Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
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|---|---|---|---|
30,800 | If there are exactly three points on the circle $x^{2}+y^{2}-2x-6y+1=0$ that are at a distance of $2$ from the line $y=kx$, determine the possible values of $k$. | \frac{4}{3} | 84.375 |
30,801 | Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one. | 153846 | 78.125 |
30,802 | Given that the four vertices A, B, C, D of the tetrahedron A-BCD are all on the surface of the sphere O, AC ⊥ the plane BCD, and AC = 2√2, BC = CD = 2, calculate the surface area of the sphere O. | 16\pi | 32.8125 |
30,803 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(b+2c,a)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the measure of angle $A$.
(2) If $a=4 \sqrt {3}$ and $b+c=8$, find the length of the altitude $h$ on edge $AC$. | 2 \sqrt {3} | 0 |
30,804 | Regular octagon $ABCDEFGH$ is divided into eight smaller isosceles triangles, with vertex angles at the center of the octagon, such as $\triangle ABJ$, by constructing lines from each vertex to the center $J$. By connecting every second vertex (skipping one vertex in between), we obtain a larger equilateral triangle $\triangle ACE$, both shown in boldface in a notional diagram. Compute the ratio $[\triangle ABJ]/[\triangle ACE]$. | \frac{1}{4} | 3.125 |
30,805 | Given points P and Q are on a circle of radius 7 and the length of chord PQ is 8. Point R is the midpoint of the minor arc PQ. Find the length of the line segment PR. | \sqrt{98 - 14\sqrt{33}} | 28.125 |
30,806 | What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common? | 62 | 2.34375 |
30,807 | Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ . | 907200 | 0 |
30,808 | The equation $x^3 + ax^2 = -30$ has only integer solutions for $x$. If $a$ is a positive integer, what is the greatest possible value of $a$? | 29 | 9.375 |
30,809 | In triangle $ABC$, medians $AF$ and $BE$ intersect at angle $\theta = 60^\circ$, where $AF = 10$ and $BE = 15$. Calculate the area of triangle $ABC$.
A) $150\sqrt{3}$
B) $200\sqrt{3}$
C) $250\sqrt{3}$
D) $300\sqrt{3}$ | 200\sqrt{3} | 7.03125 |
30,810 | Below is the graph of $y = a \sin (bx + c)$ for some constants $a$, $b$, and $c$. Assume $a>0$ and $b>0$. Find the smallest possible value of $c$ if it is given that the graph reaches its minimum at $x = 0$. | \frac{3\pi}{2} | 82.03125 |
30,811 | The volume of the geometric body formed by points whose distance to line segment AB is no greater than three units is $216 \pi$. Calculate the length of the line segment AB. | 20 | 27.34375 |
30,812 | Determine how many perfect cubes exist between \(3^6 + 1\) and \(3^{12} + 1\), inclusive. | 72 | 46.875 |
30,813 | Maria needs to provide her waist size in centimeters for a custom dress order. She knows her waist measures 28 inches. If she has been advised to add an extra 1 inch for a comfortable fit and there are 12 inches in a foot, and 31 centimeters in a foot, how many centimeters should Maria specify for her waist size? | 74.9 | 0 |
30,814 | Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 406 | 87.5 |
30,815 | In rectangle $PQRS$, $PQ = 8$ and $QR = 4$. Points $T$ and $U$ are on $\overline{RS}$ such that $RT = 2$ and $SU = 3$. Lines $PT$ and $QU$ intersect at $V$. Find the area of $\triangle PVQ$. [asy]
pair P,Q,R,S,V,T,U;
P=(0,0);
Q=(8,0);
R=(8,4);
S=(0,4);
T=(2,4);
U=(6,4);
V=(3.2,6);
draw(P--Q--R--S--cycle,linewidth(0.7));
draw(P--T--V--cycle,linewidth(0.7));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,NE);
label("$S$",S,NW);
label("$T$",T,S);
label("$U$",U,S);
label("$V$",V,N);
label("2",(1,4),N);
label("3",(7,4),N);
label("4",(8,2),E);
label("8",(4,0),S);
[/asy] | 32 | 80.46875 |
30,816 | A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year? | 13.04\% | 60.15625 |
30,817 | The water tank in the diagram is in the shape of an inverted right circular cone. The radius of its base is 20 feet, and its height is 100 feet. The water in the tank fills 50% of the tank's total capacity. Find the height of the water in the tank, which can be expressed in the form \( a\sqrt[3]{b} \), where \( a \) and \( b \) are positive integers and \( b \) is not divisible by a perfect cube greater than 1. What is \( a+b \)? | 52 | 6.25 |
30,818 | Given the ellipse $\frac{x^{2}}{m+1}+y^{2}=1$ $(m > 0)$ with two foci $F_{1}$ and $F_{2}$, and $E$ is a common point of the line $y=x+2$ and the ellipse, calculate the eccentricity of the ellipse when the sum of distances $|EF_{1}|+|EF_{2}|$ reaches its minimum value. | \frac{\sqrt{6}}{3} | 32.03125 |
30,819 | Given that $7^{-1} \equiv 55 \pmod{78}$, find $49^{-1} \pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.) | 61 | 28.90625 |
30,820 | [asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$ . | 16 | 18.75 |
30,821 | When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80. | 600 | 38.28125 |
30,822 | Given real numbers $x$ and $y$ satisfying $x^{2}+4y^{2}\leqslant 4$, find the maximum value of $|x+2y-4|+|3-x-y|$. | 12 | 0 |
30,823 | Define $F(x, y, z) = x \times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$? | 8 \cdot \sqrt[3]{2} | 7.8125 |
30,824 | The sum of the coefficients of all rational terms in the expansion of $$(2 \sqrt {x}- \frac {1}{x})^{6}$$ is \_\_\_\_\_\_ (answer with a number). | 365 | 78.90625 |
30,825 | Given an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the minor axis is $2 \sqrt{3}$, and the eccentricity $e= \frac{1}{2}$,
(1) Find the standard equation of ellipse $(C)$;
(2) If $F_{1}$ and $F_{2}$ are the left and right foci of ellipse $(C)$, respectively, a line $(l)$ passes through $F_{2}$ and intersects with ellipse $(C)$ at two distinct points $A$ and $B$, find the maximum value of the inradius of $\triangle F_{1}AB$. | \frac{3}{4} | 16.40625 |
30,826 | From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. | \frac{6}{35} | 22.65625 |
30,827 | Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 416 | 0 |
30,828 | How many non-similar regular 720-pointed stars are there, given that a regular $n$-pointed star requires its vertices to not all align with vertices of a smaller regular polygon due to common divisors other than 1 between the step size and $n$? | 96 | 21.875 |
30,829 | Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? | 8\pi - 16 | 42.96875 |
30,830 | In triangle $AHI$, which is equilateral, lines $\overline{BC}$, $\overline{DE}$, and $\overline{FG}$ are all parallel to $\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$. | \frac{3}{4} | 71.875 |
30,831 | $J K L M$ is a square. Points $P$ and $Q$ are outside the square such that triangles $J M P$ and $M L Q$ are both equilateral. The size, in degrees, of angle $P Q M$ is | 15 | 29.6875 |
30,832 | Find the coefficient of $x^{90}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{12} - 12)(x^{13} - 13).\] | -1 | 3.125 |
30,833 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 47 | 3.125 |
30,834 | A line $y = -2$ intersects the graph of $y = 5x^2 + 2x - 6$ at points $C$ and $D$. Find the distance between points $C$ and $D$, expressed in the form $\frac{\sqrt{p}}{q}$ where $p$ and $q$ are positive coprime integers. What is $p - q$? | 16 | 4.6875 |
30,835 | In triangle $\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\sin C$ and the area of $\triangle ABC$.<br/>Condition 1: $c=7$, $\cos A=-\frac{1}{7}$;<br/>Condition 2: $\cos A=\frac{1}{8}$, $\cos B=\frac{9}{16}$.<br/>Note: If both conditions 1 and 2 are answered separately, the first answer will be scored. | \frac{15\sqrt{7}}{4} | 3.90625 |
30,836 | Determine the smallest positive integer \(n\) for which there exists positive real numbers \(a\) and \(b\) such that
\[(a + 3bi)^n = (a - 3bi)^n,\]
and compute \(\frac{b}{a}\). | \frac{\sqrt{3}}{3} | 13.28125 |
30,837 | In the Cartesian coordinate system $xOy$, a moving point $M(x,y)$ always satisfies the relation $2 \sqrt {(x-1)^{2}+y^{2}}=|x-4|$.
$(1)$ What is the trajectory of point $M$? Write its standard equation.
$(2)$ The distance from the origin $O$ to the line $l$: $y=kx+m$ is $1$. The line $l$ intersects the trajectory of $M$ at two distinct points $A$ and $B$. If $\overrightarrow{OA} \cdot \overrightarrow{OB}=-\frac{3}{2}$, find the area of triangle $AOB$. | \frac{3\sqrt{7}}{5} | 25.78125 |
30,838 | Given $$a_{n}= \frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\{a_n\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\{b_n\}$. Find the value of $b_{21}$. | 861 | 78.125 |
30,839 | There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ points. If the expected value of the score $X$ obtained from drawing a ball from the bag is $\frac{5}{3}$. <br/>$(1)$ Find the value of the positive integer $a$; <br/>$(2)$ Draw $3$ balls from the bag at once, and find the probability that the sum of the scores obtained is $5$. | \frac{3}{10} | 28.90625 |
30,840 | Given that $θ$ is an angle in the second quadrant and $\tan(\begin{matrix}θ+ \frac{π}{4}\end{matrix}) = \frac{1}{2}$, find the value of $\sin(θ) + \cos(θ)$. | -\frac{\sqrt{10}}{5} | 85.15625 |
30,841 | A line that passes through the two foci of the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola. | \frac{\sqrt{5} + 1}{2} | 15.625 |
30,842 | Find $c$ such that $\lfloor c \rfloor$ satisfies
\[3x^2 - 9x - 30 = 0\]
and $\{ c \} = c - \lfloor c \rfloor$ satisfies
\[4x^2 - 8x + 1 = 0.\] | 6 - \frac{\sqrt{3}}{2} | 53.90625 |
30,843 | On the side \(AB\) of an equilateral triangle \(\mathrm{ABC}\), a right triangle \(\mathrm{AHB}\) is constructed (\(\mathrm{H}\) is the vertex of the right angle) such that \(\angle \mathrm{HBA}=60^{\circ}\). Let the point \(K\) lie on the ray \(\mathrm{BC}\) beyond the point \(\mathrm{C}\), and \(\angle \mathrm{CAK}=15^{\circ}\). Find the angle between the line \(\mathrm{HK}\) and the median of the triangle \(\mathrm{AHB}\) drawn from the vertex \(\mathrm{H}\). | 15 | 21.09375 |
30,844 | In a certain math competition, there are 10 multiple-choice questions. Each correct answer earns 4 points, no answer earns 0 points, and each wrong answer deducts 1 point. If the total score becomes negative, the grading system automatically sets the total score to zero. How many different total scores are possible? | 35 | 1.5625 |
30,845 | What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 91.40625 |
30,846 | Let $z$ be a complex number with $|z| = 2.$ Find the maximum value of
\[ |(z-2)^3(z+2)|. \] | 24\sqrt{3} | 2.34375 |
30,847 | In a class of 50 students, each student must be paired with another student for a project. Due to prior groupings in other classes, 10 students already have assigned partners. If the pairing for the remaining students is done randomly, what is the probability that Alex is paired with her best friend, Jamie? Express your answer as a common fraction. | \frac{1}{29} | 0.78125 |
30,848 | How many of the first 512 smallest positive integers written in base 8 use the digit 5 or 6 (or both)? | 296 | 0 |
30,849 | In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:
\[\begin{array}{ccccccccc}&&&&*&&&&&&&*&*&*&&&&&*&*&*&*&*&&&*&*&*&*&*&*&*& *&*&*&*&*&*&*&*&*&&&&\vdots&&&&\end{array}\]
Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly $25$ jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between $10$ and $20$ rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no marble gets put inside more than one bag.) | 45 | 54.6875 |
30,850 | Find the smallest number composed exclusively of ones that is divisible by 333...33 (where there are 100 threes in the number). | 300 | 5.46875 |
30,851 | On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse in the middle. It is known that after the truck has traveled 300 kilometers (the truck chose the optimal route), it successfully completed the task. There are warehouses on this straight road. | 25 | 27.34375 |
30,852 | A list of seven positive integers has a median of 4 and a mean of 10. What is the maximum possible value of the list's largest element? | 52 | 0 |
30,853 | A natural number greater than 1 is defined as nice if it is equal to the product of its distinct proper divisors. A number \( n \) is nice if:
1. \( n = pq \), where \( p \) and \( q \) are distinct prime numbers.
2. \( n = p^3 \), where \( p \) is a prime number.
3. \( n = p^2q \), where \( p \) and \( q \) are distinct prime numbers.
Determine the sum of the first ten nice numbers under these conditions. | 182 | 53.90625 |
30,854 | A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. | \frac{3}{5} | 13.28125 |
30,855 | Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with the sum of their first $n$ terms being $S\_n$ and $T\_n$ respectively. If $\frac{S\_n}{T\_n} = \frac{2n}{3n+1}$, find the value of $\frac{a\_{11}}{b\_{11}}$. | \frac{21}{32} | 84.375 |
30,856 | Let point O be a point inside triangle ABC with an area of 6, and it satisfies $$\overrightarrow {OA} + \overrightarrow {OB} + 2\overrightarrow {OC} = \overrightarrow {0}$$, then the area of triangle AOC is \_\_\_\_\_\_. | \frac {3}{2} | 5.46875 |
30,857 | The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$. | 60 | 26.5625 |
30,858 | Consider a large square of side length 60 units, subdivided into a grid with non-uniform rows and columns. The rows are divided into segments of 20, 20, and 20 units, and the columns are divided into segments of 15, 15, 15, and 15 units. A shaded region is created by connecting the midpoint of the leftmost vertical line to the midpoint of the top horizontal line, then to the midpoint of the rightmost vertical line, and finally to the midpoint of the bottom horizontal line. What is the ratio of the area of this shaded region to the area of the large square? | \frac{1}{4} | 11.71875 |
30,859 | The volumes of a regular tetrahedron and a regular octahedron have equal edge lengths. Find the ratio of their volumes without calculating the volume of each polyhedron. | 1/2 | 0.78125 |
30,860 | The year 2013 has arrived, and Xiao Ming's older brother sighed and said, "This is the first year in my life that has no repeated digits." It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013? | 18 | 12.5 |
30,861 | Determine the sum and product of the solutions of the quadratic equation $9x^2 - 45x + 50 = 0$. | \frac{50}{9} | 13.28125 |
30,862 | A class leader is planning to invite graduates from the class of 2016 to give speeches. Out of 8 people, labeled A, B, ..., H, the leader wants to select 4 to speak. The conditions are: (1) at least one of A and B must participate; (2) if both A and B participate, there must be exactly one person speaking between them. The number of different speaking orders is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (please answer with a number). | 1080 | 5.46875 |
30,863 | If \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that minimizes \( J(p) \)? | \frac{9}{16} | 37.5 |
30,864 | $ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle. | \frac{1+\sqrt{13}}{3} | 18.75 |
30,865 | Compute $\left(\sqrt{625681 + 1000} - \sqrt{1000}\right)^2$. | 626681 - 2 \cdot \sqrt{626681} \cdot 31.622776601683793 + 1000 | 0 |
30,866 | If the function $f(x)=x^{2}-m\cos x+m^{2}+3m-8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is \_\_\_\_\_\_. | \{2\} | 5.46875 |
30,867 | A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a "best of seven games" format (i.e., the first player to win four games wins the match and the competition ends). Assuming that the probability of A winning each game is $\frac{1}{3}$. <br/>$(1)$ Find the probability that exactly $5$ games are played when the match ends; <br/>$(2)$ If A leads with a score of $3:1$, let $X$ represent the number of games needed to finish the match. Find the probability distribution and expectation of $X$. | \frac{19}{9} | 0.78125 |
30,868 | Triangle $ABC$ has $\angle{A}=90^{\circ}$ , $AB=2$ , and $AC=4$ . Circle $\omega_1$ has center $C$ and radius $CA$ , while circle $\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\omega_2$ and in the interior of $ABC$ , such that $BM$ is parallel to $EC$ . Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$ . What is the area of quadrilateral $ZEBK$ ? | 20 | 0 |
30,869 | The radius of the Earth measures approximately $6378 \mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.
Next, suppose that the length of the wire is increased by $1 \mathrm{~m}$, so that the wire and the Equator are concentric circles around the Earth. Can a standing man, an ant, or an elephant pass underneath this wire? | 0.159 | 0 |
30,870 | Given a triangle ABC, let the lengths of the sides opposite to angles A, B, C be a, b, c, respectively. If a, b, c satisfy $a^2 + c^2 - b^2 = \sqrt{3}ac$,
(1) find angle B;
(2) if b = 2, c = $2\sqrt{3}$, find the area of triangle ABC. | 2\sqrt{3} | 53.90625 |
30,871 | Given the discrete random variable $X$ follows a two-point distribution, and $P\left(X=1\right)=p$, $D(X)=\frac{2}{9}$, determine the value of $p$. | \frac{2}{3} | 35.9375 |
30,872 | A ball with a diameter of 6 inches rolls along a complex track from start point A to endpoint B. The track comprises four semicircular arcs with radii $R_1 = 120$ inches, $R_2 = 50$ inches, $R_3 = 90$ inches, and $R_4 = 70$ inches respectively. The ball always stays in contact with the track and rolls without slipping. Calculate the distance traveled by the center of the ball from A to B.
A) $320\pi$ inches
B) $330\pi$ inches
C) $340\pi$ inches
D) $350\pi$ inches | 330\pi | 10.9375 |
30,873 | Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0.78125 |
30,874 | A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):
- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).
Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
(From the 38th American High School Mathematics Examination, 1987) | 15 | 42.1875 |
30,875 | In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho^{2}= \dfrac {16}{1+3\sin ^{2}\theta }$, and $P$ is a moving point on curve $C$, which intersects the positive half-axes of $x$ and $y$ at points $A$ and $B$ respectively.
$(1)$ Find the parametric equation of the trajectory of the midpoint $Q$ of segment $OP$;
$(2)$ If $M$ is a moving point on the trajectory of point $Q$ found in $(1)$, find the maximum value of the area of $\triangle MAB$. | 2 \sqrt {2}+4 | 0 |
30,876 | Given sets A = {-2, 1, 2} and B = {-1, 1, 3}, calculate the probability that a line represented by the equation ax - y + b = 0 will pass through the fourth quadrant. | \frac{5}{9} | 13.28125 |
30,877 | What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{1001}+1\right)^{3/2}$? | 743 | 17.96875 |
30,878 | 1. Solve the trigonometric inequality: $\cos x \geq \frac{1}{2}$
2. In $\triangle ABC$, if $\sin A + \cos A = \frac{\sqrt{2}}{2}$, find the value of $\tan A$. | -2 - \sqrt{3} | 41.40625 |
30,879 | A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a "storefront fee" of $10,000 to the JD website each month, and $5000 for repairs. What is the minimum selling price for each small refrigerator in order for the shop to maintain a monthly sales profit margin of at least 20%? | 1824 | 18.75 |
30,880 | The slope angle of the tangent line to the curve $y= \sqrt {x}$ at $x= \frac {1}{4}$ is ______. | \frac {\pi}{4} | 83.59375 |
30,881 | The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence. | 3240 | 9.375 |
30,882 | Define the ☆ operation: observe the following expressions:
$1$☆$2=1\times 3+2=5$;
$4$☆$(-1)=4\times 3-1=11$;
$(-5)$☆$3=(-5)\times 3+3=-12$;
$(-6)$☆$(-3)=(-6)\times 3-3=-21\ldots$
$(1)-1$☆$2=$______, $a$☆$b=$______;
$(2)$ If $a \lt b$, then $a$☆$b-b$☆$a$ ______ 0 (use "$ \gt $", "$ \lt $", or "$=$ to connect");
$(3)$ If $a$☆$(-2b)=4$, please calculate the value of $\left[3\left(a-b\right)\right]$☆$\left(3a+b\right)$. | 16 | 82.03125 |
30,883 | In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be? | 30 | 4.6875 |
30,884 | The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? | \sqrt{3} | 33.59375 |
30,885 | An isosceles triangle $ABP$ with sides $AB = AP = 3$ inches and $BP = 4$ inches is placed inside a square $AXYZ$ with a side length of $8$ inches, such that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. Calculate the total path length in inches traversed by vertex $P$.
A) $\frac{24\pi}{3}$
B) $\frac{28\pi}{3}$
C) $\frac{32\pi}{3}$
D) $\frac{36\pi}{3}$ | \frac{32\pi}{3} | 1.5625 |
30,886 | Given the function $f$ mapping from set $M$ to set $N$, where $M=\{a, b, c\}$ and $N=\{-3, -2, -1, 0, 1, 2, 3\}$, calculate the number of mappings $f$ that satisfy the condition $f(a) + f(b) + f(c) = 0$. | 37 | 100 |
30,887 | For what ratio of the bases of a trapezoid does there exist a line on which the six points of intersection with the diagonals, the lateral sides, and the extensions of the bases of the trapezoid form five equal segments? | 1:2 | 3.90625 |
30,888 | In quadrilateral $ABCD$, $\overrightarrow{AB}=(1,1)$, $\overrightarrow{DC}=(1,1)$, $\frac{\overrightarrow{BA}}{|\overrightarrow{BA}|}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}|}=\frac{\sqrt{3}\overrightarrow{BD}}{|\overrightarrow{BD}|}$, calculate the area of the quadrilateral. | \sqrt{3} | 11.71875 |
30,889 | Given that the terminal side of angle $α$ rotates counterclockwise by $\dfrac{π}{6}$ and intersects the unit circle at the point $\left( \dfrac{3 \sqrt{10}}{10}, \dfrac{\sqrt{10}}{10} \right)$, and $\tan (α+β)= \dfrac{2}{5}$.
$(1)$ Find the value of $\sin (2α+ \dfrac{π}{6})$,
$(2)$ Find the value of $\tan (2β- \dfrac{π}{3})$. | \dfrac{17}{144} | 0.78125 |
30,890 | In this 5x5 square array of 25 dots, five dots are to be chosen at random. What is the probability that the five dots will be collinear? Express your answer as a common fraction.
[asy]
size(59);
for(int i = 0; i < 5; ++i)
for(int j = 0; j < 5; ++j)
dot((i, j), linewidth(7));
[/asy] | \frac{12}{53130} | 0 |
30,891 | Given two congruent squares $ABCD$ and $EFGH$, each with a side length of $12$, they overlap to form a rectangle $AEHD$ with dimensions $12$ by $20$. Calculate the percent of the area of rectangle $AEHD$ that is shaded. | 20\% | 16.40625 |
30,892 | If $f(x)=x^{2}+bx+c$, and $f(1)=0$, $f(3)=0$, find
(1) the value of $f(-1)$;
(2) the maximum and minimum values of $f(x)$ on the interval $[2,4]$. | -1 | 64.84375 |
30,893 | A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________. | 80 | 3.90625 |
30,894 | The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have? | 1003 | 85.15625 |
30,895 | Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as "division exponentiation," denoted as $a^{ⓝ}$, read as "$a$ circle $n$ times." For example, $2\div 2\div 2$ is denoted as $2^{③}$, read as "$2$ circle $3$ times"; $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ is denoted as $\left(-3\right)^{④}$, read as "$-3$ circle $4$ times".<br/>$(1)$ Write down the results directly: $2^{③}=$______, $(-\frac{1}{2})^{④}=$______; <br/>$(2)$ Division exponentiation can also be converted into the form of powers, such as $2^{④}=2\div 2\div 2\div 2=2\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=(\frac{1}{2})^{2}$. Try to directly write the following operation results in the form of powers: $\left(-3\right)^{④}=$______; ($\frac{1}{2})^{⑩}=$______; $a^{ⓝ}=$______; <br/>$(3)$ Calculate: $2^{2}\times (-\frac{1}{3})^{④}\div \left(-2\right)^{③}-\left(-3\right)^{②}$. | -73 | 3.125 |
30,896 | For how many values of $n$ in the set $\{101, 102, 103, ..., 200\}$ is the tens digit of $n^2$ even? | 60 | 0 |
30,897 | In the tetrahedron \( P-ABC \), \( \triangle ABC \) is an equilateral triangle with a side length of \( 2\sqrt{3} \), \( PB = PC = \sqrt{5} \), and the dihedral angle between \( P-BC \) and \( BC-A \) is \( 45^\circ \). Find the surface area of the circumscribed sphere around the tetrahedron \( P-ABC \). | 25\pi | 2.34375 |
30,898 | On a plane, points are colored in the following way:
1. Choose any positive integer \( m \), and let \( K_{1}, K_{2}, \cdots, K_{m} \) be circles with different non-zero radii such that \( K_{i} \subset K_{j} \) or \( K_{j} \subset K_{i} \) for \( i \neq j \).
2. Points chosen inside the circles are colored differently from the points outside the circles on the plane.
Given that there are 2019 points on the plane such that no three points are collinear, determine the maximum number of different colors possible that satisfy the given conditions. | 2019 | 63.28125 |
30,899 | The first term of a geometric sequence is 1024, and the 6th term is 125. What is the positive, real value for the 4th term? | 2000 | 13.28125 |
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