task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | The range of values of $x$ that make the fraction $\frac{7}{{x-2}}$ meaningful is what set of real numbers? | x\neq2 | 31 | 5 |
math | Determine the number of $x$-intercepts and $y$-intercepts for the graph of the equation $y = -x^2 + 3x - 2$. | 1 | 39 | 1 |
math | 1. Write the angle -1120° in the form of 2kπ+α (k∈Z), where 0≤α<2π.
2. Write the set of angles β that have the same terminal side as the angle α in part 1, and write the angle β in the interval [-4π, 0]. | -\frac{20π}{9} | 73 | 9 |
math | Given the function $f(x)=x^{3}+ax^{2}+b$, where $a,b \in \mathbb{R}$.
1. Discuss the monotonicity of $f(x)$.
2. If $b=c-a$, where $c$ is a constant unrelated to $a$, and the function $f(x)$ has three distinct zeros, the range of $a$ is exactly $(-∞,-3)∪(1, \frac {3}{2})∪( \frac {3}{2},+∞)$. Find the value of $c$. | c=1 | 122 | 3 |
math | For all composite integers $n$, what is the largest integer that always divides into the difference between $n^2$ and the cube of $n^2$? | 6 | 34 | 1 |
math | Given the function $f(x) = \ln x - mx, m \in \mathbb{R}$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ If $f(x) \leqslant \frac{m-1}{x} - 2m + 1$ holds true over the interval $[1, +\infty)$, find the range of values for the positive real number $m$. | [\frac{1}{2}, +\infty) | 98 | 12 |
math | The base of a pyramid is a triangle with sides of lengths \(a\), \(a\), and \(b\). All lateral edges are inclined at an angle of \(60^{\circ}\) to the plane of the base. Determine the volume of the pyramid. | \frac{a^2 b \sqrt{3}}{12} | 56 | 16 |
math | How many perfect squares less than 5000 have a ones digit of 1, 5, or 6? | 35 | 26 | 2 |
math | Given the function $f(x)=e^{x}(x^{3}-3x+3)-ae^{x}-x$, where $e$ is the base of the natural logarithm, find the minimum value of the real number $a$ such that the inequality $f(x)\leqslant 0$ has solutions in the interval $x\in\[-2,+\infty)$. | 1-\frac{1}{e} | 83 | 8 |
math | 6 × 111 - 2 × 111 | 444 | 14 | 3 |
math | Given the function $y=2\sin \left(2x-\dfrac{\pi }{3}\right)$, determine the horizontal shift required to obtain this graph from the graph of $y=2\sin 2x$. | \dfrac{\pi}{6} | 49 | 7 |
math | Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\circ$ clockwise around vertex $D$, then it is rotated $90^\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by point $A$?
A) $\frac{3\pi(\sqrt{17} + 6)}{2}$
... | \frac{\pi(\sqrt{34} + 5)}{2} | 161 | 17 |
math | Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=60^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta = m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$ | 0 | 109 | 1 |
math | Consider each positive integer $n$, let $g_1(n)$ be thrice the number of positive integer divisors of $n$ raised to the power of 2, and for $j \ge 2$, let $g_j(n) = g_1(g_{j-1}(n))$. Determine for how many values of $n \le 30$ is $g_{50}(n) = 243$. | 0 | 93 | 1 |
math | Given a positive integer $n$ not exceeding $100$ such that if $n \leq 60$, the probability of choosing $n$ is $p$, and if $n > 60$, the probability of choosing $n$ is $2p$. Find the probability that a randomly chosen integer is a perfect square. | \frac{3}{35} | 71 | 8 |
math | Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters, and find the total number of different arrangements. | 12 | 47 | 2 |
math | In the sequence $5, 8, 15, 18, 25, 28, \cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$) | 202 | 78 | 3 |
math | For what values of the constant $k$ does the graph of $g(x) = \frac{x^2 - 3x + k}{x^2 - 2x - 8}$ have exactly two vertical asymptotes? | k \neq -4 \text{ and } k \neq -10 | 48 | 18 |
math | The difference between the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}-\frac{\pi }{3} \right)(0\leqslant x\leqslant 9)$ is ______. | 2+ \sqrt{3} | 57 | 7 |
math | Let \( ABCD \) be a convex quadrilateral with \( AC = 7 \) and \( BD = 17 \). Let \( M, P, N, Q \) be the midpoints of sides \( AB, BC, CD, DA \) respectively. Compute \( MN^{2} + PQ^{2} \). | 169 | 70 | 3 |
math | Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with the next player from the opposing team. T... | 3432 | 97 | 4 |
math | Find the sum of all positive integers $b < 1000$ such that the base- $b$ integer $36_{b}$ is a perfect square and the base- $b$ integer $27_{b}$ is a perfect cube. | 371 | 54 | 3 |
math | Six equilateral triangles, each with a side length of $3$, are arranged such that they are all on the same side of a line. This line contains one side of each triangle, where the midpoint of the base of one triangle is a vertex of the next. Calculate the area of the region that is covered by the union of the six triang... | \frac{171\sqrt{3}}{16} | 72 | 15 |
math | Find the number of ordered pairs $(x, y)$ of real numbers such that
\[9^{x^2 + y} + 9^{x + y^2} = \sqrt{2}.\] | 1 | 44 | 1 |
math | (The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row,
(1) The number of all different arrangements;
(2) The number of arrangements where exactly two male students are adjacent;
(3) The number of arrangements where male students are of different heights and are ar... | 20 | 89 | 2 |
math | In the sequence $\{a_n\}$, $a_n+a_{n+1}+a_{n+2}=(\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\{a_n\}$ (express the answer as a numerical value). | 4+9\sqrt{2} | 64 | 8 |
math | Form a four-digit number using the digits 1, 2, and 3, with the rule that all three digits must be used, but the same digit cannot be adjacent. Calculate the total number of such four-digit numbers. | 18 | 47 | 2 |
math | Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $a\_3=5$, $a\_5=9$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $S\_n$, $S\_n=2^{n+1}-2 (n∈N^{})$.
(1) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(2) If $c\_n=a\_n⋅b\_n (n∈N^{})$, and $T\_n$ is the sum of... | (2n-3)2^{n+1}+6 | 158 | 14 |
math | There are several square paper sheets of type $A$ with side length $a$, square paper sheets of type $B$ with side length $b$, and rectangular paper sheets of type $C$ with dimensions $a$ and $b$. Xiao Ming selected $2$ sheets of type $A$, $3$ sheets of type $B$, and $7$ sheets of type $C$ to form a rectangle. The perim... | 6a + 8b | 106 | 6 |
math | Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $( \overrightarrow {a}-2 \overrightarrow {b})⊥(3 \overrightarrow {a}+ \overrightarrow {b})$, and $| \overrightarrow {a}|= \frac {1}{2}| \overrightarrow {b}|$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac {2\pi}{3} | 91 | 9 |
math | The game of Rorrim 2 is played on a $4 \times 4$ board, starting with a counter in one corner. At each turn, the player moves the counter to a cell that is the reflection of its current cell in one of the six dashed lines. How many cells could the counter occupy after precisely three turns?
A) 4
B) 6
C) 8
D) 12 ... | 8 | 95 | 1 |
math | In a class of 40 students, it is known that $\frac{1}{2}$ of the students have a dog, $\frac{2}{5}$ have a cat, 8 students have a different type of pet, and 7 students have no pets at all. The Venn diagram details are as follows: 12 students only have dogs, 3 students have both dogs and other pets but no cats, and 11 s... | 5 | 107 | 1 |
math | Given proposition $P$: If the radius of the inscribed circle of a triangle is $r$, and the lengths of its sides are $a$, $b$, and $c$, then the area of the triangle $S= \dfrac{1}{2}r(a+b+c)$. Try to, inspired by proposition $P$, propose a proposition $Q$ about a tetrahedron. | V= \dfrac{1}{3}R(S_{1}+S_{2}+S_{3}+S_{4}) | 82 | 30 |
math | Given a square in the coordinate plane with vertices at $(0, 0), (3030, 0), (3030, 3030),$ and $(0, 3030)$, a point is chosen at random within the square. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Determine $d$ to the nearest tenth. | 0.3 | 92 | 3 |
math | Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 7.$ Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\] | 9 | 67 | 1 |
math | Define the sequence $(b_n)$ as follows: $b_0 = 1$, $b_1 = \sqrt[17]{3}$, and $b_n = b_{n-1}^3b_{n-2}^2$ for $n \geq 2$. Determine the smallest positive integer $k$ such that the product $b_1b_2 \cdots b_k$ is an integer. | k = 5 | 92 | 4 |
math | Calculate the value of $\sin30^\circ\sin75^\circ+\sin60^\circ\sin15^\circ$. | \frac{\sqrt{2}}{2} | 29 | 10 |
math | Given that $2^{1}×1=2$, $2^{2}×1×3=3×4$, $2^{3}×1×3×5=4×5×6$, and so on, determine the equation for the 5th term in the sequence. | 2^{5}×1×3×5×7×9=6×7×8×9×10 | 61 | 25 |
math | Let $f(x) = e^{x}(x-1)$ and $g(x) = mx$. If for any $x_{1} \in [-2,2]$, there exists $x_{2} \in [1,2]$ such that $f(x_{1}) > g(x_{2})$, then the range of real number $m$ is ______. | (-\infty, -\frac{1}{2}) | 79 | 13 |
math | Integers \(x\) and \(y\) such that \(x > y > 0\) satisfy the equation \(x + y + xy = 119\). What is the value of \(y\)? | 1 | 44 | 1 |
math | Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\). | 78 | 88 | 2 |
math | Using the definitions of square root and cube root, find the unknown variable $x$ that satisfies the following equations:<br/>$(1)\left(x-1\right)^{2}=4$;<br/>$(2)\left(x-2\right)^{3}=-125$. | x = -3 | 61 | 4 |
math | Suppose the radius of Earth at the equator is about 3950 miles. A jet flies around the Earth along the equator at a speed of 550 miles per hour. Assuming a negligible height above the equator, estimate the number of hours the flight would take. | 45 | 60 | 2 |
math | The inclination angle of the line $\sqrt{3}x - y - 1 = 0$ is \_\_\_\_\_\_. | \frac{\pi}{3} | 29 | 7 |
math | What is the greatest number of Sundays that can occur in the first 50 days of a year? | 7 | 21 | 1 |
math | The 24th Winter Olympics, the 2022 Beijing Winter Olympics, opened on February 4, 2022. The Winter Olympics mascot "Bing Dwen Dwen" made its official debut as early as September 2019 and has since become very popular, leading to the creation of many different types of mascot figurines. A certain company has undertaken ... | 1200 | 327 | 4 |
math | The diagram shows a regular octagon. What is the ratio of the area of the shaded trapezium to the area of the whole octagon?
A) $1: 4$
B) $5: 16$
C) $1: 3$
D) $\sqrt{2}: 2$
E) $3: 8$ | 1:4 | 74 | 3 |
math | Given the function $f(x)=\sin (2x- \frac {\pi}{6})$, calculate one of the axes of symmetry of the graph after it is translated to the right by $\frac {\pi}{12}$ units. | \frac{5\pi}{12} | 49 | 10 |
math | Given real numbers $a$ and $b$ satisfying $\ln(b+1) + a - 3b = 0$, and real numbers $c$ and $d$ satisfying $2d - c + \sqrt{5} = 0$, determine the minimum value of $(a-c)^2 + (b-d)^2$. | 1 | 70 | 1 |
math | The curve $x^2+y^2+y+m=0$ and its symmetric curve about the line $x+2y-1=0$ always have four common tangents. Find the range of $m$. | \left(-\frac{11}{20}, \frac{1}{4}\right) | 45 | 21 |
math | Find the largest three-digit number that is a multiple of the sum of its digits and in which the first digit matches the third digit, but does not match the second digit. | 828 | 35 | 3 |
math | The chord length cut by the line $y=kx+3$ from the circle $(x-2)^{2}+(y-3)^{2}=4$ is $2 \sqrt {3}$. Find the slope angle of the line. | \dfrac {π}{6} \text{ or } \dfrac {5π}{6} | 52 | 21 |
math | 2500 chess kings have to be placed on a $100 \times 100$ chessboard so that**(i)** no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);**(ii)** each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by ... | 2 | 102 | 1 |
math | A street has parallel curbs that are 50 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet, and each stripe is 65 feet long. Find the distance, in feet, between the stripes. | \frac{200}{13} \; \text{feet} | 64 | 18 |
math | Show the following identities for all integers $n > 0$:
$$
\begin{aligned}
\sum_{k=0}^{n}\binom{n}{k}=2^{n} & \sum_{k=0, k \text { even }}^{n}\binom{n}{k}=2^{n-1} \\
\sum_{k=0}^{n} k\binom{n}{k}=n 2^{n-1} & \sum_{k=0}^{n} \sum_{\ell=0}^{k}\binom{n}{k}\binom{k}{\ell}=3^{n}
\end{aligned}
$$ | \sum_{k=0}^{n} \sum_{\ell=0}^{k}\binom{n}{k}\binom{k}{\ell} = 3^{n} | 142 | 40 |
math | A company is developing a novelty ice cream cone shaped like a right circular cone. The cone will have a four-inch radius. Find the necessary height of the cone so that it can hold 150 cubic inches of ice cream. | 9 | 47 | 1 |
math | Evaluate the infinite sum $$\sum_{n=2}^{\infty} \log _{2}\left(\frac{1-\frac{1}{n}}{1-\frac{1}{n+1}}\right)$$ | -1 | 50 | 2 |
math | Given that the heights of the pillars at points X, Y, and Z of a regular triangle XYZ are 8, 5, and 7 meters, respectively, determine the height of the pillar directly opposite Y. | 8 | 44 | 1 |
math | Let A, B, and C be three piles of rocks. The mean weight of the rocks in A is 30 pounds, and in B it is 55 pounds. The mean weight of the rocks in the combined piles A and B is 35 pounds, and in the combined piles A and C it is 32 pounds. Find the greatest possible integer value for the mean in pounds of the rocks in t... | 62 | 93 | 2 |
math | In the rectangular coordinate system, the standard equation of the hyperbola passing through the point $P(2 \sqrt {2},- \sqrt {2})$ with an eccentricity of $\sqrt {3}$ is which equation? | \frac {x^{2}}{7}-\frac {y^{2}}{14}=1 | 48 | 22 |
math | The zeros of the function $f(x) = \begin{cases} -x-4, & (x < 0) \\ x^{2}-4, & (x > 0) \end{cases}$ are -4 and 2. | -4, 2 | 53 | 5 |
math | $N $ different numbers are written on blackboard and one of these numbers is equal to $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers betwe... | N = 2 | 107 | 5 |
math | A boat has a speed of 24 mph in still water. There is a river with a current of 6 mph, where the boat travels a distance of 3 miles downstream and then returns. Calculate the ratio of the average speed for the round trip to the speed in still water.
**A**) $\frac{12}{13}$
**B**) $\frac{15}{16}$
**C**) $\frac{17}{18}$
*... | \frac{15}{16} | 110 | 9 |
math | Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x$,
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) Find the set of values of $x$ that make $f(x)\geqslant 3$ true. | [kπ, \frac {π}{4}+kπ] | 78 | 14 |
math | A bowl contained 320 grams of pure white sugar. Mixture \( Y \) was formed by taking \( x \) grams of the white sugar out of the bowl, adding \( x \) grams of brown sugar to the bowl, and then mixing uniformly. In Mixture \( Y \), the ratio of the mass of the white sugar to the mass of the brown sugar, expressed in low... | 48 | 181 | 2 |
math | Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy? | 12 | 66 | 2 |
math | Let \( f(n) \) be the sum of all the divisors of a positive integer \( n \). If \( f(f(n)) = n + 6 \), then call \( n \) ultra-deficient. How many ultra-deficient positive integers are there? | 1 | 56 | 1 |
math | Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is divisible by $x^{2}+1$ modulo 3. | 8 | 37 | 1 |
math | Consider a 6-digit palindrome $m$ formed as $\overline{abccba}$ and chosen uniformly at random. Every digit ($a$, $b$, $c$) can be any digit from 0 to 9, except $a$, which cannot be 0 because $m$ is a 6-digit number. What is the probability that both $m$ and $m+11$ are palindromes?
A) $\frac{800}{900}$
B) $\frac{9}{10}... | \frac{8}{9} | 144 | 7 |
math | The numbers \(a, b, c\) satisfy the conditions \(2011(++)=1\) and \(a b + a c + b c = 2011 \cdot a b c\). Find \(a^{2011} \cdot b^{2011} + c^{2011}\). | \frac{1}{(2011)^{2011}} | 72 | 17 |
math | Monica decides to tile the floor of her 15-foot by 20-foot dining room. She plans to create a two-foot-wide border using one-foot by one-foot square tiles around the edges of the room and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles she will use. | 144 | 71 | 3 |
math | A circle has a diameter $AC$ of length 10 units, with $B$ lying on the circle such that $AB$ is a chord of length 8 units. If $D$ is the midpoint of $AB$ and $CD$ is perpendicular to $AB$, find the length of the line segment $CD$. | 2\sqrt{21} | 68 | 7 |
math | Given vectors $\overrightarrow{a}=(\cos α,\sin α)$, $\overrightarrow{b}=(\cos β,\sin β)$, and $\overrightarrow{c}=(-1,0)$.
(1) Find the maximum length of vector $\overrightarrow{b}+\overrightarrow{c}$.
(2) If $α=\frac{π}{4}$ and $\overrightarrow{a} \perp (\overrightarrow{b}+\overrightarrow{c})$, find the value of $\cos... | \cos β = 0 \text{ or } \cos β = 1 | 111 | 17 |
math | If a four-digit number is called a "good number" when its unit digit is 1 and it has exactly three identical digits, then how many "good numbers" are there among the four-digit numbers formed by the digits 1, 2, 3, and 4 with repetitions? | 12 | 60 | 2 |
math | Given $\tan 2\_\theta= \frac{3}{4}\left(\pi < \theta < \frac{3\pi}{2}\right)$, find the value of $\frac{2{\cos }^{2} \frac{\theta}{2}+\sin \theta-1}{ \sqrt{2}\cos \left(\theta+ \frac{\pi}{4}\right)}$. | 2 | 84 | 1 |
math | Given a triangle $\triangle ABC$ with an area of $S$, and $3 \overrightarrow{AB} \cdot \overrightarrow{AC} = 2S$.
(1) Find the value of $\sin A$;
(2) If $C = \frac{\pi}{4}$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 16$, find the length of $AC$. | AC = 8 | 90 | 4 |
math | If the measure of an exterior angle of a regular polygon is 40°, calculate the number of sides of the polygon. | 9 | 26 | 1 |
math | Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends? | 126/29 | 56 | 6 |
math | Twelve unit cubes are glued together to form a larger solid. The solid has a base layer of eight cubes arranged in a rectangle (4 by 2), with four additional cubes stacked on top at one end (forming a 2 by 2 square on top of one half of the rectangle). Calculate the surface area of the resulting solid. | 36\text{ square units} | 69 | 8 |
math | The chord length intercepted by the curve $\rho=4\cos \theta$ is what value? | 4 | 20 | 1 |
math | During the testing of two engines, it was found that the first engine consumed 300 grams of gasoline, while the second consumed 192 grams. Additionally, the second engine operated for 2 hours less than the first engine. The first engine consumes 6 grams more gasoline per hour than the second engine. How much gasoline d... | 30 \text{ and } 24 \text{ grams per hour} | 75 | 17 |
math | A dart board is a circular disc with a radius of 10 units and contains a central square target. The square has its vertices on the circle. Calculate the probability that a dart thrown randomly and landing anywhere on the board will land within the center square.
- **A)** $\frac{1}{\pi}$
- **B)** $\frac{2}{\pi}$
- **C)*... | \frac{2}{\pi} | 114 | 8 |
math | The planet Yendor follows an elliptical orbit with its sun, Solara, at one focus. At its closest approach (perigee), Yendor is 3 astronomical units (AU) from Solara, while at its farthest point (apogee) it is 15 AU away. Yendor's orbit also contains another celestial body, Lunaris, exactly at the second focus. Calculat... | 9 \, \text{AU} | 105 | 8 |
math | Lucas, Mia, and Grace are planning a hike. Lucas hikes at a speed of 5 miles per hour. If Mia hikes $\frac{3}{4}$ as fast as Lucas, and Grace hikes $\frac{6}{7}$ as fast as Mia, how fast does Grace hike? Additionally, if Liam hikes $\frac{4}{3}$ times faster than Grace, determine Liam’s hiking speed. | \frac{30}{7} \text{ mph} | 83 | 13 |
math | Find the derivative of the function $y=\cos (1+x^{2})$. | -2x\sin (1+x^{2}) | 17 | 11 |
math | Find the shortest distance from a point on the parabola $y=x^{2}$ to the line $2x-y=4$. | \frac{3\sqrt{5}}{5} | 28 | 12 |
math | Let $x$ and $y$ be two-digit positive integers with mean 60. What is the maximum value of the ratio $\frac{x}{y}$? | \frac{33}{7} | 34 | 8 |
math | Given that $\overset{\to }{a}$ and $\overset{\to }{b}$ are two mutually perpendicular unit vectors in a plane, and vector $\overset{\to }{c}$ satisfies $(\overset{\to }{a}-\overset{\to }{c})\cdot (\overset{\to }{a}-\overset{\to }{c})=0$, find the maximum value of $|\overset{\to }{c}|$. | \sqrt{2} | 101 | 5 |
math | A and B play the following game with N counters. A divides the counters into 2 piles, each with at least 2 counters. Then B divides each pile into 2 piles, each with at least one counter. B then takes 2 piles according to a rule which both of them know, and A takes the remaining 2 piles. Both A and B make their choices... | \left\lfloor \frac{N}{2} \right\rfloor | 201 | 17 |
math | Find the general solution of the equation
$$
y^{\prime \prime}+10 y^{\prime}+25 y=4 e^{-5 x}
$$ | y(x) = (C_1 + C_2 x)e^{-5x} + 2x^2 e^{-5x} | 37 | 29 |
math | In quadrilateral $PQRS,$ $PQ = 6,$ $QR = 10$, and $RS = 25$ units. Both angle $Q$ and angle $R$ are right angles. Determine the length of segment $PS$. | \sqrt{461} | 53 | 7 |
math | Let $f(x) = \ln x - \frac{1}{2}ax^2 - 2x$, where $a\le0$.
(Ⅰ) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is given by $y=2x+b$, find the value of $a-2b$.
(Ⅱ) Discuss the monotonicity of the function $f(x)$.
(Ⅲ) Let $g(x) = x^2 - 3x + 3$. If for any $x, t \in [0, 1]$, it always holds that $f(x)... | [-6, 0] | 159 | 6 |
math | Given the function $f\left(x\right)=x^{2}-\left(2m+1\right)x+m\left(m+1\right)$.
$(1)$ When $m=0$, find the zeros of $f\left(x\right)$;
$(2)$ If $f\left(x\right)$ has only one zero within the interval $\left(1,3\right)$, find the range of values for $m$. | (0,1] \cup [2,3) | 97 | 12 |
math | In the figure, $ABCD$ is a rectangle, $AZ=WC=6$ units, $AB=12$ units and the area of trapezoid $ZWCD$ is 120 square units. What is the area of triangle $BQW$? [asy]
draw((0,0)--(12,0)--(12,20)--(0,20)--(0,0)--(12,20));
draw((0,14)--(12,6));
label("$A$",(0,20),W);
label("$Z$",(0,14),W);
label("$D$",(0,0),W);
label("$Q$"... | 42 | 201 | 2 |
math | A line passing through point M(2, 1) intersects the X-axis and Y-axis at points P and Q, respectively, and $|MP| = |MQ|$. Find the equation of line L. | x + 2y - 4 = 0 | 44 | 11 |
math | Calculate the value of $\cos\left(\frac{\pi}{2} + \frac{\pi}{3}\right) + \sin\left(-\pi - \frac{\pi}{6}\right)$. | -\frac{\sqrt{3}}{2} - \frac{1}{2} | 44 | 18 |
math | It is a beautiful day at the beach and ten beach volleyball players have shown up at the volleyball courts. Each two-person volleyball team should consist of a setter and a spiker. Five of the players prefer to be a spiker, four of the players prefer to be a setter, and one player is fine either way.
In how many ways ... | 29 | 85 | 2 |
math | Two parallel tangents to a circle with center O and radius r are given, with the distance from the center O to each tangent line being d, where d > r. Determine the number of points that are equidistant from the circle and these two parallel tangents. | 0 | 55 | 1 |
math | Given the function $f(x) = (x + a)e^{x} + b(x - 2)^{2}$, the tangent line equation to the curve $y = f(x)$ at the point $(0, f(0))$ is $y = -5$.
1. Find the values of $a$ and $b$.
2. Find the extreme values of $f(x)$. | -5 | 85 | 2 |
math | Twelve chairs are evenly spaced around a round table, numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating. No one is to sit either next to, two seats away from, or across from his/her spouse. Additionally, no two men or two women can sit next to each oth... | 17280 | 123 | 5 |
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