Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
23,200 | If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $8x^3 + 125 = (ax^2 + bx + c)(d x^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? | 770 | 51.5625 |
23,201 | Club Truncator is now in a soccer league with four other teams, each of which it plays once. In any of its 4 matches, the probabilities that Club Truncator will win, lose, or tie are $\frac{1}{3}$, $\frac{1}{3}$, and $\frac{1}{3}$ respectively. The probability that Club Truncator will finish the season with more wins than losses is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 112 | 3.90625 |
23,202 | Given points $A(-2,0)$ and $B(0,2)$, let point $C$ be a moving point on the circle $x^{2}-2x+y^{2}=0$. Determine the minimum area of $\triangle ABC$. | 3-\sqrt{2} | 83.59375 |
23,203 | Given a line $l$ intersects the hyperbola $x^2 - \frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$. | 4\sqrt{2} | 45.3125 |
23,204 | Given a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder. | \frac{\sqrt{3}}{3} | 74.21875 |
23,205 | Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads. | 93.75 | 98.4375 |
23,206 | The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$. | -\frac{3}{5} | 0 |
23,207 | Given the assumption that smoking is unrelated to lung disease, calculate the confidence level that can be concluded from the chi-square statistic $K^2=5.231$, with $P(K^2 \geq 3.841) = 0.05$ and $P(K^2 \geq 6.635) = 0.01$. | 95\% | 64.84375 |
23,208 | Given vectors $\overrightarrow{a}=(-3,1)$, $\overrightarrow{b}=(1,-2)$, and $\overrightarrow{n}=\overrightarrow{a}+k\overrightarrow{b}$ ($k\in\mathbb{R}$).
$(1)$ If $\overrightarrow{n}$ is perpendicular to the vector $2\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$;
$(2)$ If vector $\overrightarrow{c}=(1,-1)$, and $\overrightarrow{n}$ is parallel to the vector $\overrightarrow{c}+k\overrightarrow{b}$, find the value of the real number $k$. | -\frac {1}{3} | 53.90625 |
23,209 | Determine the number of 6-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition and with alternating even and odd digits. | 60 | 33.59375 |
23,210 | In $\triangle ABC$, if $|\overrightarrow{AB}|=2$, $|\overrightarrow{AC}|=3$, $|\overrightarrow{BC}|=4$, and $O$ is the incenter of $\triangle ABC$, and $\overrightarrow{AO}=\lambda \overrightarrow{AB}+\mu \overrightarrow{BC}$, calculate the value of $\lambda+\mu$. | \frac{7}{9} | 68.75 |
23,211 | In the Cartesian coordinate system $xOy$, the parametric equation of line $C_1$ is $\begin{cases} & x=1+\frac{1}{2}t \\ & y=\frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter), and in the polar coordinate system with the origin as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho^2(1+2\sin^2\theta)=3$.
$(1)$ Write the general equation of $C_1$ and the Cartesian coordinate equation of $C_2$;
$(2)$ Line $C_1$ intersects curve $C_2$ at points $A$ and $B$, with point $M(1,0)$. Find $||MA|-|MB||$. | \frac{2}{5} | 56.25 |
23,212 | How many multiples of 5 are there between 5 and 205? | 41 | 96.875 |
23,213 | 1. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?
2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number must be divisible by 5? (Answer with numbers) | 216 | 82.8125 |
23,214 | Evaluate \(\left(d^d - d(d-2)^d\right)^d\) when \(d=4\). | 1358954496 | 17.96875 |
23,215 | Given \\(a > b\\), the quadratic trinomial \\(a{x}^{2}+2x+b \geqslant 0 \\) holds for all real numbers, and there exists \\(x_{0} \in \mathbb{R}\\), such that \\(ax_{0}^{2}+2{x_{0}}+b=0\\), then the minimum value of \\(\dfrac{a^{2}+b^{2}}{a-b}\\) is \_\_\_\_\_\_\_\_\_. | 2 \sqrt{2} | 57.8125 |
23,216 | Given $\sin \left( \frac{\pi}{4}-x\right)= \frac{1}{5} $, and $-\pi < x < - \frac{\pi}{2}$. Find the values of the following expressions:
$(1)\sin \left( \frac{5\pi}{4}-x\right)$;
$(2)\cos \left( \frac{3\pi}{4}+x\right)$;
$(3)\sin \left( \frac{\pi}{4}+x\right)$． | -\frac{2 \sqrt{6}}{5} | 79.6875 |
23,217 | Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\] | \frac{1}{16} | 0 |
23,218 | Let \(p\), \(q\), \(r\), and \(s\) be the roots of the polynomial \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\] | \frac{10}{3} | 89.84375 |
23,219 | Given the function $f(x)=x^{2}-6x+4\ln x$, find the x-coordinate of the quasi-symmetric point of the function. | \sqrt{2} | 15.625 |
23,220 | In $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
label("B", B, E);
label("C", C, NE);
label("P", P, NW);
label("8", 3*dir(A--C), SE);
label("6", B+3*dir(B--C), NE);
label("10", (5,0), S);
[/asy] | 4.8 | 54.6875 |
23,221 | In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total? | 140 | 25 |
23,222 | Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Simon travels west at 12 mph. How many hours will it take for them to be 130 miles apart? | \frac{65}{\sqrt{61}} | 4.6875 |
23,223 | Simplify $\sqrt[3]{8+27} \cdot \sqrt[3]{8+\sqrt{64}}$. | \sqrt[3]{560} | 31.25 |
23,224 | If six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series. | 3888 | 60.15625 |
23,225 | Given vectors $\overrightarrow {a}=(\sin(2x+ \frac {\pi}{6}), 1)$, $\overrightarrow {b}=( \sqrt {3}, \cos(2x+ \frac {\pi}{6}))$, and the function $f(x)= \overrightarrow {a} \cdot \overrightarrow {b}$.
(Ⅰ) Find the interval where the function $f(x)$ is monotonically decreasing;
(Ⅱ) In $\triangle ABC$, where $A$, $B$, and $C$ are the opposite sides of $a$, $b$, and $c$ respectively, if $f(A)= \sqrt {3}$, $\sin C= \frac {1}{3}$, and $a=3$, find the value of $b$. | \sqrt {3}+2 \sqrt {2} | 0 |
23,226 | The graph of the power function $f(x)$ passes through the point $(3, \frac{1}{9})$, find the maximum value of the function $g(x) = (x-1)f(x)$ on the interval $[1,3]$. | \frac{1}{4} | 68.75 |
23,227 | Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white. | \frac{1}{16} | 47.65625 |
23,228 | Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches? | 3375 | 97.65625 |
23,229 | The maximum value of the function $f(x) = 8\sin x - \tan x$, defined on $\left(0, \frac{\pi}{2}\right)$, is $\_\_\_\_\_\_\_\_\_\_\_\_$. | 3\sqrt{3} | 84.375 |
23,230 | Given four real numbers that form an arithmetic sequence: -9, $a_1$, $a_2$, -1, and five real numbers that form a geometric sequence: -9, $b_1$, $b_2$, $b_3$, -1, find the value of $b_2(a_2-a_1)$. | -8 | 97.65625 |
23,231 | On a 6 by 6 grid of points, what fraction of the larger square's area is inside the new shaded square? Place the bottom-left vertex of the square at grid point (3,3) and the square rotates 45 degrees (square's sides are diagonals of the smaller grid cells).
```
[asy]
size(6cm);
fill((3,3)--(4,4)--(5,3)--(4,2)--cycle,gray(0.7));
dot((1,1));
for (int i = 0; i <= 6; ++i) {
draw((0,i)--(6,i));
draw((i,0)--(i,6));
for (int j = 0; j <= 6; ++j) {
dot((i,j));
}
}
draw((3,3)--(4,4)--(5,3)--(4,2)--cycle);
[/asy]
``` | \frac{1}{18} | 64.0625 |
23,232 | The maximum value of $k$ such that the inequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution. | \sqrt{6} | 79.6875 |
23,233 | Green Valley School has 120 students enrolled, consisting of 70 boys and 50 girls. If $\frac{1}{7}$ of the boys and $\frac{1}{5}$ of the girls are absent on a particular day, what percent of the total student population is absent? | 16.67\% | 72.65625 |
23,234 | In a set of 15 different-colored markers, how many ways can Jane select five markers if the order of selection does not matter? | 3003 | 100 |
23,235 | Given that the two asymptotes of the hyperbola $\dfrac{y^2}{4}-x^2=1$ intersect with the directrix of the parabola $y^2=2px(p > 0)$ at points $A$ and $B$, and $O$ is the origin, determine the value of $p$ given that the area of $\Delta OAB$ is $1$. | \sqrt{2} | 96.875 |
23,236 | In $\triangle ABC$, two side lengths are $2$ and $3$, and the cosine value of the included angle is $\frac{1}{3}$. Find the radius of the circumscribed circle. | \frac{9\sqrt{2}}{8} | 60.15625 |
23,237 | When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(7, 8)$, and $H(7, 3)$. What is the area of trapezoid $EFGH$? | 25 | 100 |
23,238 | Suppose that $a_1 = 1$ , and that for all $n \ge 2$ , $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$ . If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$ , find $k$ .
*Proposed by Andrew Wu* | 2022 | 75 |
23,239 | Find $\tan G$ in the right triangle where side GH is 20 units, side FG is 25 units, and ∠H is the right angle.
[asy]
pair H,F,G;
H = (0,0);
G = (20,0);
F = (0,25);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$25$",(F+G)/2,NE);
label("$20$",G/2,S);
[/asy] | \frac{3}{4} | 80.46875 |
23,240 | The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both tests, the higher score of the two tests will be taken as the final score. If the student scores full marks in the first test, they will not take the second test. To prepare for the English listening and speaking test, Li Ming takes English listening and speaking mock exams every week. The table below shows his scores in $20$ English listening and speaking mock exams before the first actual test.
Assumptions:
1. The difficulty level of the mock exams is equivalent to that of the actual exam.
2. The difficulty level of the two actual listening and speaking tests is the same.
3. If Li Ming continues to practice listening and speaking after not scoring full marks in the first test, the probability of scoring full marks in the second test can reach $\frac{1}{2}$.
| $46$ | $50$ | $47$ | $48$ | $49$ | $50$ | $50$ | $47$ | $48$ | $47$ |
|------|------|------|------|------|------|------|------|------|------|
| $48$ | $49$ | $50$ | $49$ | $50$ | $50$ | $48$ | $50$ | $49$ | $50$ |
$(Ⅰ)$ Let event $A$ be "Li Ming scores full marks in the first English listening and speaking test." Estimate the probability of event $A$ using frequency.
$(Ⅱ)$ Based on the assumptions in the problem, estimate the maximum probability of Li Ming scoring full marks in the English college entrance examination listening and speaking test. | \frac{7}{10} | 0.78125 |
23,241 | Given that $3\sin \alpha - 2\cos \alpha = 0$, find the value of the following expressions:
$$(1)\ \frac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha} + \frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha};$$
$$(2)\ \sin^2\alpha - 2\sin \alpha\cos \alpha + 4\cos^2\alpha.$$ | \frac{28}{13} | 98.4375 |
23,242 | Calculate the number of terms in the simplified expression of \[(x+y+z)^{2020} + (x-y-z)^{2020},\] by expanding it and combining like terms. | 1,022,121 | 0 |
23,243 | Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 12 | 14.84375 |
23,244 | Evaluate the expression: \\( \frac {\cos 40 ^{\circ} +\sin 50 ^{\circ} (1+ \sqrt {3}\tan 10 ^{\circ} )}{\sin 70 ^{\circ} \sqrt {1+\cos 40 ^{\circ} }}\\) | \sqrt {2} | 0 |
23,245 | Given that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\angle F_{1}PF_{2}=60^{\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$. | \frac{\sqrt{7}}{4} | 100 |
23,246 | Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$. | \dfrac{e^{2}-1}{e^{2}+1} | 0 |
23,247 | Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\triangle ABC$, with $a=4$ and $(4+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum value of the area of $\triangle ABC$. | 4\sqrt{3} | 57.8125 |
23,248 | Seven balls are numbered 1 through 7 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and greater than 15? Express your answer as a common fraction. | \frac{6}{49} | 18.75 |
23,249 | Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. | 120 | 97.65625 |
23,250 | A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. What is the original ratio of the number of pairs of black socks to the number of pairs of blue socks? | 1: 4 | 3.90625 |
23,251 | Consider the geometric sequence $5$, $\dfrac{15}{4}$, $\dfrac{45}{16}$, $\dfrac{135}{64}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction. | \frac{98415}{262144} | 24.21875 |
23,252 | In $\triangle ABC$, $AB=7$, $BC=5$, $CA=6$, then $\overrightarrow{AB} \cdot \overrightarrow{BC} =$ \_\_\_\_\_\_. | -19 | 72.65625 |
23,253 | In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)? | 200 | 17.96875 |
23,254 | Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = 2 + f\left(\frac{1}{2}\right)\log_{2}x$, evaluate $f(-2)$. | -3 | 99.21875 |
23,255 | $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, where $0 < \alpha < \frac{\pi}{2}$, find the value of $\tan \alpha$. | \frac{\sqrt{15}}{15} | 10.9375 |
23,256 | For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ . | 4 + 2\sqrt{2} | 60.9375 |
23,257 | Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?
A) 16
B) 20
C) 24
D) 30
E) 45 | 20 | 75 |
23,258 | Given that α is an acute angle, cos(α+π/6) = 2/3, find the value of sinα. | \dfrac{\sqrt{15} - 2}{6} | 9.375 |
23,259 | Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$. | \frac{1}{2} | 10.9375 |
23,260 | In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$ | 17 | 78.90625 |
23,261 | Find the x-coordinate of point Q, given that point P has coordinates $(\frac{3}{5}, \frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\angle POQ = \frac{3\pi}{4}$. | -\frac{7\sqrt{2}}{10} | 73.4375 |
23,262 | Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers. | 3\sqrt{3} | 1.5625 |
23,263 | Consider the geometric sequence $5$, $\dfrac{15}{4}$, $\dfrac{45}{16}$, $\dfrac{135}{64}$, $\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction. | \frac{98415}{262144} | 28.125 |
23,264 | A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction. | \frac{1}{6561} | 78.125 |
23,265 | Given the quadratic function $f(x)=x^{2}+mx+n$.
(1) If $f(x)$ is an even function with a minimum value of $1$, find the analytic expression of $f(x)$;
(2) Under the condition of (1), for the function $g(x)= \frac {6x}{f(x)}$, solve the inequality $g(2^{x}) > 2^{x}$ with respect to $x$;
(3) For the function $h(x)=|f(x)|$, if the maximum value of $h(x)$ is $M$ when $x\in[-1,1]$, and $M\geqslant k$ holds for any real numbers $m$ and $n$, find the maximum value of $k$. | \frac {1}{2} | 2.34375 |
23,266 | Given the function $f(x)$ is a monotonic function on the real numbers $\mathbb{R}$, and for any real number $x$, it satisfies the equation $f\left[f(x)+ \frac{2}{2^{x}+1}\right]= \frac{1}{3}$, find the value of $f(\log_23)$. | \frac{1}{2} | 59.375 |
23,267 | Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares. | 153 | 83.59375 |
23,268 | Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers needed at intersections. | 51 | 66.40625 |
23,269 | Let $f$ be a mapping from set $A = \{a, b, c, d\}$ to set $B = \{0, 1, 2\}$.
(1) How many different mappings $f$ are there?
(2) If it is required that $f(a) + f(b) + f(c) + f(d) = 4$, how many different mappings $f$ are there? | 19 | 92.1875 |
23,270 | Given an ellipse in the Cartesian coordinate system $xoy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1,\frac{1}{2})$.
(1) Find the standard equation of the ellipse;
(2) If $P$ is a moving point on the ellipse, find the trajectory equation of the midpoint $M$ of the line segment $PA$;
(3) A line passing through the origin $O$ intersects the ellipse at points $B$ and $C$. Find the maximum area of $\triangle ABC$. | \sqrt{2} | 8.59375 |
23,271 | Given the ellipse $\frac{x^2}{3} + y^2 = 1$ and the line $l: y = kx + m$ intersecting the ellipse at two distinct points $A$ and $B$.
(1) If $m = 1$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ ($O$ is the origin), find the value of $k$.
(2) If the distance from the origin $O$ to the line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$. | \frac{\sqrt{3}}{2} | 18.75 |
23,272 | A polynomial with integer coefficients is of the form
\[8x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]Find the number of different possible rational roots for this polynomial. | 28 | 1.5625 |
23,273 | In $\triangle ABC$, $|AB|=5$, $|AC|=6$, if $B=2C$, then calculate the length of edge $BC$. | \frac {11}{5} | 23.4375 |
23,274 | Given the equation $x^{2}-px+q=0$ ($p > 0, q > 0$) with two distinct roots $x_{1}$, $x_{2}$, and the fact that $x_{1}$, $x_{2}$, and $-2$ can be appropriately sorted to form an arithmetic sequence as well as a geometric sequence, find the value of $p \times q$. | 20 | 37.5 |
23,275 | Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group at two separate circular tables? | 1247400 | 0.78125 |
23,276 | Determine the length of the interval of solutions of the inequality $a \le 3x + 6 \le b$ where the length of the interval is $15$. | 45 | 30.46875 |
23,277 | Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \] | 20 | 39.84375 |
23,278 | A certain department store sells suits and ties, with each suit priced at $1000$ yuan and each tie priced at $200 yuan. During the "National Day" period, the store decided to launch a promotion offering two discount options to customers.<br/>Option 1: Buy one suit and get one tie for free;<br/>Option 2: Pay 90% of the original price for both the suit and the tie.<br/>Now, a customer wants to buy 20 suits and $x$ ties $\left(x > 20\right)$.<br/>$(1)$ If the customer chooses Option 1, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$). If the customer chooses Option 2, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$).<br/>$(2)$ If $x=30$, calculate and determine which option is more cost-effective at this point.<br/>$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Please describe your purchasing method. | 21800 | 72.65625 |
23,279 | When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? | 205 | 4.6875 |
23,280 | Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by maintaining the same ratio of ingredients. | 26 | 57.03125 |
23,281 | Allison, Brian, and Noah each have a die. All of the faces on Allison's die have a 5. The faces on Brian's die are numbered 1, 2, 3, 4, 4, 5, 5, and 6. Noah's die has an 8-sided die with the numbers 2, 2, 6, 6, 3, 3, 7, and 7. All three dice are rolled. What is the probability that Allison's roll is greater than each of Brian's and Noah's? Express your answer as a common fraction. | \frac{5}{16} | 16.40625 |
23,282 | Given the functions $f(x)=(x-2)e^{x}$ and $g(x)=kx^{3}-x-2$,
(1) Find the range of $k$ such that the function $g(x)$ is not monotonic in the interval $(1,2)$;
(2) Find the maximum value of $k$ such that the inequality $f(x)\geqslant g(x)$ always holds when $x\in[0,+\infty)$. | \frac{1}{6} | 46.875 |
23,283 | Dad is $a$ years old this year, which is 4 times plus 3 years more than Xiao Hong's age this year. Xiao Hong's age expressed in an algebraic expression is ____. If Xiao Hong is 7 years old this year, then Dad's age is ____ years old. | 31 | 89.84375 |
23,284 | Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\cos \alpha,3\sin \alpha)$ respectively, and $\alpha\in\left( \frac {\pi}{2}, \frac {3\pi}{4}\right)$. If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $\frac {2\sin ^{2}\alpha-\sin 2\alpha}{1+\tan \alpha}$. | - \frac {7 \sqrt {23}}{48} | 0 |
23,285 | Given that the vertex of angle \\(θ\\) is at the origin of coordinates, its initial side coincides with the positive half-axis of \\(x\\), and its terminal side is on the line \\(4x+3y=0\\), then \\( \dfrac{\cos \left( \left. \dfrac{π}{2}+θ \right. \right)-\sin (-π-θ)}{\cos \left( \left. \dfrac{11π}{2}-θ \right. \right)+\sin \left( \left. \dfrac{9π}{2}+θ \right. \right)}=\)\_\_\_\_\_\_\_\_． | \dfrac{8}{7} | 77.34375 |
23,286 | How many integers $-12 \leq n \leq 12$ satisfy $(n-3)(n+5)(n+9)<0$? | 10 | 82.03125 |
23,287 | Given the function $f\left(x\right)=4^{x}+m\cdot 2^{x}$, where $m\in R$.
$(1)$ If $m=-3$, solve the inequality $f\left(x\right) \gt 4$ with respect to $x$.
$(2)$ If the minimum value of the function $y=f\left(x\right)+f\left(-x\right)$ is $-4$, find the value of $m$. | -3 | 55.46875 |
23,288 | Suppose $105 \cdot 77 \cdot 132 \equiv m \pmod{25}$, where $0 \le m < 25$. | 20 | 75 |
23,289 | Find the constant term in the expansion of $\left( \sqrt {x}+ \dfrac {1}{2 \sqrt {x}}\right)^{8}$. | \dfrac {35}{8} | 89.84375 |
23,290 | Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .
*Proposed by Michael Tang* | 423 | 29.6875 |
23,291 | Given that the parabola passing through points $A(2-3b, m)$ and $B(4b+c-1, m)$ is $y=-\frac{1}{2}x^{2}+bx-b^{2}+2c$, if the parabola intersects the $x$-axis, calculate the length of segment $AB$. | 12 | 7.03125 |
23,292 | Let $f(x) = a\cos x - \left(x - \frac{\pi}{2}\right)\sin x$, where $x \in \left[0, \frac{\pi}{2}\right]$.
$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$.
$(2)$ If $f(x) \leq 0$ always holds, find the maximum value of the real number $a$. | -1 | 8.59375 |
23,293 | A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.
A) $\frac{10}{3}$
B) $5$
C) $\frac{15}{4}$
D) $2$ | \frac{10}{3} | 69.53125 |
23,294 | Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients. | 84 | 6.25 |
23,295 | The left and right foci of the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\angle F_{1}PF_{2} = 30^{\circ}$. Find the area of triangle $F_{1}PF_{2}$. | 18 - 9\sqrt{3} | 46.875 |
23,296 | Given vectors $\overrightarrow{a}=(x,3)$ and $\overrightarrow{b}=(-1,y-1)$, and $\overrightarrow{a}+2\overrightarrow{b}=(0,1)$, find the value of $|\overrightarrow{a}+\overrightarrow{b}|$. | \sqrt{5} | 96.09375 |
23,297 | In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\alpha$, if $\cos(\alpha+\beta)=\frac{1}{3}$, where $\beta \in (0,\pi)$, then $\cos \beta =\_\_\_\_\_\_.$ | \frac{3 + 8\sqrt{2}}{15} | 0 |
23,298 | Let $P, Q$ be functions defined as $P(x) = 3\sqrt{x}$ and $Q(x) = x^2$. Calculate the value of $P(Q(P(Q(P(Q(5))))))$. | 135 | 78.90625 |
23,299 | Given that $a$, $b$, and $c$ are positive real numbers satisfying $a + 2b + 3c = 6$, find the maximum value of $\sqrt{a + 1} + \sqrt{2b + 1} + \sqrt{3c + 1}$. | 3\sqrt{3} | 60.9375 |
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