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 | Given the function $f(x)=\left| x+\frac{1}{2} \right|+\left| x-\frac{3}{2} \right|$.
(Ⅰ) Solve the inequality $f(x)\leqslant 3$;
(2) If the solution set of the inequality $f(x) < \frac{1}{2}\left| 1-a \right|$ about $x$ is empty, find the range of the real number $a$. | [-3,5] | 102 | 5 |
math | Given that θ is in the interval (0, π) and sin2θ = 2 - 2cos2θ, find the value of tanθ. | 1 | 34 | 1 |
math | What is the probability that a randomly chosen positive integer is relatively prime to 6? What is the probability that at least one of two randomly chosen integers is relatively prime to 6? | \frac{5}{9} | 37 | 7 |
math | Given that the vertices of $\triangle ABC$ correspond to the complex numbers $z_1, z_2, z_3$, and satisfy $\left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1$, express the orthocenter and the center of the nine-point circle of $\triangle ABC$ in terms of $z_1, z_2, z_3$. | \frac{1}{2}(z_1 + z_2 + z_3) | 96 | 19 |
math | In a math test, $15\%$ of the students scored $65$ points, $25\%$ scored $75$ points, $40\%$ scored $85$ points, and the rest scored $95$ points. Calculate the difference between the mean and median score of this test. | 3.5 | 70 | 3 |
math | A merchant buys goods at $25\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\%$ on the marked price and still clear a profit of $25\%$ on the selling price. What percent of the list price must he mark the goods? | 125\% | 69 | 5 |
math | The 31st World University Summer Games will be held in Chengdu, Sichuan from July 28th to August 8th, 2023. A company decided to evaluate a certain product under its umbrella for bidding for related endorsement activities. The original selling price of the product was $25 per unit, with an annual sales volume of 80,000... | 30 \text{ dollars} | 301 | 7 |
math | Using a steel bar with a total length of 14.8 meters to make the frame of a rectangular container, if one side of the base of the container is 0.5 meters longer than the other side, what height will maximize the volume of the container? Also, find the maximum volume. | 1.8 | 62 | 3 |
math | The function $f(x) = x^3 + ax$ ($x \in \mathbb{R}$) has an extremum at $x = l$. Then, the equation of the tangent line to the curve $y = f(x)$ at the origin is ____. | 3x + y = 0 | 57 | 7 |
math | A collection of six positive integers has a mean of 5, a unique mode of 4, and a median of 5. If a 9 is added to the collection, what is the new median? | 5 | 43 | 1 |
math | The value of the expression \(10 - 10.5 \div [5.2 \times 14.6 - (9.2 \times 5.2 + 5.4 \times 3.7 - 4.6 \times 1.5)]\) is | 9.3 | 63 | 3 |
math | Given a linear function $f(x)$ that satisfies $f(f(x))=x+4$.
$(1)$ Find the analytical expression of $f(x)$.
$(2)$ Let $g(x)=(1-a)x^{2}-x$. If for all $x_{1}\in[\frac{1}{4},4]$, there exists $x_{2}\in[-3,\frac{1}{3}]$ such that $g(x_{1})\geqslant f(x_{2})$, find the range of values for $a$. | (-\infty,\frac{3}{4}] | 116 | 11 |
math | Determine the domain of the function $$f(x) = \frac{1}{x+9} + \frac{1}{x^2+9} + \frac{1}{x^3+9} + \frac{1}{e^x + 9}.$$ | (-\infty, -9) \cup (-9, -\sqrt[3]{9}) \cup (-\sqrt[3]{9}, \infty) | 59 | 35 |
math | What is the greatest possible common divisor of the numbers \(9m + 7n\) and \(3m + 2n\) if the numbers \(m\) and \(n\) have no common divisors other than one? | 3 | 46 | 1 |
math | Given that the total number of passengers is 900, the rental costs of buses A and B are 1600 yuan/bus and 2400 yuan/bus, respectively, and the passenger capacities are 36 people/bus and 60 people/bus, determine the minimum total rental cost. | 36800 | 64 | 5 |
math | When point P moves on the circle $x^2-4x+y^2=0$, there exist two fixed points A(1,0) and B(a,0) such that $|PB|=2|PA|$. Find the coordinates of point B. | (-2,0) | 55 | 5 |
math | Let's define strange numbers in the following way: A single-digit prime number is considered strange. A prime number with at least two digits is considered strange if, by removing either its first or last digit, the resulting number is also a strange number. Determine all the strange numbers. | 2, \quad 3, \quad 5, \quad 7, \quad 23, \quad 37, \quad 53, \quad 73, \quad 373 | 56 | 47 |
math | Let the sides opposite to the angles $A$, $B$, $C$ in $\triangle ABC$ be $a$, $b$, $c$, respectively. Let $S$ be the area of $\triangle ABC$, satisfying $S= \frac{ \sqrt{3}}{4}(a^{2}+c^{2}-b^{2})$
(I) Find $B$;
(II) If $b=\sqrt{3}$, find the maximum value of $(\sqrt{3}-1)a+2c$. | 2 \sqrt {6} | 112 | 6 |
math | In the Cartesian coordinate plane, the area of the region formed by the points \((x, y)\) that satisfy \( |x| + |y| + |x - 2| \leqslant 4 \) is ______. | 12 | 51 | 2 |
math | A point $P$ starts from the origin and moves along the $x$-axis with a velocity $v(t) = 2 - t$ (the positive direction of velocity is consistent with the positive direction of the $x$-axis). Find the distance the point $P$ has moved at $t = 3$. | \frac{5}{2} | 68 | 7 |
math | Find the sum of all integral values of \( c \) with \( c \leq 30 \) for which the equation \( y = x^2 - 9x - c \) has two rational roots. | -28 | 46 | 3 |
math | In triangle \(PQR,\) \(PQ = 10,\) \(QR = 12,\) \(PR = 14,\) and point \(M\) is the intersection of the medians. Points \(P',\) \(Q',\) and \(R',\) are the images of \(P,\) \(Q,\) and \(R,\) respectively, after a \(180^\circ\) rotation about \(M.\) What is the area of the union of the two regions enclosed by the triangl... | 24\sqrt{6} | 120 | 7 |
math | Two fair dice, each with a different number of faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of $8$ is half the probability of rolling a sum of $11$, and the probability of rolling a sum of $15$ is $\frac{... | 19 | 128 | 2 |
math | How many positive integers less than 100 that are either prime, or have an even number of positive divisors? | 90 | 25 | 2 |
math | In a circle with center $O$, the measure of $\angle RIP$ is $45^\circ$ and $OR=12$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$. | 6\pi | 54 | 3 |
math | Circles $\omega_1$ , $\omega_2$ , and $\omega_3$ are centered at $M$ , $N$ , and $O$ , respectively. The points of tangency between $\omega_2$ and $\omega_3$ , $\omega_3$ and $\omega_1$ , and $\omega_1$ and $\omega_2$ are tangent at $A$ , $B$ , and $C$ , respectively. Line $MO$ intersects $\ome... | \sqrt{3} | 210 | 5 |
math | Given a regular nonagon \(N\), with \(O\) as the center of its circumcircle, \(PQ\) and \(QR\) as adjacent sides of \(N\). Let \(A\) be the midpoint of \(PQ\), and \(B\) be the midpoint of the radius perpendicular to \(QR\). Find the angle between \(AO\) and \(AB\). | 30^{\circ} | 78 | 6 |
math | A parallelepiped $PQRS TUVW$ is generated by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PT},$ as defined below. Compute the ratio:
\[\frac{PT^2 + QU^2 + RV^2 + SW^2}{PQ^2 + PR^2 + PT^2}.\] | 4 | 81 | 1 |
math | Given that Vertex E of right triangle ABE (where ∠AEB = 90°) is in the interior of unit square ABCD. Let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/4 and 1/2. What is the area of R? | \frac{1}{8} | 71 | 7 |
math | If $10 + 9 + 8 \times 7 \div \square + 6 - 5 \times 4 - 3 \times 2 = 1$, then $\square=$ $\qquad$. | 28 | 47 | 2 |
math | Given $p:\frac{2x}{x+1}＜1$ is a necessary but not sufficient condition for $q:a \lt x \lt a+1$ to hold, solve for the range of the real number $a$. | [-1, 0] | 50 | 6 |
math | Given that the moving circle $M$ is tangent to the line $y=2$ and externally tangent to the fixed circle $C$: $x^{2}+(y+3)^{2}=1$, find the equation of the trajectory of the center of the moving circle $M$. | x^{2}=-12y | 59 | 8 |
math | Determine the value of $\sin 36^{\circ}\cos 6^{\circ}-\sin 54^{\circ}\cos 84^{\circ}$. | \frac{1}{2} | 39 | 7 |
math | A circle intersects the line $l_1: x - 6y - 10 = 0$ at point P(4, -1) and the center of the circle lies on the line $l_2: 5x - 3y = 0$.
(I) Find the equation of the circle;
(II) Find the length of the shortest chord that the circle cuts off from lines passing through the origin. | 2 \sqrt{3} | 90 | 6 |
math | Determine the smallest natural number \( k \) such that for any \( a \in [0,1] \) and any \( n \in \mathbf{N}^{*} \), it always holds that \( a^{k} (1-a)^{n} < \frac{1}{(n+1)^{3}} \). | 4 | 74 | 1 |
math | Points $D, E, F$ lie on circle $O$ such that the line tangent to $O$ at $D$ intersects ray $\overrightarrow{E F}$ at $P$. Given that $P D=4, P F=2$, and $\angle F P D=60^{\circ}$, determine the area of circle $O$. | 12 \pi | 75 | 4 |
math | Chewbacca has 25 pieces of cherry gum and 40 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly \(y\) pieces of gum. If Chewbacca loses two packs of cherry gum, then the ratio of the number of pieces of cherry gum he has to the number of pieces of grape... | 2.5 | 101 | 3 |
math | Two semicircles, each with radius \(\sqrt{2}\), are tangent to each other. If \( AB \parallel CD \), determine the length of segment \( AD \). | 4\sqrt{2} | 38 | 6 |
math | In the polar coordinate system, the polar equation of curve $C$ is given by $ρ=4 \sqrt {2}\sin (θ+ \dfrac {π}{4})$. Establish a Cartesian coordinate system with point $O$ as the origin and the non-negative half of the polar axis as the $x$-axis. The parametric equations of line $l$ are given by $ \begin{cases} x=-2+ \d... | 33 | 197 | 2 |
math | Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$ . | 165 | 43 | 3 |
math | Given the area of a circle is doubled when its radius $r$ is increased by $n$, determine the relationship between $r$ and $n$. | n(\sqrt{2} + 1) | 31 | 10 |
math | A cube with 4-inch edges is made using 64 cubes with 1-inch edges. Forty-eight of the smaller cubes are white and sixteen are black. If the sixteen black cubes are placed so that each face of the larger cube has two black cubes along one edge, calculate the fraction of the surface area of the larger cube that is white. | \frac{7}{8} | 71 | 7 |
math | In a store, there are candies priced at 2 p. per kg and 3 p. per kg, with an equal total cost for each type. At what price should the mixture of these candies be sold so that the total cost remains the same? | 2.40 \, \text{rubles per kg} | 52 | 14 |
math | Find the integer $m$, $0 \le m \le 180$, such that $\cos m^\circ = \cos 1234^\circ$. | 154 | 36 | 3 |
math | Given the function $f(x) = (m+1)x^{2} - (m-1)x + m-1$.
$(1)$ Solve the inequality with respect to $x$: $f(x) \geq (m+1)x$;
$(2)$ If the inequality $f(x) \geq 0$ holds for all $x \in [-\frac{1}{2}, \frac{1}{2}]$, find the range of values for $m$. | [1, +\infty) | 103 | 8 |
math | Find the smallest prime number that can be expressed as the sum of five different prime numbers. | 43 | 18 | 2 |
math | For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . | 47 | 73 | 2 |
math | Given an arithmetic sequence $\{a_{n}\}$ with a common difference $d > 0$, and $a_{1}⋅a_{6}=11$, $a_{3}+a_{4}=12$.
1. Find the general term formula of the sequence $\{a_{n}\}$.
2. Find the sum of the first $n$ terms, denoted as $T_{n}$, of the sequence $\{\frac{a_{n+1}-2a_{n}}{2^{n+1}}\}$. | \frac{2n+1}{2^{n+1}} - \frac{1}{2} | 117 | 22 |
math | In a positive geometric sequence $\{a_{n}\}$, it is known that $a_{1}a_{2}a_{3}=4$, $a_{4}a_{5}a_{6}=8$, and $a_{n}a_{n+1}a_{n+2}=128$. Find the value of $n$. | 16 | 76 | 2 |
math | Given the function $f(x)=\cos 2x+2\sin x$.
(Ⅰ) Find the value of $f\left(-\frac{\pi}{6}\right)$;
(Ⅱ) When $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$ and the corresponding value of $x$. | \frac{3}{2} | 90 | 7 |
math | In $\triangle ABC$, given $BC=2$, $AC=\sqrt{7}$, $B=\dfrac{2\pi}{3}$, find the area of $\triangle ABC$. | \dfrac{\sqrt{3}}{2} | 40 | 10 |
math | Let $g$ be a function defined by $g\left(\dfrac{x}{3}\right)=x^3-x^2+2x+3$. Find the sum of all values of $y$ for which $g(3y)=9$. | \frac{1}{3} | 54 | 7 |
math | Given the line $l\_1$: $\begin{cases} x=t \\ y= \sqrt {3}t \end{cases}$ (where $t$ is a parameter), establish a rectangular coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The circle $C\_1$ is defined as $ρ^{2}-2 \sqrt {3}ρ\cos θ-4ρ\sin θ+6=0$.
1. Find ... | \dfrac {\sqrt {3}}{4} | 168 | 10 |
math | Calculate the value of $9.98^5$ using the binomial theorem. | 99004 | 18 | 5 |
math | Given $S_n$ be the sum of the first $n$ terms of a positive sequence $\{ a_n \}$, given $a_1 = 2$ and $S_{n+1}(S_{n+1} - 2S_n + 1) = 3S_n(S_n + 1)$, find the value of $a_{100}$. | 4 \times 3^{98} | 82 | 9 |
math | Find all values of \( c \) for which the inequality \( a + \sqrt{b + c} > b + \sqrt{a + c} \) holds for any positive \( a \) and \( b \) with \( a > b \). | c = \frac{1}{4} | 54 | 9 |
math | Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$ , different from $n$ , is bigger than $n$ . Determine the largest natural number $k$ , which divides any number from $S$ .
(A natural number is a positive integer) | 6 | 75 | 1 |
math | 11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the length of the queue decrease? Assume people ... | 2.2 | 85 | 3 |
math | Find all functions \( g(x) \) that satisfy the equation:
\[ \sin x + \cos y = f(x) + f(y) + g(x) - g(y), \; x, y \in \mathbf{R} \] | g(x) = \frac{\sin x - \cos x}{2} + C | 54 | 18 |
math | Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = |\overrightarrow{b}| = 2$, and $\overrightarrow{a} \cdot (\overrightarrow{b} - \overrightarrow{a}) = -6$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 78 | 9 |
math | In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, "Not including yourself, are there an equal number of knights and liars on your isla... | 1013 | 165 | 4 |
math | Alice and Bob each arrive at a meeting at a random time between 12:00 and 1:00. If Alice arrives after Bob, what is the probability that Bob arrived before 12:30? | \frac{1}{4} | 47 | 7 |
math | Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds: $$ (n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d) $$ | 2 | 96 | 1 |
math | The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no ... | 294 | 82 | 3 |
math | The quadratic $x^2 + 520x + 600$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$? | -258 | 52 | 4 |
math | Find the number of scalene triangles having all sides of integral lengths, at least one side even, and a perimeter less than 16. | 6 | 29 | 1 |
math | Given that \\(\{a_n\}\) is a geometric sequence satisfying \\(a_1=2\), and \\(a_2\), \\(a_3+2\), \\(a_4\) form an arithmetic sequence. The sequence \\(\{b_n\}\) satisfies \\(b_1+ \frac{1}{2}b_2+ \frac{1}{3}b_3+\ldots+ \frac{1}{n}b_n=2n\) for all natural numbers \\(n\).
\\((1)\\) Find the general formula for \\(\{a_n\}\... | S_{2n} = \frac{2^{2n+1} - 2}{3} - 2n | 198 | 26 |
math | Given the function $f(x)=\cos x(\sin x- \sqrt {3}\cos x)+ \frac { \sqrt {3}}{2}$, where $x\in\mathbb{R}$.
- (I) Find the smallest positive period of $f(x)$ and the intervals of monotonic increase;
- (II) If the function $g(x)=f(x+a)$ is even, find the minimum value of $|a|$. | \frac {\pi}{12} | 96 | 8 |
math | Suppose nine circles of radius 1 are centered at every integer coordinate point within the square defined by vertices (0,0), (0,3), (3,0), and (3,3) in the first quadrant of the coordinate plane. Let $\mathcal{S}$ be the union of these circular regions. Line $m$, with slope 2, divides region $\mathcal{S}$ into two regi... | 5 | 143 | 1 |
math | Given a sequence $\left\{ a_n \right\}$ that satisfies for any $n\in N^{*}$, we have $a_1^3+a_2^3+\cdots +a_n^3=(a_1+a_2+\cdots +a_n)^2$, and $a_n > 0$.
(1) Find the general formula for the sequence $\left\{ a_n \right\}$.
(2) Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n\cdot a_{n+2}} \... | (0, \frac{1}{2}) | 176 | 10 |
math | Given a triangle $\triangle ABC$ with the equations of the lines containing two of its altitudes being $2x - 3y + 1 = 0$ and $x + y = 0$, and the coordinates of vertex $A$ being $(1,2)$, find the equation of the line containing side $BC$. | 2x + 3y + 7 = 0 | 69 | 12 |
math | Describe a regular octahedron around a sphere of radius \( R \) and find its volume. | 4R^3 \sqrt{3} | 21 | 9 |
math | Arc $AC$ is a quarter-circle with center $B$. The shaded region $ABC$ is "rolled" along another quarter-circle path $PQ$ (with the same radius as $AC$) until it reaches its original orientation for the first time with point $B$ landing at point $B^{\prime}$. If $BC = \frac{4}{\pi}$ cm, what is the length of the path th... | 8\text{ cm} | 102 | 6 |
math | Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$ . Find the $ x$ -coordinate of the point $ X$ on the $ x$ -axis maximizing $ \angle AXB$ . | 5\sqrt{2} - 3 | 69 | 9 |
math | Compute $11^{-1} \pmod{1021}$. Express your answer as a residue from $0$ to $1020$, inclusive. | 557 | 36 | 3 |
math | Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$ -gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other. | n-3 | 56 | 3 |
math | In the polar coordinate system, the coordinates of point $P$ is $(1,0)$, and the equation of curve $C$ is $\rho =2\sqrt{2}\cos (\theta -\dfrac{\pi }{4})$. Establish a rectangular coordinate system with the pole as the coordinate origin and the positive semi-axis of the polar axis as the $x$-axis. A line $l$ with a slop... | |PA|^{2}+|PB|^{2}=4 | 170 | 14 |
math | A sequence of numbers \( t_{1}, t_{2}, t_{3}, \ldots \) has its terms defined by \( t_{n}=\frac{1}{n}-\frac{1}{n+2} \) for every integer \( n \geq 1 \). For example, \( t_{4}=\frac{1}{4}-\frac{1}{6} \). What is the largest positive integer \( k \) for which the sum of the first \( k \) terms (that is, \( t_{1}+t_{2}+\c... | 1998 | 141 | 4 |
math | Given the one-variable quadratic equation $x^{2}+2x-(k-1)=0$, determine the range of values for $k$ such that the equation has real roots with respect to $x$. | k \geqslant 0 | 43 | 8 |
math | Find the range of values for $x$ such that $\frac{1}{x} < 3$ and $\frac{1}{x} > -2$, provided also that $2x - 5 > 0$.
A) $x > \frac{5}{2}$
B) $x < -\frac{1}{2}$
C) $x < \frac{1}{3}$
D) $x < \frac{5}{2}$
E) $x > -\frac{1}{2}$ | x > \frac{5}{2} | 114 | 9 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\sin C = \sqrt{3}$, $\angle B = \frac{\pi}{4}$.
$(1)$ Find the value of side $c$;
$(2)$ If the area of $\triangle ABC$ is $\frac{9}{2}$, find the value of side $b$. | \sqrt{15} | 101 | 6 |
math | A recent survey indicated that 70% of men and 75% of women endorse a new environmental policy. The survey included 200 men and 800 women. What percentage of the surveyed population supports the new policy? | 74\% | 50 | 4 |
math | Using only the digits $2,3$ and $9$ , how many six-digit numbers can be formed which are divisible by $6$ ? | 81 | 34 | 2 |
math | Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xi... | 72 | 113 | 2 |
math | Let the function $y=f(x)$ have its first derivative $f'(x)$ and its second derivative $f''(x)$ on the interval $(a, b)$. If $f''(x)<0$ holds for all $x$ in $(a,b)$, then the function $f(x)$ is called a "convex function" on $(a,b)$. Given $f(x) = \frac{1}{12}x^4 - \frac{1}{6}mx^3 - \frac{3}{2}x^2$,
(1) Find $f'(x)$ an... | b - a = 1 - (-1) = 2 | 225 | 13 |
math | Two pirates were playing with gold coins. First, the first pirate lost half of his coins (gave them to the second one), then the second pirate lost half of his coins, then the first pirate lost half of his coins again. As a result, the first pirate had 15 coins, and the second pirate had 33 coins. How many coins did th... | 24 | 86 | 2 |
math | Find the integer $n$, $0 \le n \le 180$, such that $\cos n^\circ = \cos 812^\circ$. | 92 | 35 | 2 |
math | We are allowed to remove exactly one integer from the list $$ -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 $$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12? | 6 | 104 | 1 |
math | A set of composite numbers from the set $\{1,2,3,4, \ldots, 2016\}$ is called good if any two numbers in this set do not have common divisors (other than 1). What is the maximum number of numbers that a good set can have? | 14 | 65 | 2 |
math | In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \). | 6\pi | 54 | 3 |
math | Triangle PQR has P=(0,0), R=(8,0), and Q in the first quadrant. Additionally, ∠QRP=90° and ∠QPR=45°. Determine the coordinates of the image of Q after P is rotated 120° counterclockwise about P. | (-4 - 4\sqrt{3}, 4\sqrt{3} - 4) | 67 | 21 |
math | Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$.
$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$.
$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. | a = \frac{3}{2} | 97 | 9 |
math |
Compute the volumes of the bodies bounded by the surfaces:
\[ z = 4x^2 + 9y^2 \]
\[ z = 6 \] | 3\pi | 35 | 3 |
math | The domain of the function $f(x)=\frac{1}{\sqrt{1-\log_{2}x}}$ is (0, 2]. | (0,2) | 33 | 5 |
math | In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $\sqrt{2}r$ units. From $O$, a perpendicular to $AB$ meets $AB$ at point $M$. From $M$, a perpendicular to $OA$ meets $OA$ at point $D$. What is the area of triangle $MDA$ expressed in terms of $r$?
A) $\frac{3r^2}{16}$
B) $\frac{\pi r... | \frac{r^2}{4\sqrt{3}} | 166 | 13 |
math | Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\frac{1}{3}$, find the eccentricity of the hyperbola $\frac{y^2}{m^2}-\frac{x^2}{n^2}=1$. | \frac{\sqrt{5}}{2} | 96 | 10 |
math | Given the function $f(x)=A\sin (2x+\varphi)$ ($A > 0$, $0 < \varphi < \pi$), $x\in\mathbb{R}$ has a maximum value of $1$, and its graph passes through the point $M\left( \frac {\pi}{6}, \frac { \sqrt {3}}{2}\right)$.
(I) Find $\varphi$;
(II) Find the intervals of monotonic increase for $f(x)$;
(III) Through what ... | \frac {\pi}{6} | 140 | 7 |
math | It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the \( x \) and \( y \) axes and so that the medians to the midpoints of the legs lie on the lines \( y = 4x + 1 \) and \( y = nx + 2 \). Determine the number of different constants \( n \) for which such a triangle exi... | 2 | 85 | 1 |
math | Five identical cylindrical pipes are stacked in a crate with two on the bottom and three resting on top of the first two. Each pipe has a diameter of $12\text{ cm}.$ Determine the height, $h,$ of this pile of $5$ pipes as shown below.
[asy]
draw(circle((10,10),12),black+linewidth(1));
draw(circle((30,10),12),black+lin... | 12 + 6\sqrt{3}\text{ cm} | 253 | 14 |
math | Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=\frac{3^{n+1} a_{n}}{a_{n}+3^{n+1}}, \, a_{1}=3$, find the general term of the sequence $\left\{a_{n}\right\}$. | \frac{2 \cdot 3^n}{3^n - 1} | 74 | 16 |
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