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27,300 | The diagram shows a shaded semicircle of diameter 4, from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded? | $\frac{1}{2}$ | 0 |
27,301 | When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______. | 2618 | 3.90625 |
27,302 | In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If acosB - bcosA = $$\frac {c}{3}$$, then the minimum value of $$\frac {acosA + bcosB}{acosB}$$ is \_\_\_\_\_\_. | \sqrt {2} | 0 |
27,303 | Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ . | 22 | 75 |
27,304 | An entrepreneur invested \$12,000 in a three-month savings certificate that paid a simple annual interest rate of $8\%$. After three months, she invested the total value of her investment in another three-month certificate. After three more months, the investment was worth \$12,980. If the annual interest rate of the second certificate is $s\%$, what is $s?$ | 24\% | 9.375 |
27,305 | Express $0.7\overline{32}$ as a common fraction. | \frac{1013}{990} | 0 |
27,306 | In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$? | 12 \sqrt[4]{3} | 7.03125 |
27,307 | Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{3}{5}$ and a minor axis length of $8$,
(1) Find the standard equation of the ellipse $C$;
(2) Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$, respectively. A line $l$ passing through $F_{2}$ intersects the ellipse $C$ at two distinct points $M$ and $N$. If the circumference of the inscribed circle of $\triangle F_{1}MN$ is $π$, and $M(x_{1},y_{1})$, $N(x_{2},y_{2})$, find the value of $|y_{1}-y_{2}|$. | \frac{5}{3} | 20.3125 |
27,308 | When a certain unfair die is rolled, an even number is $5$ times as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd. | \frac{5}{18} | 60.9375 |
27,309 | Let $r$ be the speed in miles per hour at which a wheel, $15$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $4$ miles per hour. Determine the original speed $r$.
A) 9
B) 10
C) 11
D) 12
E) 13 | 12 | 14.84375 |
27,310 | Mom decides to take Xiaohua on a road trip to 10 cities during the vacation. After checking the map, Xiaohua is surprised to find that for any three cities among these 10, either all three pairs of cities are connected by highways, or exactly two pairs of cities among the three are not connected by highways. What is the minimum number of highways that must be opened among these 10 cities? (Note: There can be at most one highway between any two cities.) | 40 | 5.46875 |
27,311 | Given an isosceles triangle \(ABC\) with \(\angle A = 30^\circ\) and \(AB = AC\). Point \(D\) is the midpoint of \(BC\). Point \(P\) is chosen on segment \(AD\), and point \(Q\) is chosen on side \(AB\) such that \(PB = PQ\). What is the measure of angle \(PQC\)? | 15 | 17.96875 |
27,312 | How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $16$? | 23 | 41.40625 |
27,313 | The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | 12 | 26.5625 |
27,314 | Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | \frac{29}{120} | 4.6875 |
27,315 | Given that the center of circle $C$ lies on the $x$-axis and circle $C$ is tangent to the line $x + \sqrt{3}y + n = 0$ at the point $(\frac{3}{2}, \frac{\sqrt{3}}{2})$, find:
1. The value of $n$ and the equation of circle $C$.
2. If circle $M: x^2 + (y - \sqrt{15})^2 = r^2 (r > 0)$ is tangent to circle $C$, find the length of the chord formed by intersecting circle $M$ with the line $\sqrt{3}x - \sqrt{2}y = 0$. | 2\sqrt{19} | 1.5625 |
27,316 | How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \le x \le 5$ and $1 \le y \le 5$? | 2160 | 4.6875 |
27,317 | There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? | 18 | 2.34375 |
27,318 | Company $W$'s product $p$ produced by line $D$ follows a normal distribution $N(80, 0.25)$ in terms of size. $400$ randomly selected products were tested from the current production line, and the size distribution is summarized in the table below:
| Product Size ($mm$) | $[76,78.5]$ | $(78.5,79]$ | $(79,79.5]$ | $(79.5,80.5]$ | $(80.5,81]$ | $(81,81.5]$ | $(81.5,83]$ |
|---------------------|-------------|------------|-------------|--------------|------------|-------------|------------|
| Number of Items | $8$ | $54$ | $54$ | $160$ | $72$ | $40$ | $12$ |
According to the product quality standards and the actual situation of the production line, events outside the range $(\mu - 3\sigma, \mu + 3\sigma]$ are considered rare events. If a rare event occurs, it is considered an abnormality in the production line. Products within $(\mu - 3\sigma, \mu + 3\sigma]$ are considered as qualified products, while those outside are defective. $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$.
$(1)$ Determine if the production line is working normally and explain the reason.
$(2)$ Express the probability in terms of frequency. If $3$ more products are randomly selected from the production line for retesting, with a testing cost of $20$ dollars per qualified product and $30$ dollars per defective product, let the testing cost of these $3$ products be a random variable $X$. Find the expected value and variance of $X. | \frac{57}{4} | 0 |
27,319 | A rabbit and a hedgehog participated in a running race on a 550 m long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of 10 m/s and the hedgehog ran anticlockwise at a speed of 1 m/s. When they met, the rabbit continued as before, but the hedgehog turned around and ran clockwise. How many seconds after the rabbit did the hedgehog reach the finish? | 545 | 0.78125 |
27,320 | The slope angle of the tangent line to the curve $f\left(x\right)=- \frac{ \sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\tan^{-1}\left( \frac{f'\left(1\right)}{\mid f'\left(1\right) \mid} \right)$, where $f'\left(x\right)$ is the derivative of $f\left(x\right)$. | \frac{2\pi}{3} | 2.34375 |
27,321 | Given vectors $\overrightarrow{a}=(\sin (\frac{\omega}{2}x+\varphi), 1)$ and $\overrightarrow{b}=(1, \cos (\frac{\omega}{2}x+\varphi))$, where $\omega > 0$ and $0 < \varphi < \frac{\pi}{4}$, define the function $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}-\overrightarrow{b})$. If the function $y=f(x)$ has a period of $4$ and passes through point $M(1, \frac{1}{2})$,
(1) Find the value of $\omega$;
(2) Find the maximum and minimum values of the function $f(x)$ when $-1 \leq x \leq 1$. | \frac{1}{2} | 0.78125 |
27,322 | How many three-digit numbers exist that are 5 times the product of their digits? | 175 | 0.78125 |
27,323 | Suppose the roots of the polynomial $Q(x) = x^3 + px^2 + qx + r$ are $\sin \frac{3\pi}{7}, \sin \frac{5\pi}{7},$ and $\sin \frac{\pi}{7}$. Calculate the product $pqr$.
A) $0.725$
B) $1.45$
C) $1.0$
D) $1.725$ | 0.725 | 17.1875 |
27,324 | How many two digit numbers have exactly $4$ positive factors? $($ Here $1$ and the number $n$ are also considered as factors of $n. )$ | 31 | 4.6875 |
27,325 | A quadrilateral has vertices $P(a,b)$, $Q(2b,a)$, $R(-a, -b)$, and $S(-2b, -a)$, where $a$ and $b$ are integers and $a>b>0$. Find the area of quadrilateral $PQRS$.
**A)** $16$ square units
**B)** $20$ square units
**C)** $24$ square units
**D)** $28$ square units
**E)** $32$ square units | 24 | 26.5625 |
27,326 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sin \frac{B}{2}-\cos \frac{B}{2}= \frac{1}{5}$.
(I) Find the value of $\cos B$;
(II) If $b^{2}-a^{2}=ac$, find the value of $\frac{\sin C}{\sin A}$. | \frac{11}{25} | 7.8125 |
27,327 | Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x) = x^{3} + ax^{2} + bx + c\) has three real roots \(x_{1}, x_{2}, x_{3}\), and satisfy:
(1) \(x_{2} - x_{1} = \lambda\);
(2) \(x_{3} > \frac{1}{2}\left(x_{1} + x_{2}\right)\).
Find the maximum value of \(\frac{2a^{3} + 27c - 9ab}{\lambda^{3}}\). | \frac{3\sqrt{3}}{2} | 1.5625 |
27,328 | A rectangular prism has a volume of \(8 \mathrm{~cm}^{3}\), total surface area of \(32 \mathrm{~cm}^{2}\), and its length, width, and height are in geometric progression. Find the sum of all its edge lengths (in cm). | 28 | 2.34375 |
27,329 | For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | 231 | 7.8125 |
27,330 | An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after working a normal 8 hour day by that slow running clock, is | $2.60 | 0 |
27,331 | The sum of two numbers is \( t \) and the positive difference between the squares of these two numbers is 208. What is the larger of the two numbers? | 53 | 3.90625 |
27,332 | Given that $\alpha$ and $\beta$ are acute angles, $\tan\alpha= \frac {1}{7}$, $\sin\beta= \frac { \sqrt {10}}{10}$, find $\alpha+2\beta$. | \frac {\pi}{4} | 96.09375 |
27,333 | In the plane Cartesian coordinate system \( xOy \), a moving line \( l \) is tangent to the parabola \( \Gamma: y^{2} = 4x \), and intersects the hyperbola \( \Omega: x^{2} - y^{2} = 1 \) at one point on each of its branches, left and right, labeled \( A \) and \( B \). Find the minimum area of \(\triangle AOB\). | 2\sqrt{5} | 0 |
27,334 | Consider the function \( g(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x-1} & \quad \text{ if } x \ge 5 \end{aligned} \right. \). Find the value of \( g^{-1}(-6) + g^{-1}(-5) + \dots + g^{-1}(4) + g^{-1}(5) \). | 58 | 3.125 |
27,335 | Evaluate: $(12345679^2 \times 81 - 1) \div 11111111 \div 10 \times 9 - 8$ in billions. (Answer in billions) | 10 | 6.25 |
27,336 | In $\triangle ABC$, the side lengths are: $AB = 17, BC = 20$ and $CA = 21$. $M$ is the midpoint of side $AB$. The incircle of $\triangle ABC$ touches $BC$ at point $D$. Calculate the length of segment $MD$.
A) $7.5$
B) $8.5$
C) $\sqrt{8.75}$
D) $9.5$ | \sqrt{8.75} | 0.78125 |
27,337 | There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ unique integers $b_k$ ($1\le k\le s$) with each $b_k$ either $1$ or $- 1$ such that\[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 2012.\]Find $m_1 + m_2 + \cdots + m_s$. | 22 | 2.34375 |
27,338 | Let $\{a_n\}$ be an arithmetic sequence. If we select any 4 different numbers from $\{a_1, a_2, a_3, \ldots, a_{10}\}$ such that these 4 numbers still form an arithmetic sequence, then there are at most \_\_\_\_\_\_ such arithmetic sequences. | 24 | 0 |
27,339 | Compute $\sqrt[4]{256000000}$. | 40\sqrt{10} | 7.03125 |
27,340 | If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). | 42 | 23.4375 |
27,341 | Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area? | 45 \sqrt{3} | 10.15625 |
27,342 | Let \( p, q, r, s, t, u, v, w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[ (pt)^2 + (qu)^2 + (rv)^2 + (sw)^2. \] | 400 | 1.5625 |
27,343 | Solve the equation using the completing the square method: $2x^{2}-4x-1=0$. | \frac{2-\sqrt{6}}{2} | 0.78125 |
27,344 | Given eight cubes with volumes \(1, 8, 27, 64, 125, 216, 343,\) and \(512\) cubic units, and each cube having its volume decreasing upwards, determine the overall external surface area of this arrangement. | 1021 | 3.90625 |
27,345 | Find the smallest possible sum of two perfect squares such that their difference is 175 and both squares are greater or equal to 36. | 625 | 5.46875 |
27,346 | In a trapezoid $ABCD$ with bases $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \perp \overline{CD}$, suppose that $CD = 10$, $\tan C = 2$, and $\tan D = 1$. Calculate the length of $AB$ and determine the area of the trapezoid. | 300 | 2.34375 |
27,347 | Let $\triangle ABC$ be a triangle with $AB=85$ , $BC=125$ , $CA=140$ , and incircle $\omega$ . Let $D$ , $E$ , $F$ be the points of tangency of $\omega$ with $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ respectively, and furthermore denote by $X$ , $Y$ , and $Z$ the incenters of $\triangle AEF$ , $\triangle BFD$ , and $\triangle CDE$ , also respectively. Find the circumradius of $\triangle XYZ$ .
*Proposed by David Altizio* | 30 | 15.625 |
27,348 | Find the number of solutions to the equation
\[\tan (7 \pi \cos \theta) = \cot (7 \pi \sin \theta)\]
where $\theta \in (0, 4 \pi).$ | 28 | 0 |
27,349 | Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 8.59375 |
27,350 | Distribute 6 volunteers into 4 groups, with each group having at least 1 and at most 2 people, and assign them to four different exhibition areas of the fifth Asia-Europe Expo. The number of different allocation schemes is ______ (answer with a number). | 1080 | 57.03125 |
27,351 | The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot? | $\frac{1}{8}$ | 0 |
27,352 | Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$. | \dfrac{\sqrt{5}}{5} | 11.71875 |
27,353 | Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis. | \frac{\sqrt{2}}{8} | 47.65625 |
27,354 | Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).
Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \). | -4 | 11.71875 |
27,355 | In the Cartesian coordinate system Oxyz, given points A(2, 0, 0), B(2, 2, 0), C(0, 2, 0), and D(1, 1, $\sqrt{2}$), calculate the relationship between the areas of the orthogonal projections of the tetrahedron DABC onto the xOy, yOz, and zOx coordinate planes. | \sqrt{2} | 0 |
27,356 | Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a-b+c=7$, $b-c+d=8$, $c-d+a=4$, and $d-a+b=1$. Determine the value of $2a+2b+2c+2d$. | 20 | 15.625 |
27,357 | Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half. | 10 | 10.9375 |
27,358 | Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$. | \frac{\sqrt{3}}{4} | 7.03125 |
27,359 | Given that the side length of square $ABCD$ is 1, point $M$ is the midpoint of side $AD$, and with $M$ as the center and $AD$ as the diameter, a circle $\Gamma$ is drawn. Point $E$ is on segment $AB$, and line $CE$ is tangent to circle $\Gamma$. Find the area of $\triangle CBE$. | 1/4 | 26.5625 |
27,360 | Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? | 27.55 | 17.96875 |
27,361 | Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} | 6.25 |
27,362 | Given the data set $10$, $6$, $8$, $5$, $6$, calculate the variance $s^{2}=$ \_\_\_\_\_\_. | \frac{16}{5} | 0 |
27,363 | Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? | \frac{5}{42} | 3.90625 |
27,364 |
The square of a natural number \( a \) gives a remainder of 8 when divided by a natural number \( n \). The cube of the number \( a \) gives a remainder of 25 when divided by \( n \). Find \( n \). | 113 | 7.03125 |
27,365 | Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n? | 23 | 0 |
27,366 | How many two-digit numbers have digits whose sum is a prime number? | 35 | 0 |
27,367 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty-cent piece. What is the probability that at least 40 cents worth of coins land on heads? | \frac{1}{2} | 9.375 |
27,368 | Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation. | \frac{64\pi}{105} | 0.78125 |
27,369 | Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^4 b^6 = a^8 b^3 = 1.\] | 24 | 6.25 |
27,370 | Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum at $x = 1$ with the value of 10, find the values of $a$ and $b$. | -11 | 0 |
27,371 | The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points respectively), with a total score of 120 points. The second round consists of 4 problem-solving questions covering plane geometry, algebra, number theory, and combinatorics. The first two questions are worth 40 points each, and the last two questions are worth 50 points each, with a total score of 180 points.
It is known that a certain math competition participant has a probability of $\frac{4}{5}$ of correctly answering each fill-in-the-blank question in the first round, and a probability of $\frac{3}{5}$ of correctly answering each problem-solving question in the first round. In the second round, the participant has a probability of $\frac{3}{5}$ of correctly answering each of the first two questions, and a probability of $\frac{2}{5}$ of correctly answering each of the last two questions. Assuming full marks for correct answers and 0 points for incorrect answers:
1. Let $X$ denote the participant's score in the second round. Find $P(X \geq 100)$.
2. Based on the historical competition results in the participant's province, if a participant scores 100 points or above in the first round, the probability of winning the provincial first prize is $\frac{9}{10}$, while if the score is below 100 points, the probability is $\frac{2}{5}$. Can the probability of the participant winning the provincial first prize reach $\frac{1}{2}$, and explain the reason.
(Reference data: $(\frac{4}{5})^8 \approx 0.168$, $(\frac{4}{5})^7 \approx 0.21$, $(\frac{4}{5})^6 \approx 0.262$.) | \frac{1}{2} | 17.1875 |
27,372 | From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers. | 525 | 0 |
27,373 | Maria wants to purchase a book which costs \$35.50. She checks her purse and discovers she has two \$20 bills, and twelve quarters, and a bunch of nickels. Determine the minimum number of nickels that Maria needs with her to buy the book. | 150 | 24.21875 |
27,374 | Let $g_{1}(x) = \sqrt{2 - x}$, and for integers $n \geq 2$, define \[g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right).\] Find the largest value of $n$, denoted as $M$, for which the domain of $g_n$ is nonempty. For this value of $M$, if the domain of $g_M$ consists of a single point $\{d\}$, compute $d$. | 25 | 2.34375 |
27,375 | Determine the area enclosed by the curve of $y = \arccos(\cos x)$ and the $x$-axis over the interval $\frac{\pi}{4} \le x \le \frac{9\pi}{4}.$ | \frac{3\pi^2}{2} | 5.46875 |
27,376 | Compute
\[
\frac{(1 + 23) \left( 1 + \dfrac{23}{2} \right) \left( 1 + \dfrac{23}{3} \right) \dotsm \left( 1 + \dfrac{23}{25} \right)}{(1 + 27) \left( 1 + \dfrac{27}{2} \right) \left( 1 + \dfrac{27}{3} \right) \dotsm \left( 1 + \dfrac{27}{21} \right)}.
\] | 421200 | 0 |
27,377 | If $x$ and $y$ are positive integers such that $xy - 8x + 9y = 632$, what is the minimal possible value of $|x - y|$? | 27 | 1.5625 |
27,378 | As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) square centimeters. | 1176 | 0.78125 |
27,379 | Find the sum of the digits of the greatest prime number that is a divisor of $16,385$. | 13 | 4.6875 |
27,380 | \( P \) is a moving point on the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\). The tangent line to the ellipse at point \( P \) intersects the circle \(\odot O\): \(x^{2}+y^{2}=12\) at points \( M \) and \( N \). The tangents to \(\odot O\) at \( M \) and \( N \) intersect at point \( Q \).
(1) Find the equation of the locus of point \( Q \);
(2) If \( P \) is in the first quadrant, find the maximum area of \(\triangle O P Q\). | \frac{\sqrt{3}}{2} | 3.90625 |
27,381 | A rectangular prism has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \le i \le 4$. The dimensions of the prism are given by lengths 2 along the x-axis, 3 along the y-axis, and 1 along the z-axis. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. Find the side length of the octahedron. | \frac{\sqrt{14}}{2} | 1.5625 |
27,382 | Points $ K$ , $ L$ , $ M$ , and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$ , $ BLC$ , $ CMD$ , and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$ , find the area of $ KLMN$. | 32 + 16\sqrt{3} | 0 |
27,383 | In Zuminglish-Advanced, all words still consist only of the letters $M, O,$ and $P$; however, there is a new rule that any occurrence of $M$ must be immediately followed by $P$ before any $O$ can occur again. Also, between any two $O's$, there must appear at least two consonants. Determine the number of $8$-letter words in Zuminglish-Advanced. Let $X$ denote this number and find $X \mod 100$. | 24 | 1.5625 |
27,384 | Jake will roll two standard six-sided dice and make a two-digit number from the numbers he rolls. If he rolls a 4 and a 2, he can form either 42 or 24. What is the probability that he will be able to make an integer between 30 and 40, inclusive? Express your answer as a common fraction. | \frac{11}{36} | 9.375 |
27,385 | Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu* | 42 | 67.96875 |
27,386 | How many non-congruent triangles can be formed by selecting vertices from the ten points in the triangular array, where the bottom row has four points, the next row has three points directly above the gaps of the previous row, followed by two points, and finally one point at the top? | 11 | 0 |
27,387 | In the equilateral triangle \(PQR\), \(S\) is the midpoint of \(PR\), and \(T\) is on \(PQ\) such that \(PT=1\) and \(TQ=3\). Many circles can be drawn inside the quadrilateral \(QRST\) such that no part extends outside of \(QRST\). The radius of the largest such circle is closest to: | 1.10 | 0 |
27,388 | A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | \frac{7}{256} | 2.34375 |
27,389 | The sequence $\{a_{n}\}$ is an increasing sequence of integers, and $a_{1}\geqslant 3$, $a_{1}+a_{2}+a_{3}+\ldots +a_{n}=100$. Determine the maximum value of $n$. | 10 | 1.5625 |
27,390 | In the diagram, \(P Q R S T\) is a pentagon with \(P Q=8\), \(Q R=2\), \(R S=13\), \(S T=13\), and \(T P=8\). Also, \(\angle T P Q=\angle P Q R=90^\circ\). What is the area of pentagon \(P Q R S T\) ? | 100 | 0 |
27,391 | Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 950 | 3.90625 |
27,392 | A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? | \frac{5}{8} | 10.15625 |
27,393 | Fill the nine numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid, placing one number in each cell, such that the numbers in each row increase from left to right and the numbers in each column decrease from top to bottom. How many different ways are there to achieve this arrangement? | 42 | 89.0625 |
27,394 |
A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit? | 23/90 | 58.59375 |
27,395 | Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). | 75600 | 0 |
27,396 | Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________. | \frac{1}{3} | 60.15625 |
27,397 | Given the equation of the line $y = mx + 3$, find the maximum possible value of $a$ such that the line passes through no lattice point with $0 < x \leq 150$ for all $m$ satisfying $\frac{2}{3} < m < a$. | \frac{101}{151} | 7.03125 |
27,398 | If the probability of producing a Grade B product is $0.03$, and the probability of producing a Grade C product is $0.02$, calculate the probability of randomly inspecting a product and finding it to be a qualified product. | 0.95 | 16.40625 |
27,399 | Determine the value of $1 - 2 - 3 + 4 + 5 + 6 + 7 + 8 - 9 - 10 - \dots + 9801$, where the signs change after each perfect square and repeat every two perfect squares. | -9801 | 0 |
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