Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
32,400 | In the diagram, $ABCD$ is a trapezoid with an area of $18$. $CD$ is three times the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(1,3)--(10,3)--(15,0)--cycle);
draw((10,3)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,3),NW);
label("$B$",(10,3),NE);
label("$C$",(15,0),E);
[/asy] | 4.5 | 78.90625 |
32,401 | A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole column, we found that there were a total of 1223 columns. We also noticed that the number of sections was 5 more than twice the number of whole columns in each section.
How many columns were there in each section? | 23 | 68.75 |
32,402 | A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly 60 seconds, all of the living cells simultaneously split into two identical copies of itself, and this continues to happen every 60 seconds thereafter. After performing the experiment for awhile, the scientist realizes the population of the culture will be unbounded and quickly shuts down the experiment before the cells take over the world. What is the smallest number of cells that the experiment could have started with? | 61 | 17.96875 |
32,403 | In a diagram, $\triangle ABC$ and $\triangle BDC$ are right-angled, with $\angle ABC = \angle BDC = 45^\circ$, and $AB = 16$. Determine the length of $BC$. | 8\sqrt{2} | 25.78125 |
32,404 | The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yuan, and each student computer costs 7000 yuan. It is known that the total investment for purchasing computers in both labs is equal and is between 200,000 yuan and 210,000 yuan. How many student computers should be prepared for each lab? | 27 | 10.9375 |
32,405 | Given the random variable $\xi$ follows a binomial distribution $B(5,0.5)$, and $\eta=5\xi$, calculate the values of $E\eta$ and $D\eta$ respectively. | \frac{125}{4} | 0 |
32,406 | Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\text A$ , $\text B$ or $\text C$ , possibly repeated. The lock has three buttons, labeled $\text A$ , $\text B$ and $\text C$ . When the most recent $3$ button-presses form the code, the lock opens. What is the minimum number of total button presses Alex needs to guarantee opening the lock? | 29 | 86.71875 |
32,407 | Determine the value of
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\] | 9006 | 0 |
32,408 | A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to randomly generate integer values between 0 and 3, with 0, 1, 2, and 3 representing "惠", "州", "美", and "丽" respectively. Each group of three random numbers represents the result of three draws. The following 16 groups of random numbers were generated:
232 321 230 023 123 021 132 220
231 130 133 231 331 320 122 233
Estimate the probability that the drawing stops exactly on the third draw. | \frac{1}{8} | 0.78125 |
32,409 | Given an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $4$, and the point $P(1,\frac{3}{2})$ lies on the ellipse $E$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ A line $l$ passing through the right focus $F$ of the ellipse $E$ is drawn such that it does not coincide with the two coordinate axes. The line intersects $E$ at two distinct points $M$ and $N$. The perpendicular bisector of segment $MN$ intersects the $y$-axis at point $T$. Find the minimum value of $\frac{|MN|}{|OT|}$ (where $O$ is the origin) and determine the equation of line $l$ at this point. | 24 | 10.9375 |
32,410 | Given the set $\{a, b, c\} = \{0, 1, 2\}$, and the following three conditions: ① $a \neq 2$; ② $b = 2$; ③ $c \neq 0$ are correct for only one of them, then $10a + 2b + c$ equals $\boxed{?}$. | 21 | 52.34375 |
32,411 | A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught?
*Author: Anderson Wang* | 25 | 17.96875 |
32,412 | Let \( X = \{1, 2, \ldots, 98\} \). Call a subset of \( X \) good if it satisfies the following conditions:
1. It has 10 elements.
2. If it is partitioned in any way into two subsets of 5 elements each, then one subset has an element coprime to each of the other 4, and the other subset has an element which is not coprime to any of the other 4.
Find the smallest \( n \) such that any subset of \( X \) of \( n \) elements has a good subset. | 50 | 14.0625 |
32,413 | A block of wood has the shape of a right circular cylinder with a radius of $8$ and a height of $10$. The entire surface of the block is painted red. Points $P$ and $Q$ are chosen on the edge of one of the circular faces such that the arc $\overarc{PQ}$ measures $180^\text{o}$. The block is then sliced in half along the plane passing through points $P$, $Q$, and the center of the cylinder, revealing a flat, unpainted face on each half. Determine the area of one of these unpainted faces expressed in terms of $\pi$ and $\sqrt{d}$. Find $a + b + d$ where the area is expressed as $a \pi + b\sqrt{d}$, with $a$, $b$, $d$ as integers, and $d$ not being divisible by the square of any prime. | 193 | 7.03125 |
32,414 | (Convert the following binary number to decimal: 101111011<sub>(2)</sub>) | 379 | 84.375 |
32,415 | In the following diagram (not to scale), $A$ , $B$ , $C$ , $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$ . Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$ . Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy] | 30 | 44.53125 |
32,416 | Given the regular octagon $ABCDEFGH$ with its center at $J$, and each of the vertices and the center associated with the digits 1 through 9, with each digit used once, such that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal, determine the number of ways in which this can be done. | 1152 | 84.375 |
32,417 | It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.
[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label(" $1$ ",(1,3));
label(" $9$ ",(2,2));
label(" $14$ ",(3,1));
label(" $5$ ",(4,2));[/asy] | 10 | 0.78125 |
32,418 | Given the sequence of even counting numbers starting from $2$, find the sum of the first $3000$ terms and the sequence of odd counting numbers starting from $3$, find the sum of the first $3000$ terms, and then calculate their difference. | -3000 | 29.6875 |
32,419 | Given $\sin \alpha + \cos \alpha = \frac{1}{5}$, and $- \frac{\pi}{2} \leqslant \alpha \leqslant \frac{\pi}{2}$, find the value of $\tan \alpha$. | - \frac{3}{4} | 43.75 |
32,420 | A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces? | 4:3 | 0.78125 |
32,421 | A clothing retailer offered a discount of $\frac{1}{4}$ on all jackets tagged at a specific price. If the cost of the jackets was $\frac{2}{3}$ of the price they were actually sold for and considering this price included a sales tax of $\frac{1}{10}$, what would be the ratio of the cost to the tagged price?
**A)** $\frac{1}{3}$
**B)** $\frac{2}{5}$
**C)** $\frac{11}{30}$
**D)** $\frac{3}{10}$
**E)** $\frac{1}{2}$ | \frac{11}{30} | 8.59375 |
32,422 | Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity of ellipse $C$. | \frac{\sqrt{5} - 1}{2} | 1.5625 |
32,423 | Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola. | \frac{\sqrt{17}}{2} | 1.5625 |
32,424 | Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$. | 4032 | 31.25 |
32,425 | Square $ABCD$ has an area of $256$ square units. Point $E$ lies on side $\overline{BC}$ and divides it in the ratio $3:1$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has an area of $48$ square units, what is the area of triangle $GCD$? | 48 | 8.59375 |
32,426 | A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$. | 5 + 10\sqrt{5} | 2.34375 |
32,427 | Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.
(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason. | m = 4 | 30.46875 |
32,428 | Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$ . Compute the radius of circle $E$ .
*2015 CCA Math Bonanza Individual Round #5* | 3\sqrt{3} - 6 | 2.34375 |
32,429 | A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm). | 300 | 48.4375 |
32,430 | Given the equation $x^{2}+4ax+3a+1=0 (a > 1)$, whose two roots are $\tan \alpha$ and $\tan \beta$, with $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$, find $\tan \frac{\alpha + \beta}{2}$. | -2 | 0.78125 |
32,431 | What is the value of $2 \cdot \sqrt[4]{2^7 + 2^7 + 2^8}$? | 8 \cdot \sqrt[4]{2} | 75.78125 |
32,432 | Given a point $P$ on the curve $y= \frac{1}{2}e^{x}$ and a point $Q$ on the curve $y=\ln (2x)$, determine the minimum value of $|PQ|$. | \sqrt{2}(1-\ln 2) | 0.78125 |
32,433 | A square is completely covered by a large circle and each corner of the square touches a smaller circle of radius \( r \). The side length of the square is 6 units. What is the radius \( R \) of the large circle? | 3\sqrt{2} | 46.09375 |
32,434 | What is the smallest positive value of $m$ such that the equation $10x^2 - mx + 660 = 0$ has integral solutions? | 170 | 8.59375 |
32,435 | Let $a,$ $b,$ and $c$ be three positive real numbers whose sum is 1. If no one of these numbers is more than three times any other, find the minimum value of the product $abc.$ | \frac{9}{343} | 7.03125 |
32,436 | Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 30$, find the length of $BE$. | 20 | 3.90625 |
32,437 | Find the largest solution to \[
\lfloor x \rfloor = 7 + 150 \{ x \},
\] where $\{x\} = x - \lfloor x \rfloor$. | 156.9933 | 0 |
32,438 | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2a\sin B= \sqrt{3}b$ and $\cos C = \frac{5}{13}$:
(1) Find the value of $\sin A$;
(2) Find the value of $\cos B$. | \frac{12\sqrt{3} - 5}{26} | 56.25 |
32,439 | In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is $\frac{1}{4}$ length from $R$. Given $LM = 4$, and $LO = MO = 3$. Find the area of quadrilateral $PRSQ$. | 0.75 | 0 |
32,440 | Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $D(2, \frac{3}{2} \sqrt{3})$ and the distance from point $P$ to the $y$-axis. | \frac{5}{2} | 15.625 |
32,441 | Given the function $f(x) = \frac{\ln x}{x}$.
(1) Find the monotonic intervals of the function $f(x)$;
(2) Let $m > 0$, find the maximum value of $f(x)$ on $[m, 2m]$. | \frac{1}{e} | 0.78125 |
32,442 | In triangle $ABC$, $AB = 12$, $AC = 10$, and $BC = 16$. The centroid $G$ of triangle $ABC$ divides each median in the ratio $2:1$. Calculate the length $GP$, where $P$ is the foot of the perpendicular from point $G$ to side $BC$. | \frac{\sqrt{3591}}{24} | 3.125 |
32,443 | How many positive integers, not exceeding 200, are multiples of 5 or 7 but not 10? | 43 | 24.21875 |
32,444 | Given vectors $\overrightarrow {a}, \overrightarrow {b}$ that satisfy $\overrightarrow {a}\cdot ( \overrightarrow {a}+ \overrightarrow {b})=5$, and $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{\pi}{3} | 100 |
32,445 | Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | 3\sqrt{2} | 8.59375 |
32,446 | There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately? | 1152 | 23.4375 |
32,447 | Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack. | 60 | 42.1875 |
32,448 | Last year, Australian Suzy Walsham won the annual women's race up the 1576 steps of the Empire State Building in New York for a record fifth time. Her winning time was 11 minutes 57 seconds. Approximately how many steps did she climb per minute? | 130 | 3.90625 |
32,449 | Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y=g(x) is symmetric about the line x=2, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22. | -24 | 0.78125 |
32,450 | In our daily life, we often use passwords, such as when making payments through Alipay. There is a type of password generated using the "factorization" method, which is easy to remember. The principle is to factorize a polynomial. For example, the polynomial $x^{3}+2x^{2}-x-2$ can be factorized as $\left(x-1\right)\left(x+1\right)\left(x+2\right)$. When $x=29$, $x-1=28$, $x+1=30$, $x+2=31$, and the numerical password obtained is $283031$.<br/>$(1)$ According to the above method, when $x=15$ and $y=5$, for the polynomial $x^{3}-xy^{2}$, after factorization, what numerical passwords can be formed?<br/>$(2)$ Given a right-angled triangle with a perimeter of $24$, a hypotenuse of $11$, and the two legs being $x$ and $y$, find a numerical password obtained by factorizing the polynomial $x^{3}y+xy^{3}$ (only one is needed). | 24121 | 13.28125 |
32,451 | Last month, Xiao Ming's household expenses were 500 yuan for food, 200 yuan for education, and 300 yuan for other expenses. This month, the costs of these three categories increased by 6%, 20%, and 10%, respectively. What is the percentage increase in Xiao Ming's household expenses for this month compared to last month? | 10\% | 69.53125 |
32,452 | Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If A does not stand at either end, calculate the number of different arrangements where C and D are adjacent. | 24 | 25 |
32,453 | An $10 \times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle. | 10 | 44.53125 |
32,454 | Let $P$ be the portion of the graph of $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$ located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$ . Find $\lfloor 1000d \rfloor$ .
*Proposed by **Th3Numb3rThr33*** | 433 | 14.84375 |
32,455 | How many positive integer multiples of $77$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | 182 | 6.25 |
32,456 | Let the set $I = \{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$? | 49 | 10.15625 |
32,457 | Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box. | 176 | 60.15625 |
32,458 | For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$? | 30 | 63.28125 |
32,459 | Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9. | 417 | 3.125 |
32,460 | The altitude of an equilateral triangle is $\sqrt{12}$ units. What is the area and the perimeter of the triangle, expressed in simplest radical form? | 12 | 69.53125 |
32,461 | Given that \( r, s, t \) are integers, and the set \( \{a \mid a = 2^r + 2^s + 2^t, 0 \leq t < s < r\} \) forms a sequence \(\{a_n\} \) from smallest to largest as \(7, 11, 13, 14, \cdots\), find \( a_{36} \). | 131 | 95.3125 |
32,462 | A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction. | \frac{60}{11} | 57.03125 |
32,463 | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{5}{7} | 6.25 |
32,464 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 20.3125 |
32,465 | The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed. | 15 | 77.34375 |
32,466 | If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another example is the four-digit number $5324$, since $53-32=21\neq 24$, is not a "decreasing number". If a "decreasing number" is $\overline{a312}$, then this number is ______; if the sum of the three-digit number $\overline{abc}$ formed by the first three digits and the three-digit number $\overline{bcd}$ formed by the last three digits of a "decreasing number" is divisible by $9$, then the maximum value of the number that satisfies the condition is ______. | 8165 | 75.78125 |
32,467 | In the Cartesian coordinate system, the center of circle $C$ is at $(2,0)$, and its radius is $\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is:
$$
\begin{cases}
x=-t \\
y=1+t
\end{cases} \quad (t \text{ is a parameter}).
$$
$(1)$ Find the polar coordinate equations of circle $C$ and line $l$;
$(2)$ The polar coordinates of point $P$ are $(1,\frac{\pi}{2})$, line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $|PA|+|PB|$. | 3\sqrt{2} | 16.40625 |
32,468 | In $\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\triangle BPC$? | 12 | 17.1875 |
32,469 | Given that the parabola $y^2=4x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a > 0, b > 0)$ have the same focus $F$, $O$ is the coordinate origin, points $A$ and $B$ are the intersection points of the two curves. If $(\overrightarrow{OA} + \overrightarrow{OB}) \cdot \overrightarrow{AF} = 0$, find the length of the real axis of the hyperbola. | 2\sqrt{2}-2 | 0.78125 |
32,470 | A company buys an assortment of 150 pens from a catalog for \$15.00. Shipping costs an additional \$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen? | 12 | 16.40625 |
32,471 | A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals. | 1139.71 | 11.71875 |
32,472 | A farmer has a right-angled triangular farm with legs of lengths 3 and 4. At the right-angle corner, the farmer leaves an unplanted square area $S$. The shortest distance from area $S$ to the hypotenuse of the triangle is 2. What is the ratio of the area planted with crops to the total area of the farm? | $\frac{145}{147}$ | 0 |
32,473 | In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), the value of \( x \) is: | 38 | 45.3125 |
32,474 | Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5. | \frac{493}{1848} | 7.8125 |
32,475 | Four-digit "progressive numbers" are arranged in ascending order, determine the 30th number. | 1359 | 28.90625 |
32,476 | A sequence \( b_1, b_2, b_3, \ldots \) is defined recursively by \( b_1 = 2 \), \( b_2 = 3 \), and for \( k \geq 3 \),
\[ b_k = \frac{1}{2} b_{k-1} + \frac{1}{3} b_{k-2}. \]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \) | 24 | 37.5 |
32,477 | A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of component B, and 5 kg of component C. It has been determined that for healthy development of the animals, the farm needs at least 45 kg of component A, 60 kg of component B, and 5 kg of component C daily. How much of each feed mixture should they purchase to minimize the cost while meeting the nutritional requirements? | 165 | 4.6875 |
32,478 | Two adjacent faces of a tetrahedron, representing equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing this edge. | \frac{9\sqrt{3}}{4} | 19.53125 |
32,479 | In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is | 48 | 21.09375 |
32,480 | Given the parametric equation of curve \\(C_{1}\\) as \\(\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases} (\alpha\\) is the parameter\\()\\), and taking the origin \\(O\\) of the Cartesian coordinate system \\(xOy\\) as the pole and the positive half-axis of \\(x\\) as the polar axis to establish a polar coordinate system, the polar equation of curve \\(C_{2}\\) is \\(\rho\cos \left(\theta+ \dfrac{\pi}{4}\right)= \sqrt{2} \\).
\\((\\)Ⅰ\\()\\) Find the Cartesian equation of curve \\(C_{2}\\) and the maximum value of the distance \\(|OP|\\) from the moving point \\(P\\) on curve \\(C_{1}\\) to the origin \\(O\\);
\\((\\)Ⅱ\\()\\) If curve \\(C_{2}\\) intersects curve \\(C_{1}\\) at points \\(A\\) and \\(B\\), and intersects the \\(x\\)-axis at point \\(E\\), find the value of \\(|EA|+|EB|\\). | \dfrac{6 \sqrt{3}}{5} | 2.34375 |
32,481 | Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0 |
32,482 | Given a sequence $\{a_n\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\geqslant 2, n\in \mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then $S_{40}=$_______. | 440 | 36.71875 |
32,483 | In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second match and the loser of the first match; the fourth match is between the winner of the third match and the loser of the second match. Based on past records, the probability of A winning over B is 0.4, B winning over C is 0.5, and C winning over A is 0.6.
(1) Calculate the probability of B winning four consecutive matches to end the competition;
(2) Calculate the probability of C winning three consecutive matches to end the competition. | 0.162 | 10.15625 |
32,484 | A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen people in total in the club. How many people will be in the club after five years? | 3164 | 1.5625 |
32,485 | (1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 2146 and 1813.
(2) Use the Horner's method to calculate the value of $v_4$ for the function $f(x) = 2x^5 + 3x^4 + 2x^3 - 4x + 5$ when $x = 2$. | 60 | 10.15625 |
32,486 | Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after his name "Gauss". For $x\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\{x\}=x-[x]$ represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_{n}\}$ satisfies: $a_{1}=\sqrt{3}$, $a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}$, $(n∈N^{*})$, then $a_{2019}=$\_\_\_\_\_\_\_\_. | 3027+ \sqrt{3} | 0.78125 |
32,487 | The altitudes of an acute isosceles triangle, where \(AB = BC\), intersect at point \(H\). Find the area of triangle \(ABC\), given \(AH = 5\) and the altitude \(AD\) is 8. | 40 | 9.375 |
32,488 | Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to the three trick-or-treaters given these restrictions? | 72 | 35.15625 |
32,489 | Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$ , and $T_3$ is a subset of $\{1, 2, 3, 4\}$ ,
2. $T_1 \subseteq T_2 \cup T_3$ ,
3. $T_2 \subseteq T_1 \cup T_3$ , and
4. $T_3\subseteq T_1 \cup T_2$ . | 625 | 85.9375 |
32,490 | If the equation with respect to \( x \), \(\frac{x \lg^2 a - 1}{x + \lg a} = x\), has a solution set that contains only one element, then \( a \) equals \(\quad\) . | 10 | 1.5625 |
32,491 | For any sequence of real numbers $A=\{a_1, a_2, a_3, \ldots\}$, define $\triangle A$ as the sequence $\{a_2 - a_1, a_3 - a_2, a_4 - a_3, \ldots\}$, where the $n$-th term is $a_{n+1} - a_n$. Assume that all terms of the sequence $\triangle (\triangle A)$ are $1$ and $a_{18} = a_{2017} = 0$, find the value of $a_{2018}$. | 1000 | 18.75 |
32,492 | The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. | -1 | 21.09375 |
32,493 | In duck language, only letters $q$ , $a$ , and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter $a$ is a vowel, while $q$ and $k$ are consonants. | 21 | 10.15625 |
32,494 | Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500. | 1587 | 25.78125 |
32,495 | 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} | 27.34375 |
32,496 | Five years from now, Billy's age will be twice Joe's current age. Currently, the sum of their ages is 60. How old is Billy right now? | 38\frac{1}{3} | 0 |
32,497 | What are the mode and median of the set of ages $29$, $27$, $31$, $31$, $31$, $29$, $29$, and $31$? | 30 | 16.40625 |
32,498 | Given the triangular pyramid $P-ABC$ where $PA\bot $ plane $ABC$, $PA=AB=2$, and $\angle ACB=30^{\circ}$, find the surface area of the circumscribed sphere of the triangular pyramid $P-ABC$. | 20\pi | 36.71875 |
32,499 | Given that the first term of a geometric sequence $\{{a_n}\}$ is $\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \in \mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an arithmetic sequence.
(I) Find the general term formula of the sequence $\{{a_n}\}$.
(II) Find the maximum and minimum values of $S_n$ $(n \in \mathbb{N^*})$. | \frac{3}{4} | 26.5625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.