problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Given the function $f(x)=|x-a|$, if the solution set of the inequality $f(x) \leqslant 3$ is $\{|x|-1 \leqslant x \leqslant 5\}$.
(Ⅰ) Find the value of the real number $a$:
(Ⅱ) If the inequality $f(3x)+f(x+3) \geqslant m$ holds for all real numbers $x$, find the range of values for the real number $m$. | (-\infty, \frac{5}{3}] | math | 113 |
Given that $\{a\_n\}$ is an arithmetic sequence, where $a\_1=25$, $a\_4=16$.
(1) Find the general term formula for the sequence $\{a\_n\}$;
(2) For what value of $n$ does the sum of the first $n$ terms of the sequence $\{a\_n\}$, denoted by $S\_n$, reach its maximum value. | n = 9 | math | 95 |
Given the function $f(x)=\cos (2x-\varphi )-\sqrt{3}\sin (2x-\varphi )(|\varphi | < \frac{\pi }{2})$, the graph of which is translated to the right by $\frac{\pi }{12}$ units and then becomes symmetric about the $y$-axis, determine the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi }{2},0 \right]$. | -\sqrt{3} | math | 105 |
Find the least possible value of \( f(x) = \frac{9}{1 + \cos 2x} + \frac{25}{1 - \cos 2x} \), where \( x \) ranges over all real numbers for which \( f(x) \) is defined. | 32 | math | 62 |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p +q + r + s = 10$ and
\[pq + pr + ps + qr + qs + rs = 20.\]Find the largest possible value of $s.$ | \frac{5 + \sqrt{105}}{2} | math | 61 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, and point $P$ is a moving point on ellipse $C$. Determine the locus equation of the centroid $G$ of $\triangle PF\_1F\_2$. | \frac{9x^2}{4} + 3y^2 = 1 | math | 84 |
Triangle $ABC$ has vertices at $A(0, 8)$, $B(2, 0)$, and $C(10, 0)$. A horizontal line with equation $y = t$ intersects line segment $\overline{AB}$ at $T$ and line segment $\overline{AC}$ at $U$, forming triangle $\triangle ATU$ with area 20. Compute $t$. | 8 - 2\sqrt{10} | math | 89 |
In the geometric sequence $\{a_n\}$, if $a_1+a_2=3$ and $a_3+a_4=6$, then $a_7+a_8=$ ? | 24 | math | 43 |
Given the equation of motion of an object, $s=t^{2}+ \frac {3}{t}$, determine the velocity $v$ of the object at time $t=2$. | \frac {13}{4} | math | 40 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a=2$ and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$. | \sqrt {3} | math | 67 |
Let $P= \left\{ x|x+2 \geqslant x^{2} \right\}$ and $Q=\left\{ {x∈N|\left| {x} \right|\leqslant3} \right\}$. Find the intersection of P and Q. | \{0,1,2\} | math | 64 |
Given that for any non-zero real number $m$, the inequality $|2m-1|+|1-m| \geqslant |m|(|x-1|-|2x+3|)$ always holds, find the range of the real number $x$. | (-\infty,-3] \cup [-1,+\infty) | math | 58 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The vectors $\overrightarrow{m}=(b,\sqrt{3}a)$ and $\overrightarrow{n}=(\sin B,\sin 2A)$ are given, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
$(Ⅰ)$ Find the value of angle $A$.
$(Ⅱ)$ If the a... | \sqrt{7} | math | 213 |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point, with the integer after the decimal point being a multiple of 25. What is the larger of the two integers? | 50 | math | 63 |
For which integers $m$ and $n$ does the equation
$$
(5 + 3\sqrt{2})^m = (3 + 5\sqrt{2})^n
$$
hold true? | (m, n) = (0, 0) | math | 47 |
In how many ways can $420$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | 7 | math | 27 |
Given that θ is in the interval (0, π) and sin2θ = 2 - 2cos2θ, find the value of tanθ. | 1 | math | 34 |
Given a sequence $\left\{a_{n}\right\}$ with the nth partial sum $S_{n}=2 a_{n}-1$ (for $n=1,2, \cdots$) and another sequence $\left\{b_{n}\right\}$ satisfying $b_{1}=3$ and $b_{k+1}=a_{k}+b_{k}$ (for $k=1,2, \cdots$), find the nth partial sum of the sequence $\left\{b_{n}\right\}$. | 2^n + 2n - 1 | math | 118 |
What is the largest possible median for the five number set \(\{x, y, 4, 3, 7\}\) if \(x\) and \(y\) can be any integers, and \( y = 2x\)? | 4 | math | 51 |
Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = \arcsin x, \quad y = \arccos x, \quad y = 0
$$ | \frac{\pi}{2} | math | 51 |
Given the function $f(x)= \dfrac {2-\cos \left( \dfrac {\pi}{4}(1-x)\right)+\sin \left( \dfrac {\pi}{4}(1-x)\right)}{x^{2}+4x+5}(-4\leqslant x\leqslant 0)$, find the maximum value of $f(x)$. | 2+ \sqrt {2} | math | 86 |
Given that \(5^{-1} \equiv 39 \pmod{79}\), find \(125^{-1} \pmod{79}\), and provide the answer as a residue modulo 79. | 69 | math | 48 |
Given that $x+y=3-\cos 4θ$ and $x-y=4\sin 2θ$, find the value of $\sqrt{x}+\sqrt{y}$. | 2 | math | 39 |
Dima and Vlad play a game: first, they take turns naming a number from 1 to 97 (Dima goes first, and the numbers must be different). Then, each counts the number of distinct rectangles with integer sides whose perimeter is equal to the named number. The winner is the one with the greater number of rectangles. Which num... | 96 | math | 112 |
Find the number of triples $(x,y,z)$ of real numbers such that
\begin{align*}
x &= 3000 - 3001 \operatorname{sign}(y + z + 3), \\
y &= 3000 - 3001 \operatorname{sign}(x + z + 3), \\
z &= 3000 - 3001 \operatorname{sign}(x + y + 3).
\end{align*}
Note:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a ... | 3 | math | 183 |
Two circles with radii 1 and 2 have a common center \( O \). The area of the shaded region is three times smaller than the area of the larger circle. Find the angle \( \angle AOB \). | \frac{8\pi}{9} | math | 46 |
Both roots of the quadratic equation $x^2 - 65x + k = 0$ are consecutive prime numbers. How many possible values of $k$ are there? | 0 | math | 37 |
In a certain class, the average score of students who received an excellent grade in a math exam is 95, while the average score of those who did not receive an excellent grade is 80. Given that the average score of the entire class is at least 90, what is the minimum proportion of students who received an excellent gra... | \frac{2}{3} | math | 74 |
Given a sequence $\{a_n\}$ where $a_n^2 + 2a_n - n^2 + 2n = 0$ ($n \in \mathbb{N}^+$):
(Ⅰ) Find the general formula for the terms of the sequence $\{a_n\}$.
(Ⅱ) Calculate the sum of the first $n$ terms of the sequence $S_n$. | \frac{n(n - 3)}{2} | math | 87 |
Circle $C_{1}$: $x^{2}+y^{2}-4x+2y+1=0$ intersects with circle $C_{2}$: $x^{2}+y^{2}-2y-3=0$ at points $A$ and $B$. The length of $|AB|$ is ______. | 2\sqrt{2} | math | 72 |
Suppose that $a,$ $b,$ and $c$ are three positive numbers that satisfy the equations $abc = 1,$ $a + \frac {1}{c} = 8,$ and $b + \frac {1}{a} = 20.$ Find $c + \frac {1}{b}.$ | \frac{10}{53} | math | 69 |
Let
\[f(x) = \left\{
\begin{array}{cl}
x^2 + 3 & \text{if $x < 15$}, \\
3x - 2 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $f^{-1}(10) + f^{-1}(49).$ | \sqrt{7} + 17 | math | 84 |
Given the function $f(x) = \log_{3} \frac{x + a}{x - 1} (a > 0)$, find the value of $a$ such that the function is odd. | 1 | math | 45 |
Given that $\frac{1}{4}$ of ninth graders are assigned exactly one sixth-grade buddy and this group constitutes $\frac{1}{3}$ of all sixth graders, determine the fraction of the total number of sixth graders and ninth graders who have a buddy. | \frac{2}{7} | math | 57 |
Find the range of the function $y = \log_{3}x + \frac{1}{\log_{3}x} - 1$. | (-\infty, -3] \cup [1, +\infty) | math | 32 |
Given the function $f(x) = -tx^2 + 2x + 1$ (where $t < 0$, and $t$ is a constant), for any two different $x_1$, $x_2$, when $x_1$, $x_2 \in [-2, 2]$, it always holds that $|f(x_1) - f(x_2)| \leq k|x_1 - x_2|$ (where $k$ is a constant, $k \in \mathbb{R}$). Then, the range of values for the real number $k$ is. | [-4t + 2, +\infty) | math | 133 |
To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 27 | math | 84 |
Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms? | 15 | math | 46 |
If the equation \( a^{2x} + (1 + \lg m) a^{x} + 1 = 0 \) (where \( a > 0 \) and \( a \neq 1 \)) has a solution, then the range of values for \( m \) is $\qquad$ . | (0, 10^{-3}] | math | 68 |
Given vectors $p = (a_n, 2^n)$ and $q = (2^{n+1}, -a_{n+1})$, where $n \in \mathbb{N}^*$, $p$ is perpendicular to $q$, and $a_1 = 1$.
(1) Find the general formula for the sequence $\{a_n\}$;
(2) If the sequence $\{b_n\}$ satisfies $b_n = \log_2 a_n + 1$, find the sum of the first $n$ terms of the sequence $\{a_n \cd... | 1 + (n - 1)2^n | math | 141 |
Given the function $f(x)=x^{3}-ax-1$.
$(1)$ If $f(x)$ is an increasing function on $\mathbb{R}$, find the range of the real number $a$;
$(2)$ If the function $f(x)$ is a decreasing function on $(-1,1)$, find the range of the real number $a$;
$(3)$ If the decreasing interval of the function $f(x)$ is $(-1,1)$, find t... | (0,3) | math | 147 |
Given that $\tan\theta=a$, where $a>1$, find the value of $$\frac{\sin(\frac{\pi}{4}+\theta)}{\sin(\frac{\pi}{2}-\theta)}\cdot\tan2\theta$$. | \frac{\sqrt{2}a}{1-a} | math | 54 |
Determine the solution to $15y + 4 \equiv 7 \pmod{18}$ where $y \equiv b \pmod{n}$ for some positive integers $n \geq 2$ and $b < n$. Find $b+n$. | 11 | math | 57 |
The distance from the origin to the line $3x + 4y + 5 = 0$ can be calculated. | 1 | math | 26 |
Given a random variable $X \sim B(n, 0.8)$, and $D(X) = 1.6$, calculate the value of $n$. | 10 | math | 35 |
Let \( C \) be the curve \( y^2 = x^3 \) (where \( x \) takes all non-negative real values). Let \( O \) be the origin, and \( A \) be the point where the gradient is 1. Find the length of the curve from \( O \) to \( A \). | \frac{8}{27} (2\sqrt{2} - 1) | math | 71 |
Given the functions $y=\sin x+\cos x$ and $y=2 \sqrt {2}\sin x\cos x$, determine the correct conclusion(s) from the following options:
$①$ Both functions' graphs are centrally symmetric about the point $(-\frac{π}{4},0)$;
$②$ Both functions' graphs are axially symmetric about the line $x=-\frac{π}{4}$;
$③$ Both func... | ③⑤ | math | 148 |
Given that $\frac{\cos 2\alpha}{\sqrt{2}\sin\left(\alpha+\frac{\pi}{4}\right)}=\frac{\sqrt{5}}{2}$, find the value of $\tan\alpha+\frac{1}{\tan\alpha}$. | -8 | math | 60 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is:
$$
\begin{cases}
x=1+ \sqrt {3}\cos\phi \\
y= \sqrt {3}\sin\phi
\end{cases}
$$
($\phi$ is the parameter, $0 \leq \phi \leq \pi$). Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the po... | 5 | math | 224 |
The function $f^{-1}(x)$ is the inverse function of $f(x)=\log _{2}\left(2^{x}+1\right)$.
(1) If the equation $f^{-1}(x)=m+f(x)$ has a solution in the interval $[1,2]$, find the range of values for the real number $m$.
(2) If the inequality $f^{-1}(x)>m+f(x)$ has a solution in the interval $[1,2]$, find the range of ... | \left(-\infty, \log_2 \left( \frac{1}{3} \right) \right) | math | 164 |
Let $f(x) = x + 3$ and $g(x) = x/2.$ Also denote the inverses to these functions as $f^{-1}$ and $g^{-1}.$ Compute
$$f(g^{-1}(f^{-1}(g^{-1}(f^{-1}(g(f(15))))))).$$ | 21 | math | 70 |
Given Miki has 15 apples and 10 bananas, She uses her juicer to extract 9 ounces of apple juice from 3 apples and 10 ounces of banana juice from 2 bananas. She makes a banana-apple juice blend using 5 apples and 4 bananas. Calculate the percent of the blend that is apple juice. | 42.86\% | math | 72 |
In triangle $ABC,$ $AB = 3,$ $AC = 6,$ and $\cos \angle A = \frac{1}{8}.$ Find the length of angle bisector $\overline{AD}.$ | 3 | math | 47 |
Contracting a certain project. Team A alone needs 36 days to complete; Team B alone needs 24 days to complete; Team C alone only needs 18 days to complete. In actual construction, the three teams first complete half of the project together, and then Teams A and C continue to complete the remaining half of the project t... | 20000 | math | 114 |
Calculate the volume in cubic feet of a round swimming pool which is 20 feet in diameter and has a depth that starts at 3 feet on one end and increases linearly to 6 feet at the other end. Express your answer in terms of $\pi$. | 450\pi | math | 54 |
Given a sequence $\{a_n\}$ that satisfies $(a_1 + 4a_2 + 4^2a_3 + \cdots + 4^{n-1}a_n = \frac{n}{4})$ $(n\in \mathbb{N}^*)$.
(I) Find the general formula for the sequence $\{a_n\}$;
(II) Let $(b_n = 2^n\log_4 a_n)$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n = (1-n)2^{n+1} - 2 | math | 127 |
In the interval $[-1,1]$, two numbers $x$ and $y$ are randomly selected. Find the probability that $x^2 + y^2 < \frac{1}{4}$. | \frac{\pi}{16} | math | 44 |
The function $g(x) = x^3 - 3ax - a$ is not monotonic in the interval $(0,1)$. The range of $a$ is | (0,1) | math | 37 |
Given that the coefficient of the third term in the expansion of $(\sqrt{x} - \frac{2}{x})^n$ is 162 larger than the coefficient of the second term, find:
(1) the value of $n$;
(2) the term containing $x^3$ in the expansion. | T_2 = C_9^1 \cdot (-2) \cdot x^3 = -18x^3 | math | 69 |
Given that the line $2x+3y-9=0$ is parallel to the line $6x+my+12=0$, determine the distance between the two lines. | \sqrt{13} | math | 39 |
Find the largest real number $c$ such that \[x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 \geq cM^2\] whenever $x_1,x_2,x_3,x_4,x_5$ are real numbers such that $x_1+x_2+x_3+x_4+x_5=0$ and $M$ is the median of $x_1,x_2,x_3,x_4,x_5.$ | 2 | math | 117 |
Let $x$ be a value such that $9x^2 + 8x - 1 = 0$ and $27x^2 + 65x - 8 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. | \frac{1}{9} | math | 62 |
What is the value of $x^2 + y^2 - z^2 + 2yz$ when $x = 4$, $y = -3$, and $z = 5$? | -30 | math | 43 |
There are two arithmetic sequences: $2$, $6$, $10$, $\ldots$, $190$ and $2$, $8$, $14$, $\ldots$, $200$. A new sequence is formed by the common terms of these two sequences in ascending order. The sum of the terms in this new sequence is ______. | 1472 | math | 74 |
There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these six people, with each person getting one letter, what is the probability that none of the six people receives the correct letter? | \frac{265}{720} | math | 52 |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, its left and right foci are $F\_1$ and $F\_2$ respectively. An equilateral triangle $PF\_1F\_2$ intersects the hyperbola at points $M$ and $N$. If $M$ and $N$ are the midpoints of the line segments $PF\_1$ and $PF\_2$, find the eccentri... | \sqrt{3} + 1 | math | 127 |
A high school with 30 classes conducted a survey to understand the psychological state of its students. Each class was assigned a number from 1 to 30. Using systematic sampling, 5 classes were selected for the survey. If the sum of the numbers of the selected classes is 75, what is the smallest number among the selecte... | 3 | math | 72 |
In the arithmetic sequence $\{a_n\}$, $a_5=5$ and $a_{10}=15$. Calculate the value of $a_{15}$. | 25 | math | 39 |
Find the total area of metal wasted when a circular disc is cut out from a square metallic sheet of side length \( s \), and then a rectangle (with the rectangle's longer side equal to the circle's diameter and the shorter side half of the longer side) is cut from this circular disc. | \frac{s^2}{2} | math | 60 |
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$ | (2, 3, 4, 1) \text{ and } (3, 2, 4, 4) | math | 51 |
Given the function $f(x)=|x-3|-|x-a|$.
(I) When $a=2$, solve the inequality $f(x)\leqslant - \frac {1}{2}$.
(II) If there exists a real number $x$ such that the inequality $f(x)\geqslant a$ holds, find the range of the real number $a$. | (-\infty, \frac {3}{2}] | math | 85 |
Find the equation of the tangent line to the curve $f(x)=e^{x}+5\sin x$ at the point $(0,1)$. | y=6x+1 | math | 33 |
Given that $\sin\alpha = \frac{1}{2} + \cos\alpha$, and $\alpha \in (0, \frac{\pi}{2})$, find the value of $\frac{\cos 2\alpha}{\sin(\alpha - \frac{\pi}{4})}$. | -\frac{\sqrt{14}}{2} | math | 63 |
My friend thought of an integer between 10 and 19. To guess the number he thought of, I can ask him questions that he will answer with "yes" or "no": What is the minimum number of questions, and which questions should I ask to determine the number he thought of? | 3 | math | 62 |
Triangle $ABC$ has a right angle at $B$, and within it is a point $P$ such that $PA = 12$, $PB = 8$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$. | 16 | math | 55 |
Given the expression $3^5 \cdot 6^5 \cdot 3^6 \cdot 6^6$, evaluate the expression. | 18^{11} | math | 30 |
The volume of the tetrahedron \(ABCD\) is 5. A plane passes through the midpoints of edges \(AD\) and \(BC\) and intersects edge \(CD\) at point \(M\). The ratio of the length of segment \(DM\) to the length of segment \(CM\) is \(2/3\). Calculate the area of the cross-section of the tetrahedron formed by this plane, g... | 3 | math | 103 |
If the fractional equation $\frac{3}{{x-2}}+1=\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. | -6 | math | 42 |
Magda cut out two identical isosceles triangles, each with a perimeter of $100 \mathrm{~cm}$. First, she created a quadrilateral by placing the triangles together by their legs. Then, she created another quadrilateral by placing them together by their bases. In the first case, the perimeter of the quadrilateral was $4 ... | r = 34 \, \text{cm}, \, z = 32 \, \text{cm} | math | 99 |
If $y$ is a real number, and $|y-2| + |y-5| < b$ where $b > 1$, find the range of $b$. | b > 3 | math | 39 |
Let $x$ and $y$ be positive real numbers such that $5x + 6y < 90$. Find the maximum value of
\[xy (90 - 5x - 6y).\] | 900 | math | 47 |
Given \(1990 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}}\), where \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}\) are distinct non-negative integers. Find \(\alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}\). | 43 | math | 91 |
Given the function $f(x)=2\sin \left(x+ \frac{\pi}{3}\right)\cdot\cos x$.
$(1)$ If $0\leqslant x\leqslant \frac{\pi}{2}$, find the range of the function $f(x)$.
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ corresponding to sides $a$, $b$, $c$ respectively. If $A$ is an acute angle and $f(A)= \frac{\sq... | \frac{5\sqrt{7}}{14} | math | 146 |
Solve for \( x \): \(\sqrt[3]{30x + \sqrt[3]{30x + 18}} = 18.\) | \frac{2907}{15} | math | 36 |
Given $$\cos\alpha= \frac {4}{5}$$ and $$\cos(\alpha+\beta)= \frac {5}{13}$$, where $\alpha$ and $\beta$ are acute angles.
(1) Find the value of $\sin2\alpha$;
(2) Find the value of $\sin\beta$. | \frac {33}{65} | math | 74 |
Given the right focus F of the ellipse C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a>b>0) and a point N(0, \sqrt{2}b), let M be a moving point on C. If the maximum perimeter of \triangle MNF is (\sqrt{6}+2)a, find the eccentricity of C. | \frac{\sqrt{2}}{2} | math | 92 |
The straight line passing through the point (3, 2) intersects the positive axes at points A and B. The minimum value of the area of triangle AOB is \_\_\_\_\_\_. | 12 | math | 41 |
Suppose the function $f(x)$ is a differentiable function defined on $(-\infty, 0)$ with its derivative denoted by $f'(x)$. Given that $3f(x) + xf'(x) > 0$, find the solution set for the inequality $(x+2015)^3f(x+2015) + 27f(-3) > 0$. | (-2018, -2015) | math | 88 |
Given two sequences $\{a_n\}$ and $\{b_n\}$, where $a_1=1$, $a_n+a_{n+1}=b_n$, and $a_na_{n+1}=2^n$, find the value of $b_{10}$. | 64 | math | 60 |
Parallelogram $EFGH$ has vertices $E(5,4)$, $F(-1,-4)$, $G(-7,-2)$, and $H(1,6)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis? Express your answer as a common fraction. | \frac{1}{2} | math | 86 |
Given that $F_{1}(-4,0)$ and $F_{2}(4,0)$ are the two foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, and $P$ is a point on the ellipse such that the area of $\triangle PF_{1}F_{2}$ is $3\sqrt{3}$, find the value of $\cos\angle F_{1}PF_{2}$. | \frac{1}{2} | math | 101 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1, |\overrightarrow{b}| = 4$, and $\overrightarrow{a} \cdot \overrightarrow{b} = 2$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | math | 75 |
A geologist challenges participants in a contest to guess the weight of a meteorite by providing clues. The weight, in grams, must consist of the digits 1, 1, 3, 5, 5, and 8, with the condition that the weight starts with an odd number.
How many valid guesses are possible for the meteorite's weight? | 180 | math | 75 |
Given the following functions:①$y=-3x+2$；②$y=\frac{1}{x}$；③$y=-x^{2}+1$. Choose one function randomly from the above. The probability that the selected function satisfies the condition "when $x \lt -1$, the value of the function $y$ decreases as $x$ increases" is ____. | \frac{2}{3} | math | 84 |
A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching? | \frac{3}{20} | math | 59 |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 | math | 49 |
The condition for $m=0$ to represent a circle in the equation $x^2 + y^2 - 4x + 2y + m = 0$ is when the equation satisfies the condition for a circle. | 0 | math | 48 |
Ben throws six identical darts. Each hits one of five identical dartboards on the wall. After throwing the six darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible? | 11 | math | 50 |
If the equation $mx^2+2x+1=0$ has at least one negative root, then the range of the real number $m$ is \_\_\_\_\_\_. | (-\infty, 1] | math | 40 |
Let \( P \) be a regular 2006-sided polygon. If a diagonal of \( P \), whose endpoints divide the boundary of \( P \) into two parts each containing an odd number of sides, is called a "good diagonal". Note that each side of \( P \) is considered a "good diagonal". Given that 2003 non-intersecting diagonals within \( P... | 1003 | math | 127 |
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