pi - 5 \arctan 2 + 5 \arctan \frac{3}{4}}{6} |
A map represents a square-shaped estate with each side marked as 12 inches. If the scale of the map is 1 inch to 100 miles, determine the actual area of the estate in square miles.
A) 1200000 square miles
B) 1440000 square miles
C) 1600000 square miles
D) 1800000 square miles | The correct answer is B) 1440000 square miles. |
What is $\left(\frac{5}{6}\right)^4$? | \frac{625}{1296} |
A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines. | 200 |
Let $a$ and $b$ be the roots of $x^2 - 6x + 8 = 0.$ Compute
\[a^5 + a^3 b^3 + b^5.\] | -568 |
Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube. | 1059 |
How many five-digit positive integers are multiples of 5? | 18000 |
Given an even function $f(x)$ defined on $\mathbb{R}$ that satisfies the condition: $f(x+1) = -f(x)$, and it is an increasing function on the interval $[-1, 0]$. Consider the following statements about $f(x)$:
① $f(x)$ is a periodic function;
② $f(x)$ is an increasing function on the interval $[0, 1]$;
③ $f(x)$ i... | \text{①④} |
Find the minimum and maximum values of the function $y=2x^2-6x+1$ for $-1 \leq x \leq 1$. | -3; 9 |
Simplify $\sqrt[3]{3 \cdot 5} \cdot \sqrt{5^2 \cdot 3^4}$. | 15 |
What is the remainder when $4x^3 - 9x^2 + 12x - 14$ is divided by $2x-4$? | 6 |
Find the sum of the squares of all real numbers that satisfy \( x^{256}-256^{32}=0 \). | 8 |
Fill in the blank with the same natural number to make the equation true:
(□-□) + □×□ + □÷□ = 50. | 7 |
Find all real values of \(x\) that satisfy \(x + \frac{36}{x-5} = -9.\) | -9, \; 5 |
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the de... | 120^\circ |
Given points A, B, C, and D lie along a line in that order, and AB : AC = 1 : 5 and BC : CD = 2 : 1, calculate the ratio AB : CD. | 1 : 2 |
Reading and Thinking: If a pair of numbers $m$, $n$ satisfy $\frac{m}{2}+\frac{n}{5}=\frac{m+n}{2+5}$, we call this pair of numbers $m$, $n$ as "adjacent number pairs", denoted as $\left(m,n\right)$.
$(1)$ If $\left(2,n\right)$ is an "adjacent number pair", then $n=$______;
$(2)$ If $\left(m,n\right)$ is an "adjace... | n = -2 |
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$ . | (n, p) = (6, 3) |
For what real value of \(v\) is \(\frac{-26-\sqrt{450}}{10}\) a root of the quadratic equation \(8x^2 + 26x + v = 0\)? | \frac{113}{16} |
Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\] | \sqrt{2} |
Calculate:
1. $(-2)^{2} - (7-\pi)^{0} - \left(\frac{1}{3}\right)^{-1}$;
2. $2m^{3} \cdot 3m - (2m^{2})^{2} + \frac{m^{6}}{m^{2}}$;
3. $(a+1)^{2} + (a+1)(a-2)$;
4. $(x+y-1)(x-y-1)$. | x^{2} - 2x + 1 - y^{2} |
Given the following propositions:
① There does not exist real numbers $a$, $b$ such that the domain and range of $f(x) = \lg(x^2 + bx + c)$ are all real numbers;
② The graph of the function $y = f(x + 2)$ is symmetric to the graph of the function $y = f(2 - x)$ about the line $x = 2$;
③ The equation $\ln x + x = ... | ①③④ |
Rabbits are sawing a log. They made 10 cuts. How many pieces of wood did they get? | 11 |
What is $\frac{3}{4}$ of 48 minus 12? | 24 |
Let \[ f(x) =
\begin{cases}
-x^2 - 2x & \text{if } x \geq 0,\\
x+7& \text{if } x <0.
\end{cases}
\]Compute $f(f(f(f(f(2)))))$. | -41 |
To support the Hope Project in impoverished mountainous areas, a school organized students to prepare 1710 notebooks, 664 pens, and several sets of protractors. The students divided these learning supplies into three types of packages, labeled as A, B, and C, for mailing. Each A-type package contains 10 notebooks, 8 pe... | 680 |
Given the function $f(x) = x^5 - 2x^4 + 3x^3 - 7x^2 + 6x - 3$, calculate the function value at $x = 2$ using Horner's method, and find the result of the third step, $v_3$. | -1 |
A circle passing through the three points $A(1,3)$, $B(4,2)$, and $C(1,-7)$ intersects the $y$-axis at points $M$ and $N$. Find the length of $|MN|$. | 4\sqrt{6} |
Given the function $f(x)=\ln x+\frac{1}{2}x^2-ax$, where $a\in \mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$.
(I) When $a=3$, find the extreme values of the function $f(x)$.
(II) If $x_2\geqslant ex_1$ ($e$ is the base of the natural logarithm), find the maximum value of $f(x_2)-f(x_... | 1-\frac{e}{2}+\frac{1}{2e} |
Calculate the sum of $4321 + 3214 + 2143 + 1432$.
A) 11000
B) 11100
C) 11110
D) 11210
E) 11310 | C) 11110 |
Find the distance between the vertices of the hyperbola
\[\frac{x^2}{144} - \frac{y^2}{49} = 1.\] | 24 |
If the slope angle of the line $x+my-2=0$ is $30^\circ$, calculate the value of the real number $m$. | -\sqrt{3} |
Calculate the double integral
$$
\iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y
$$
where the region \(D\) is bounded by the lines \(x=1, y=x^{3}\), and \(y=-\sqrt{x}\). | 11 |
To cultivate students' interest in reading, a certain middle school organized a reading club on World Book Day, where students could share their favorite books. Xiaoying's favorite four books are "Journey to the West," "Romance of the Three Kingdoms," "How Steel Is Made," and "Education of Love." It is assumed that Xia... | \frac{1}{6} |
What is the radius of a circle inscribed in a rhombus with diagonals of length $14$ and $30$? | \frac{105\sqrt{274}}{274} |
Given the hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ $(a \gt 0, b \gt 0)$ with left and right foci $F_{1}$, $F_{2}$, and $|F_{1}F_{2}|=4$. A line $l:y=-\frac{\sqrt{3}}{3}x$ is perpendicular to one of the asymptotes of $C$. <br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ Let point $M$ be ... | -\frac{10}{3} |
Given the function $f(x) = x\sin x - \cos x$, find the value of $f'(-\frac{\pi}{2})$. | -2 |
Given that the derivative of the function $f(x)$ is $f′(x)$, $e$ is the base of the natural logarithm, and the function $f(x)$ satisfies $xf′(x)+f(x)= \frac{\ln x}{x}$, with $f(e)= \frac{1}{e}$, determine the solution set of the inequality $f(x)-x > \frac{1}{e}-e$. | (0,e) |
If $4(-3) = \Delta - 3$, then what does $\Delta$ equal? | -9 |
Given vectors $\overrightarrow {m}=(\sin x，-1)$ and $\overrightarrow {n}=( \sqrt {3}\cos x，- \frac {1}{2})$, and the function $f(x)= \overrightarrow {m}^{2}+ \overrightarrow {m}\cdot \overrightarrow {n}-2$.
(I) Find the maximum value of $f(x)$ and the set of values of $x$ at which the maximum is attained.
(II) Given th... | \frac{2\sqrt{3}}{3} |
Find all angles $\theta,$ $0 \le \theta \le 2 \pi,$ such that for all real numbers $x,$ $0 \le x \le 2,$
\[
x^2 \cos \theta - 2x(1 - x) + (2 - x)^2 \sin \theta > 0.
\] | \left(\frac{\pi}{12}, \frac{5\pi}{12}\right) |
Determine the sum of all single-digit replacements for $z$ such that the number ${35{,}z91}$ is divisible by 9. | 9 |
A certain triangle can be cut from a strip of paper of unit width, but cannot be cut from any strip of smaller width. What area can this triangle have? | \frac{1}{\sqrt{3}} |
Define the relationship \( \triangle \) by \( X \triangle Y = X^2 + 3Y^2 \). If \( 9 \triangle Y = 360 \), what is the positive value of \( Y \)? | Y = \sqrt{93} |
Given the sets $M={x|x^{2}=2}$ and $N={x|ax=1}$, if $N⊆M$, then the value of $a$ is _____. | 0,- \dfrac { \sqrt {2}}{2}, \dfrac { \sqrt {2}}{2} |
In the Cartesian coordinate system $xOy$, given two points $A(\cos 110^{\circ}, \sin 110^{\circ})$ and $B(\cos 50^{\circ}, \sin 50^{\circ})$, calculate the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$. | \frac{1}{2} |
Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$ 10.50 $ worth of cupcakes. Roy bakes cupcakes at a rate of $ 10$ cupcakes an hour. How many minutes will it take Roy to complete the order? | 90 \text{ minutes} |
Find the number of positive integers $x$ for which the inequality $\log_{10}(x-30) + \log_{10}(90-x) < 3$ holds. | 50 |
How many non-empty subsets $T$ of $\{1, 2, 3, \ldots, 20\}$ have the following characteristics?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $m$ elements, then $T$ contains no number less than $m+2$. | 1278 |
Consider a \(2 \times n\) grid of points and a path consisting of \(2n-1\) straight line segments connecting all these \(2n\) points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through once and no two line segments intersect. H... | \binom{4030}{2015} |
A ball is dropped from a height of 2000 feet and bounces back up to one-third the distance it just fell. After how many bounces will the ball first reach a maximum height less than 6 feet? | 6 |
Given the planar vectors $\overrightarrow{PA}, \overrightarrow{PB}$ that satisfy $|\overrightarrow{PA}| = |\overrightarrow{PB}| = 1, \overrightarrow{PA} \cdot \overrightarrow{PB} = -\frac{1}{2}$, if $|\overrightarrow{BC}| = 1$, find the maximum value of $|\overrightarrow{AC}|$. | \sqrt{3} + 1 |
Let the distance from point \( P \) to plane \( \alpha \) be \( \sqrt{3} \). Point \( Q \) is on plane \( \alpha \) such that the angle between line \( PQ \) and plane \( \alpha \) is no less than \( 30^\circ \) and no greater than \( 60^\circ \). The area of the region formed by such points \( Q \) is \(\qquad\) | 8\pi |
The function $f(x)$ is determined by a mapping $f$ from the vector set $\overrightarrow{A}$ to $\overrightarrow{A}$, and $f(x) = x - 2(x \cdot \overrightarrow{a})\overrightarrow{a}$. If there exists a non-zero constant vector $\overrightarrow{a}$ such that $f[f(x)] = f(x)$ always holds,
(1) find $|\overrightarrow{a}|... | (x - 1)^2 + (y + 2)^2 = \frac{1}{8} |
Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with the denominator equal to 3. | 9900 |
Given vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\|\mathbf{u}\| = 5,$ $\|\mathbf{v}\| = 7,$ and $\|\mathbf{u} + \mathbf{v}\| = 10.$ Find $\cos \phi,$ where $\phi$ is the angle between $\mathbf{u}$ and $\mathbf{v}.$ | \cos \phi = \frac{13}{35} |
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How man... | 22 |
Three counterfeit coins of equal weight are mixed with 12 identical genuine coins. The weight of each counterfeit coin is different from the weight of each genuine coin. Two pairs of coins are selected at random without replacement from the 15 coins, first a pair of two and then another pair of two from the remaining. ... | \frac{33}{91} |
Determine the range of the real number $m$ such that the proposition $p \vee q$ is true and $p \wedge q$ is false, given that $p$ is the proposition "The equation $x^2 + 2mx + 1 = 0$ has two distinct positive roots" and $q$ is the proposition "The equation $x^2 + 2(m-2)x - 3m + 10 = 0$ has no real roots". | (-\infty, -2] \cup [-1, 3) |
Let $x$ and $y$ be two distinct positive real numbers. Define the sequences $(A_n), (G_n), (H_n)$ as follows:
- $A_1, G_1,$ and $H_1$ are the arithmetic mean, geometric mean (of $x$ and $x+y$), and harmonic mean of $x$ and $y$, respectively.
- For $n \ge 2$, $A_n, G_n, H_n$ are the arithmetic mean, geometric mean, and ... | 81 |
Calculate \((18 \div (3 + 9 - 6)) \cdot 4\). | 12 |
Given the function $f(x)= \begin{cases} x+4 & x < 0 \\ x-4 & x > 0 \end{cases}$, find the value of $f[f(-3)]$. | -3 |
A rectangular prism has dimensions 5 inches by 4 inches by 40 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? | 600 |
The current time is 9 o'clock. On a 12-hour clock, what time will it be 2023 hours from now? | 8 |
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