Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
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stringlengths 1
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float64 0
100
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|---|---|---|---|
21,600 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b=1$, $c= \sqrt {3}$, and $\angle C= \frac {2}{3}\pi$, find the area $S_{\triangle ABC}$. | \frac { \sqrt {3}}{4} | 0 |
21,601 | Suppose the function $f(x) = ax + \frac{x}{x-1}$ where $x > 1$.
(1) If $a > 0$, find the minimum value of the function $f(x)$.
(2) If $a$ is chosen from the set \{1, 2, 3\} and $b$ is chosen from the set \{2, 3, 4, 5\}, find the probability that $f(x) > b$ always holds true. | \frac{5}{6} | 17.1875 |
21,602 | A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$ | 1024 | 0 |
21,603 | Let $ABCD$ be a square of side length $1$ , and let $P$ be a variable point on $\overline{CD}$ . Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$ . The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment? | 3 - 2\sqrt{2} | 39.0625 |
21,604 | Given a right square prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a base edge length of $1$, and $AB_{1}$ forms a $60^{\circ}$ angle with the base $ABCD$, find the distance from $A_{1}C_{1}$ to the base $ABCD$. | \sqrt{3} | 92.96875 |
21,605 | What is GCF(LCM(16, 21), LCM(14, 18))? | 14 | 0.78125 |
21,606 | Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$ -gon is an integer.
*2016 CCA Math Bonanza Team #3* | 1167 | 75 |
21,607 | Five dice with faces numbered 1 through 6 are stacked in a similar manner to the original problem. Ten of the thirty faces are visible, leaving twenty faces hidden. The visible numbers are 1, 2, 2, 3, 3, 3, 4, 4, 5, and 6. What is the total number of dots NOT visible in this view? | 72 | 51.5625 |
21,608 | Given that $y=f(x)$ is an odd function, if $f(x)=g(x)+x^{2}$ and $g(1)=1$, then $g(-1)=$ _____ . | -3 | 98.4375 |
21,609 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$. | \frac{90}{11} | 0 |
21,610 | What is the smallest positive integer $n$ such that $\frac{n}{n+53}$ is equal to a terminating decimal? | 11 | 31.25 |
21,611 | (1) Use the Euclidean algorithm to find the greatest common divisor of 117 and 182, and verify it using the subtraction method.
(2) Use the Horner's method to evaluate the polynomial $f(x) = 1 - 9x + 8x^2 - 4x^4 + 5x^5 + 3x^6$ at $x = -1$. | 12 | 50.78125 |
21,612 | A club consists of three board members and a certain number of regular members. Every year, the board members retire and are not replaced. Each regular member recruits one new person to join as a regular member. Initially, there are nine people in the club total. How many people total will be in the club after four years? | 96 | 30.46875 |
21,613 | The ecology club at a school has 30 members: 12 boys and 18 girls. A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \dfrac{530}{609} | 17.96875 |
21,614 | In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 20$, $\tan D = 2$, and $\tan B = 2.5$, then what is $BC$?
[asy]
pair A,B,C,D;
C = (0,0);
D = (20,0);
A = (20,40);
B= (30,40);
draw(A--B--C--D--A);
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,S);
[/asy] | 4\sqrt{116} | 0 |
21,615 | At a painting club meeting, 7 friends are present. They need to create two separate teams: one team of 4 members, and another team of 2 members for different competitions. How many distinct ways can they form these teams? | 105 | 94.53125 |
21,616 | The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices. | 15 | 63.28125 |
21,617 | Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $ . Compute $ n $ . | 1024 | 77.34375 |
21,618 | What is the smallest four-digit positive integer which has four different digits? | 1023 | 89.84375 |
21,619 | Given \( a=\underset{2016 \uparrow}{55 \cdots 5} \), determine the remainder when \( a \) is divided by 84. | 63 | 53.125 |
21,620 | (a) A natural number \( n \) is less than 120. What is the maximum remainder that the number 209 can leave when divided by \( n \)?
(b) A natural number \( n \) is less than 90. What is the maximum remainder that the number 209 can leave when divided by \( n \)? | 69 | 20.3125 |
21,621 | Given that there are 6 teachers with IDs $A$, $B$, $C$, $D$, $E$, $F$ and 4 different schools, with the constraints that each school must have at least 1 teacher and $B$ and $D$ must be arranged in the same school, calculate the total number of different arrangements. | 240 | 40.625 |
21,622 | Yvon has 4 different notebooks and 5 different pens. Determine the number of different possible combinations of notebooks and pens he could bring. | 20 | 99.21875 |
21,623 | The graph of the function $y=\sin (2x+\varphi) (0 < \varphi < \pi)$ is shifted to the right by $\frac{\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\varphi$. | \frac{3\pi}{4} | 75 |
21,624 | The first term of a given sequence is 2, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10000? | 16384 | 95.3125 |
21,625 | Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie's prism. Find the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction. | \frac{1}{16} | 3.90625 |
21,626 | Someone bought 5 consecutive train ticket numbers, and the sum of these 5 ticket numbers is 120. What is the product of these 5 ticket numbers? | 7893600 | 18.75 |
21,627 | Given the function $f(x)=\sin \frac {x}{2}\cos \frac {x}{2}+\cos ^{2} \frac {x}{2}-1$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the interval where it is monotonically decreasing;
$(2)$ Find the minimum value of the function $f(x)$ on the interval $\left[ \frac {\pi}{4}, \frac {3\pi}{2}\right]$. | - \frac { \sqrt {2}+1}{2} | 0 |
21,628 | Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a}|=2$, $| \overrightarrow {b}|=1$, and $\overrightarrow {b} \perp ( \overrightarrow {a}+ \overrightarrow {b})$, determine the projection of vector $\overrightarrow {a}$ onto vector $\overrightarrow {b}$. | -1 | 26.5625 |
21,629 | In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \in \{70,85,88,90,98,100\}$ and it satisfies $f(1)<f(2) \leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 students is \_\_\_\_\_\_\_\_. | 35 | 0.78125 |
21,630 | Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$, where the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively, and $\left(2n+3\right)S_{n}=nT_{n}$, calculate the value of $\frac{{{a_5}}}{{{b_6}}}$. | \frac{9}{25} | 5.46875 |
21,631 | Given the line $l: ax+y+b=0$ intersects with the circle $O: x^{2}+y^{2}=4$ at points $A$ and $B$, and $M(\sqrt{3},-1)$, and $\overrightarrow{OA}+ \overrightarrow{OB}= \frac{2}{3} \overrightarrow{OM}$, calculate the value of $\sqrt{3}ab$. | -4 | 57.8125 |
21,632 | Given that $a$ and $b$ are real numbers, and $\frac{a}{1-i} + \frac{b}{2-i} = \frac{1}{3-i}$, find the sum of the first 100 terms of the arithmetic sequence ${an + b}$. | -910 | 52.34375 |
21,633 | A book has a total of 100 pages, numbered sequentially from 1, 2, 3, 4…100. The digit “2” appears in the page numbers a total of \_\_\_\_\_\_ times. | 20 | 96.09375 |
21,634 | Given the ellipse $C:\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1 \left( a > b > 0 \right)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and point $A(1,\sqrt{2})$ is on the ellipse.
$(1)$ Find the equation of ellipse $C$;
$(2)$ If a line $l$ with a slope of $\sqrt{2}$ intersects the ellipse $C$ at two distinct points $B$ and $C$, find the maximum area of $\Delta ABC$. | \sqrt{2} | 35.9375 |
21,635 | A hotpot restaurant in Chongqing operates through three methods: dining in, takeout, and setting up a stall outside (referred to as stall). In June, the ratio of revenue from dining in, takeout, and stall for this hotpot restaurant was $3:5:2$. With the introduction of policies to promote consumption, the owner of the hotpot restaurant expects the total revenue in July to increase. It is projected that the increase in revenue from the stall will account for $\frac{2}{5}$ of the total revenue increase. The revenue from the stall in July will then reach $\frac{7}{20}$ of the total revenue in July. In order for the ratio of revenue from dining in to takeout in July to be $8:5$, the additional revenue from takeout in July compared to the total revenue in July will be ______. | \frac{1}{8} | 12.5 |
21,636 | There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys? | 45 | 35.15625 |
21,637 | Each of two wheels contains numbers from 1 to 8. When the wheels are spun, a number is selected from each wheel. Find the probability that the sum of the two selected numbers is divisible by 4. | \frac{1}{4} | 54.6875 |
21,638 | When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? | 114 | 0 |
21,639 | What is the smallest integer greater than $-\frac{17}{3}$? | -5 | 97.65625 |
21,640 | In a new sequence, the first term is \(a_1 = 5000\) and the second term is \(a_2 = 5001\). Furthermore, the values of the remaining terms are designed so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \( n \geq 1 \). Determine \(a_{1000}\). | 5666 | 15.625 |
21,641 | Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for her to see him. | 16.2 | 64.0625 |
21,642 | Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending at points $P_{1}(-1,0)$, $P_{2}(-1,1)$, $P_{3}(0,1)$, $P_{4}(1,1)$, $P_{5}(1,0)$, to form $5$ vectors. By randomly selecting any two vectors and calculating the dot product $y$ of these two vectors, if $y > 0$, she will review history; if $y=0$, she will review geography; if $y < 0$, she will review politics.
$(1)$ List all possible values of $y$;
$(2)$ Calculate the probability of Xiaoli reviewing history and the probability of reviewing geography. | \dfrac{3}{10} | 42.96875 |
21,643 | The school committee has organized a "Chinese Dream, My Dream" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not selected by any of the 4 contestants is ______. | 324 | 0 |
21,644 | Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$ hours per week studying. If the university invests $b$ million dollars in better food, the students will spend an additional $3b$ hours per week studying. Finally, if the $c$ million dollars are invested in parties, students will be more relaxed and spend $11c - c^2$ more hours per week studying. The university wants to invest its $5$ million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study? | 34 | 17.96875 |
21,645 | A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:
*The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.*As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond? | 165 | 39.0625 |
21,646 | Inside a pentagon, 1000 points were marked and the pentagon was divided into triangles such that each of the marked points became a vertex of at least one of them. What is the minimum number of triangles that could be formed? | 1003 | 21.875 |
21,647 | Four spheres of radius 1 are placed so that each touches the other three. What is the radius of the smallest sphere that contains all four spheres? | \sqrt{\frac{3}{2}} + 1 | 0 |
21,648 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 42.96875 |
21,649 | Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors. | 130 | 96.09375 |
21,650 | Use the Horner's method to calculate the value of the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$ at $x = -4$. What is the value of $V_4$? | 220 | 67.96875 |
21,651 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 13.28125 |
21,652 | Given points P(-2, -2), Q(0, -1), and a point R(2, m) is chosen such that PR + PQ is minimized. What is the value of the real number $m$? | -2 | 38.28125 |
21,653 | The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______. | \sqrt{2} | 74.21875 |
21,654 | Six students are to be arranged into two classes, with two students in each class, and there are six classes in total. Calculate the number of different arrangement plans. | 90 | 8.59375 |
21,655 | A certain store sells a batch of thermal shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to increase sales and profits, the store has taken appropriate price reduction measures. After investigation, it was found that within a certain range, for every $1 decrease in the unit price of the thermal shirts, the store can sell an additional 2 pieces per day on average. If the store aims to make a daily profit of $1200 by selling this batch of thermal shirts and minimizing inventory, the unit price of the thermal shirts should be reduced by ______ dollars. | 20 | 57.03125 |
21,656 | John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John. | 45 | 30.46875 |
21,657 | Given that $ 2^{2004}$ is a $ 604$ -digit number whose first digit is $ 1$ , how many elements of the set $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$ have a first digit of $ 4$ ? | 194 | 12.5 |
21,658 | Find the maximum and minimum values of the function $f(x)=x^{3}-2x^{2}+5$ on the interval $[-2,2]$. | -11 | 30.46875 |
21,659 | In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches. | \frac{9}{2} | 0 |
21,660 | How many positive integers less than $800$ are either a perfect cube or a perfect square? | 35 | 5.46875 |
21,661 | Find minimum of $x+y+z$ where $x$ , $y$ and $z$ are real numbers such that $x \geq 4$ , $y \geq 5$ , $z \geq 6$ and $x^2+y^2+z^2 \geq 90$ | 16 | 68.75 |
21,662 | Given the function $f(x)=2\ln(3x)+8x$, find the value of $\lim_{\triangle x \to 0}\frac{f(1-2\triangle x)-f(1)}{\triangle x}$. | -20 | 90.625 |
21,663 | Find the positive value of $x$ which satisfies
\[\log_5 (x + 2) + \log_{\sqrt{5}} (x^2 + 2) + \log_{\frac{1}{5}} (x + 2) = 3.\] | \sqrt{\sqrt{125} - 2} | 0 |
21,664 | Given that the four roots of the equation $(x^{2}-mx+27)(x^{2}-nx+27)=0$ form a geometric sequence with 1 as the first term, find $|m-n|$. | 16 | 65.625 |
21,665 | The lengths of the three sides of $\triangle ABC$ are 5, 7, and 8, respectively. The radius of its circumcircle is ______, and the radius of its incircle is ______. | \sqrt{3} | 27.34375 |
21,666 | If \( \frac{10+11+12}{3} = \frac{2010+2011+2012+N}{4} \), then find the value of \(N\). | -5989 | 85.15625 |
21,667 | The number of recommendation plans the principal can make for a certain high school with 4 students and 3 universities can accept at most 2 students from that school is to be determined. | 54 | 36.71875 |
21,668 | Let $d$ be a positive number such that when $145$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | 47 | 84.375 |
21,669 | In the diagram, $ABCD$ is a trapezoid with bases $AB$ and $CD$ such that $AB$ is parallel to $CD$ and $CD$ is three times the length of $AB$. The area of $ABCD$ is $27$. Find the area of $\triangle ABC$.
[asy]
draw((0,0)--(3,6)--(9,6)--(12,0)--cycle);
draw((3,6)--(0,0));
label("$A$",(0,0),W);
label("$B$",(3,6),NW);
label("$C$",(9,6),NE);
label("$D$",(12,0),E);
[/asy] | 6.75 | 22.65625 |
21,670 | Given $f(x)=\sin \left( 2x+ \frac{π}{6} \right)+ \frac{3}{2}$, $x\in R$.
(1) Find the minimum positive period of the function $f(x)$;
(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(3) Find the maximum value of the function and the corresponding $x$ value(s). | \frac{5}{2} | 5.46875 |
21,671 | If $C_n^2A_2^2=42$, then $\frac{n!}{3!(n-3)!}=\_\_\_\_\_\_\_\_.$ | 35 | 94.53125 |
21,672 | An ellipse has a focus at coordinates $\left(0,-\sqrt {2}\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$. | -\dfrac{2}{5} | 24.21875 |
21,673 | What is the greatest common factor of 180, 240, and 300? | 60 | 100 |
21,674 | The inverse of $f(x) = \frac{3x - 2}{x + 4}$ may be written in the form $f^{-1}(x)=\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$. | -4 | 96.09375 |
21,675 | Given that the terminal side of angle θ passes through point P(-x, -6) and $$cosθ=- \frac {5}{13}$$, find the value of $$tan(θ+ \frac {π}{4})$$. | -\frac {17}{7} | 82.8125 |
21,676 | Three circles have the same center O. Point X divides segment OP in the ratio 1:3, with X being closer to O. Calculate the ratio of the area of the circle with radius OX to the area of the circle with radius OP. Next, find the area ratio of a third circle with radius 2*OX to the circle with radius OP. | \frac{1}{4} | 90.625 |
21,677 | The square $BCDE$ is inscribed in circle $\omega$ with center $O$ . Point $A$ is the reflection of $O$ over $B$ . A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$ ). Assume $BCDE$ has area $200$ . To the nearest integer, what is the length of the hook?
*Proposed by Evan Chen* | 67 | 67.1875 |
21,678 | Given $\sin\left( \frac{\pi}{3} + a \right) = \frac{5}{13}$, and $a \in \left( \frac{\pi}{6}, \frac{2\pi}{3} \right)$, find the value of $\sin\left( \frac{\pi}{12} + a \right)$. | \frac{17\sqrt{2}}{26} | 82.8125 |
21,679 | The polynomial $\frac{1}{5}{x^2}{y^{|m|}}-(m+1)y+\frac{1}{7}$ is a cubic binomial in terms of $x$ and $y$. Find the value of $m$. | -1 | 41.40625 |
21,680 | Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 7? Express your answer as a common fraction. | \frac{107}{150} | 54.6875 |
21,681 | Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2). | -1 | 30.46875 |
21,682 | In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid, one in each quadrant. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$? | 180 | 42.1875 |
21,683 | Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$ , $b$ , $c$ . Define $Q(x)$ to be the monic cubic polynomial with roots $ab$ , $bc$ , $ca$ . Compute $|Q(1)|$ .
*Proposed by Nathan Xiong* | 75 | 84.375 |
21,684 | In trapezoid $ABCD$, $CD$ is three times the length of $AB$. If the area of trapezoid $ABCD$ is $18$ square units, what is the area of $\triangle ABC?$
[asy]
draw((0,0)--(1,6)--(10,6)--(15,0)--cycle);
draw((10,6)--(0,0));
label("$C$",(0,0),W);
label("$A$",(1,6),NW);
label("$B$",(10,6),NE);
label("$D$",(15,0),E);
[/asy] | 4.5 | 80.46875 |
21,685 | In trapezoid \(PQRS\), the lengths of the bases \(PQ\) and \(RS\) are 10 and 23 respectively. The legs of the trapezoid are extended beyond \(P\) and \(Q\) to meet at point \(T\). What is the ratio of the area of triangle \(TPQ\) to the area of trapezoid \(PQRS\)? Express your answer as a common fraction. | \frac{100}{429} | 71.09375 |
21,686 | A larger square contains two non-overlapping shapes: a circle with diameter $2$ and a rectangle with side lengths $2$ and $4$. Find the smallest possible side length of the larger square such that these shapes can fit without overlapping, and then, find the area of the square $S$ that can be inscribed precisely in the remaining free space inside the larger square.
A) $\sqrt{16-2\pi}$
B) $\pi - \sqrt{8}$
C) $\sqrt{8 - \pi}$
D) $\sqrt{10 - \pi}$ | \sqrt{8 - \pi} | 15.625 |
21,687 | Two identical test tubes were filled with 200 ml of a liquid substance each. From the first test tube, $1 / 4$ of the content was poured out and replaced with the same amount of water. This procedure was repeated 3 more times, each time pouring out a quarter of the content and refilling with the same amount of water. A similar procedure was conducted twice for the second test tube, each time pouring out a certain amount of content and refilling with the same amount of water. As a result, the concentration of the mixtures in the first and second test tubes related to each other as 9/16. Determine the amount of mixture poured out from the second test tube each time.
(12 points) | 50 | 39.84375 |
21,688 | Given triangle \( \triangle ABC \) with internal angles \( A, B, C \) opposite to sides \( a, b, c \) respectively, vectors \( \vec{m} = (1, 1 - \sqrt{3} \sin A) \) and \( \vec{n} = (\cos A, 1) \), and \( \vec{m} \perp \vec{n} \):
(1) Find angle \( A \);
(2) Given \( b + c = \sqrt{3} a \), find the value of \( \sin \left(B + \frac{\pi}{6} \right) \). | \frac{\sqrt{3}}{2} | 67.1875 |
21,689 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The yard's remaining area forms a trapezoidal shape, as shown. The lengths of the parallel sides of the trapezoid are $20$ and $30$ meters, respectively. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{5}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$ | \frac{1}{6} | 40.625 |
21,690 | In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 23, respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction. | \frac{100}{429} | 67.96875 |
21,691 | Given vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$.
- (I) Find the interval where $f(x)$ is monotonically decreasing;
- (II) Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$. | 2\sqrt{3} | 44.53125 |
21,692 | Find the sum of $432_7$, $54_7$, and $6_7$ in base $7$. | 525_7 | 79.6875 |
21,693 | Express the sum of fractions $\frac{3}{8} + \frac{5}{32}$ as a decimal. | 0.53125 | 98.4375 |
21,694 | Let $g(x) = ax^7 + bx^3 + dx^2 + cx - 8$. If $g(-7) = 3$, then find $g(7)$. | -19 | 66.40625 |
21,695 | A'Niu is riding a horse to cross a river. There are four horses named A, B, C, and D. It takes 2 minutes for horse A to cross the river, 3 minutes for horse B, 7 minutes for horse C, and 6 minutes for horse D. Only two horses can be driven across the river at a time. The question is: what is the minimum number of minutes required to get all four horses across the river? | 18 | 1.5625 |
21,696 | Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \in \mathbb{R}$).
(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$.
(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the shortest? And what is the shortest distance? | \sqrt{2} | 54.6875 |
21,697 | Consider the sequence $\sqrt{2}, \sqrt{5}, 2\sqrt{2}, \sqrt{11}, \ldots$. Determine the position of $\sqrt{41}$ in this sequence. | 14 | 85.9375 |
21,698 | Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\sqrt{5}$, find the minimum value of $3a+b+2c$. | 2\sqrt{10}+2\sqrt{2} | 0 |
21,699 | Given the curve
\[
(x - \arcsin \alpha)(x - \arccos \alpha) + (y - \arcsin \alpha)(y + \arccos \alpha) = 0
\]
is intersected by the line \( x = \frac{\pi}{4} \), determine the minimum value of the length of the chord intercepted as \( \alpha \) varies. | \frac{\pi}{2} | 48.4375 |
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