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27,400 | A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd numbers? | 154 | 2.34375 |
27,401 | A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively. | \frac{880}{29} | 14.84375 |
27,402 | A rectangle is inscribed in a triangle if its vertices all lie on the boundary of the triangle. Given a triangle \( T \), let \( d \) be the shortest diagonal for any rectangle inscribed in \( T \). Find the maximum value of \( \frac{d^2}{\text{area } T} \) for all triangles \( T \). | \frac{4\sqrt{3}}{7} | 0 |
27,403 | Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$ ), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$ ?
*2018 CCA Math Bonanza Tiebreaker Round #3* | 607 | 6.25 |
27,404 | Given a positive sequence $\{a_n\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \frac{5}{2}a_{n+1}a_n$, where $n \in \mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If $a_2 = \frac{3}{2}$, $a_3 = x$, and $a_4 = 4$, find the range of $x$.
2. Suppose the sequence $\{a_n\}$ is a geometric sequence with a common ratio of $q$. If $\frac{1}{2}S_n < S_{n+1} < 2S_n$ for $n \in \mathbb{N}^*$, find the range of $q$.
3. If $a_1, a_2, \ldots, a_k$ ($k \geq 3$) form an arithmetic sequence, and $a_1 + a_2 + \ldots + a_k = 120$, find the minimum value of the positive integer $k$, and the corresponding sequence $a_1, a_2, \ldots, a_k$ when $k$ takes the minimum value. | 16 | 0 |
27,405 | Regular decagon \( ABCDEFGHIJ \) has its center at \( K \). Each of the vertices and the center are to be associated with one of the digits \( 1 \) through \( 10 \), with each digit used exactly once, in such a way that the sums of the numbers on the lines \( AKF \), \( BKG \), \( CKH \), \( DKI \), and \( EKJ \) are all equal. Find the number of valid ways to associate the numbers. | 3840 | 0 |
27,406 | Suppose F_1 and F_2 are the two foci of a hyperbola C, and there exists a point P on the curve C that is symmetric to F_1 with respect to an asymptote of C. Calculate the eccentricity of the hyperbola C. | \sqrt{5} | 28.90625 |
27,407 | Timur and Alexander are counting the trees growing around the house. Both move in the same direction, but they start counting from different trees. What is the number of trees growing around the house if the tree that Timur called the 12th, Alexander counted as $33-m$, and the tree that Timur called the $105-m$, Alexander counted as the 8th? | 76 | 9.375 |
27,408 | What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | 63_{10} | 0 |
27,409 | In response to the national call for "entrepreneurship and innovation for everyone," Xiao Wang decided to start a business in his field after graduating from college. After market research, Xiao Wang found that the annual fixed cost for producing a certain small electronic product is 20,000 yuan, and the variable cost C(x) in ten thousands yuan for producing x ten thousands units is given as follows: $$C(x)= \frac {1}{3}x^{2}+2x$$ for annual production less than 80,000 units, and $$C(x)=7x+ \frac {100}{x}-37$$ for annual production of at least 80,000 units. Each product is sold for 6 yuan. It is assumed that all products produced by Xiao Wang are sold within the year.
(Ⅰ) Write the function expression P(x) for the annual profit in ten thousands yuan with respect to the annual production x in ten thousands units (Note: Annual profit = Annual sales income - Fixed cost - Variable cost);
(Ⅱ) At what annual production quantity (in ten thousands units) is Xiao Wang's profit maximized for this product, and what is the maximum profit? | 15 | 16.40625 |
27,410 | Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) that satisfies the following properties: The distances from three of the points \(A, B, C, D\) to \(\alpha\) are equal, while the distance from the fourth point to \(\alpha\) is twice the distance of one of the three aforementioned points. Determine the number of such planes \(\alpha\). | 32 | 5.46875 |
27,411 | What is the length of side $y$ in the following diagram?
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(3))--cycle); // modified triangle lengths
draw((0,0)--(-2,0)--(0,2*sqrt(3))--cycle);
label("10",(-1,2*sqrt(3)/2),NW); // changed label
label("$y$",(2/2,2*sqrt(3)/2),NE);
draw("$30^{\circ}$",(2.5,0),NW); // modified angle
draw("$45^{\circ}$",(-1.9,0),NE);
draw(rightanglemark((0,2*sqrt(3)),(0,0),(2,0),4));
[/asy] | 10\sqrt{3} | 1.5625 |
27,412 | There are 5 balls of the same shape and size in a bag, including 3 red balls and 2 yellow balls. Now, balls are randomly drawn from the bag one at a time until two different colors of balls are drawn. Let the random variable $\xi$ be the number of balls drawn at this time. Find $E(\xi)=$____. | \frac{5}{2} | 33.59375 |
27,413 |
A client of a brokerage company deposited 12,000 rubles into a brokerage account at a rate of 60 rubles per dollar with instructions to the broker to invest the amount in bonds of foreign banks, which have a guaranteed return of 12% per annum in dollars.
(a) Determine the amount in rubles that the client withdrew from their account after a year if the ruble exchange rate was 80 rubles per dollar, the currency conversion fee was 4%, and the broker's commission was 25% of the profit in the currency.
(b) Determine the effective (actual) annual rate of return on investments in rubles.
(c) Explain why the actual annual rate of return may differ from the one you found in point (b). In which direction will it differ from the above value? | 39.52\% | 0.78125 |
27,414 | Suppose \( a_{n} \) denotes the last two digits of \( 7^{n} \). For example, \( a_{2} = 49 \), \( a_{3} = 43 \). Find the value of \( a_{1} + a_{2} + a_{3} + \cdots + a_{2007} \). | 50199 | 63.28125 |
27,415 | Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers satisfying $0 \leq x \leq 1$, $0 \leq y \leq 3$, and $0 \leq z \leq 5$. Two distinct points are chosen at random from $S$. The probability that the midpoint of the segment connecting them also belongs to $S$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime. Find $p + q$. | 52 | 0 |
27,416 | What is the largest $5$ digit integer congruent to $17 \pmod{26}$? | 99997 | 3.90625 |
27,417 | The FISS World Cup is a very popular football event among high school students worldwide. China successfully obtained the hosting rights for the International Middle School Sports Federation (FISS) World Cup in 2024, 2026, and 2028. After actively bidding by Dalian City and official recommendation by the Ministry of Education, Dalian ultimately became the host city for the 2024 FISS World Cup. During the preparation period, the organizing committee commissioned Factory A to produce a certain type of souvenir. The production of this souvenir requires an annual fixed cost of 30,000 yuan. For each x thousand pieces produced, an additional variable cost of P(x) yuan is required. When the annual production is less than 90,000 pieces, P(x) = 1/2x^2 + 2x (in thousand yuan). When the annual production is not less than 90,000 pieces, P(x) = 11x + 100/x - 53 (in thousand yuan). The selling price of each souvenir is 10 yuan. Through market analysis, it is determined that all souvenirs can be sold out in the same year.
$(1)$ Write the analytical expression of the function of annual profit $L(x)$ (in thousand yuan) with respect to the annual production $x$ (in thousand pieces). (Note: Annual profit = Annual sales revenue - Fixed cost - Variable cost)
$(2)$ For how many thousand pieces of annual production does the factory maximize its profit in the production of this souvenir? What is the maximum profit? | 10 | 3.90625 |
27,418 | A magician writes the numbers 1 to 16 on 16 positions of a spinning wheel. Four audience members, A, B, C, and D, participate in the magic show. The magician closes his eyes, and then A selects a number from the wheel. B, C, and D, in that order, each choose the next number in a clockwise direction. Only A and D end up with even numbers on their hands. The magician then declares that he knows the numbers they picked. What is the product of the numbers chosen by A and D? | 120 | 1.5625 |
27,419 | What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence? | 51 | 38.28125 |
27,420 | Given that a 4-digit positive integer has four different digits, the leading digit is not zero, the integer is a multiple of 4, and 6 is the largest digit, determine the total count of such integers. | 56 | 3.90625 |
27,421 | Using the numbers 3, 0, 4, 8, and a decimal point to form decimals, the largest three-digit decimal is \_\_\_\_\_\_, the smallest decimal is \_\_\_\_\_\_, and their difference is \_\_\_\_\_\_. | 8.082 | 0.78125 |
27,422 | An isosceles trapezoid \(ABCD\) is circumscribed around a circle. The lateral sides \(AB\) and \(CD\) are tangent to the circle at points \(M\) and \(N\), respectively, and \(K\) is the midpoint of \(AD\). In what ratio does the line \(BK\) divide the segment \(MN\)? | 1:3 | 0.78125 |
27,423 | Calculate how many numbers from 1 to 30030 are not divisible by any of the numbers between 2 and 16. | 5760 | 82.8125 |
27,424 | For any real number $x$, the symbol $[x]$ represents the integer part of $x$, i.e., $[x]$ is the largest integer not exceeding $x$. For example, $[2]=2$, $[2.1]=2$, $[-2.2]=-3$. This function $[x]$ is called the "floor function", which has wide applications in mathematics and practical production. Given that the function $f(x) (x \in \mathbb{R})$ satisfies $f(x)=f(2-x)$, and when $x \geqslant 1$, $f(x)=\log _{2}x$, find the value of $[f(-16)]+[f(-15)]+\ldots+[f(15)]+[f(16)]$. | 84 | 95.3125 |
27,425 | The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . | 23 | 51.5625 |
27,426 | The value of \( 2 \frac{1}{10} + 3 \frac{11}{100} \) is: | 5.21 | 0.78125 |
27,427 | Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | 504 | 2.34375 |
27,428 | A company needs 500 tons of raw materials to produce a batch of Product A, and each ton of raw material can generate a profit of 1.2 million yuan. Through equipment upgrades, the raw materials required to produce this batch of Product A were reduced by $x (x > 0)$ tons, and the profit generated per ton of raw material increased by $0.5x\%$. If the $x$ tons of raw materials saved are all used to produce the company's newly developed Product B, the profit generated per ton of raw material is $12(a-\frac{13}{1000}x)$ million yuan, where $a > 0$.
$(1)$ If the profit from producing this batch of Product A after the equipment upgrade is not less than the profit from producing this batch of Product A before the upgrade, find the range of values for $x$;
$(2)$ If the profit from producing this batch of Product B is always not higher than the profit from producing this batch of Product A after the equipment upgrade, find the maximum value of $a$. | 5.5 | 0 |
27,429 | Quadrilateral $ABCD$ has mid-segments $EF$ and $GH$ such that $EF$ goes from midpoint of $AB$ to midpoint of $CD$, and $GH$ from midpoint of $BC$ to midpoint of $AD$. Given that $EF$ and $GH$ are perpendicular, and the lengths are $EF = 18$ and $GH = 24$, find the area of $ABCD$. | 864 | 13.28125 |
27,430 | The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction. | \frac{39}{50} | 9.375 |
27,431 | Simplify completely: $$\sqrt[3]{80^3 + 100^3 + 120^3}.$$ | 20\sqrt[3]{405} | 0 |
27,432 | Six orange candies and four purple candies are available to create different flavors. A flavor is considered different if the percentage of orange candies is different. Combine some or all of these ten candies to determine how many unique flavors can be created based on their ratios. | 14 | 0 |
27,433 | There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers)
(1) The male students must stand together;
(2) The female students cannot stand next to each other;
(3) If the 4 female students have different heights, they must stand from left to right in order from tallest to shortest;
(4) The teachers cannot stand at the ends, and the male students must stand in the middle. | 1728 | 13.28125 |
27,434 | The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150. | 1000 | 49.21875 |
27,435 | Six students stand in a row for a photo. Among them, student A and student B are next to each other, student C is not next to either student A or student B. The number of different ways the students can stand is ______ (express the result in numbers). | 144 | 46.875 |
27,436 | What is the least positive integer with exactly $12$ positive factors? | 72 | 0 |
27,437 | Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.
$(1)$ Find the equation of the ellipse $M$;
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and a parallelogram $OAPB$ is formed with $OA$ and $OB$ as adjacent sides, where point $P$ is on the ellipse $M$ and $O$ is the origin. Find the minimum distance from point $O$ to line $l$. | \frac{\sqrt{2}}{2} | 10.9375 |
27,438 | Let \( a, b, c, x, y, z \) be nonzero complex numbers such that
\[ a = \frac{b+c}{x-3}, \quad b = \frac{a+c}{y-3}, \quad c = \frac{a+b}{z-3}, \]
and \( xy + xz + yz = 10 \) and \( x + y + z = 6 \), find \( xyz \). | 15 | 1.5625 |
27,439 | Given that the numbers - 2, 5, 8, 11, and 14 are arranged in a specific cross-like structure, find the maximum possible sum for the numbers in either the row or the column. | 36 | 0.78125 |
27,440 | $Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. | \sqrt{6} | 27.34375 |
27,441 | The pony wants to cross a bridge where there are two monsters, A and B. Monster A is awake for 2 hours and rests for 1 hour. Monster B is awake for 3 hours and rests for 2 hours. The pony can only cross the bridge when both monsters are resting; otherwise, it will be eaten by the awake monster. When the pony arrives at the bridge, both monsters have just finished their rest periods. How long does the pony need to wait, in hours, to cross the bridge with the least amount of waiting time? | 15 | 80.46875 |
27,442 | Consider a five-digit integer in the form $AB,BCA$, where $A$, $B$, and $C$ are distinct digits. What is the largest possible value of $AB,BCA$ that is divisible by 7? | 98,879 | 0 |
27,443 | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0, S_{2017} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, find the value of $k$. | 1009 | 43.75 |
27,444 | Find all real numbers \( p \) such that the cubic equation \( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \) has two roots that are natural numbers. | 76 | 64.84375 |
27,445 | For the function $f(x) = x - 2 - \ln x$, we know that $f(3) = 1 - \ln 3 < 0$, $f(4) = 2 - \ln 4 > 0$. Using the bisection method to find the approximate value of the root of $f(x)$ within the interval $(3, 4)$, we first calculate the function value $f(3.5)$. Given that $\ln 3.5 = 1.25$, the next function value we need to find is $f(\quad)$. | 3.25 | 30.46875 |
27,446 | Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ and $0$ respectively. | 0.525 | 2.34375 |
27,447 | Given that Kira needs to store 25 files onto disks, each with 2.0 MB of space, where 5 files take up 0.6 MB each, 10 files take up 1.0 MB each, and the rest take up 0.3 MB each, determine the minimum number of disks needed to store all 25 files. | 10 | 9.375 |
27,448 | Given the sequence $\{a\_n\}$ that satisfies: $(a\_1=-13$, $a\_6+a\_8=-2)$, and $(a_{n-1}=2a\_n-a_{n+1}\ (n\geqslant 2))$, find the sum of the first 13 terms of the sequence $\left\{\frac{1}{a\_n a_{n+1}}\right\}$. | -\frac{1}{13} | 42.96875 |
27,449 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 8.8\% | 0 |
27,450 | Given a school library with four types of books: A, B, C, and D, and a student limit of borrowing at most 3 books, determine the minimum number of students $m$ such that there must be at least two students who have borrowed the same type and number of books. | 15 | 31.25 |
27,451 | Given the Fibonacci sequence $\{a_n\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$. | 2018 | 0 |
27,452 | Given a function $f(x)$ that satisfies: For any $x \in (0, +\infty)$, it always holds that $f(2x) = 2f(x)$; (2) When $x \in (1, 2]$, $f(x) = 2 - x$. If $f(a) = f(2020)$, find the smallest positive real number $a$. | 36 | 14.0625 |
27,453 | Let $m = 2^{20}5^{15}.$ How many positive integer divisors of $m^2$ are less than $m$ but do not divide $m$? | 299 | 86.71875 |
27,454 | Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous path following the stripes that goes from one vertex of the cube to the diagonally opposite vertex, passing through consecutively adjacent faces?
- A) $\frac{1}{128}$
- B) $\frac{1}{32}$
- C) $\frac{3}{64}$
- D) $\frac{1}{16}$
- E) $\frac{1}{8}$ | \frac{3}{64} | 13.28125 |
27,455 | What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | 10 | 0.78125 |
27,456 | Xiao Ming has multiple 1 yuan, 2 yuan, and 5 yuan banknotes. He wants to buy a kite priced at 18 yuan using no more than 10 of these banknotes and must use at least two different denominations. How many different ways can he pay? | 11 | 7.03125 |
27,457 | A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | \frac{1}{8} | 7.8125 |
27,458 | Add 75.892 to 34.5167 and then multiply the sum by 2. Round the final result to the nearest thousandth. | 220.817 | 6.25 |
27,459 | Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 60 | 0 |
27,460 | Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \] for all real numbers \( x \) and \( y \).
Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \). | -2 | 21.875 |
27,461 | Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? | 16 | 38.28125 |
27,462 | Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
*Bulgaria* | 11 | 0 |
27,463 | For how many integers $a$ with $|a| \leq 2005$ , does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | 90 | 53.90625 |
27,464 | A sphere with a radius of $1$ is placed inside a cone and touches the base of the cone. The minimum volume of the cone is \_\_\_\_\_\_. | \dfrac{8\pi}{3} | 9.375 |
27,465 | Given that there is a point P (x, -1) on the terminal side of ∠Q (x ≠ 0), and $\tan\angle Q = -x$, find the value of $\sin\angle Q + \cos\angle Q$. | -\sqrt{2} | 15.625 |
27,466 | A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many kilometers do we have left to ride to reach point C?" B answered, "We have a distance left to ride equivalent to one-third of the distance from here to point B." What is the distance between point A and point C? (Answer should be in fraction form.) | \frac{40}{3} | 1.5625 |
27,467 | Determine the maximal size of a set of positive integers with the following properties:
(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).
(2) No digit occurs more than once in the same integer.
(3) The digits in each integer are in increasing order.
(4) Any two integers have at least one digit in common (possibly at different positions).
(5) There is no digit which appears in all the integers. | 32 | 58.59375 |
27,468 | The orthocenter of triangle $DEF$ divides altitude $\overline{DM}$ into segments with lengths $HM = 10$ and $HD = 24.$ Calculate $\tan E \tan F.$ | 3.4 | 0 |
27,469 | In triangle ABC, medians AD and BE intersect at centroid G. The midpoint of segment AB is F. Given that the area of triangle GFC is l times the area of triangle ABC, find the value of l. | \frac{1}{3} | 2.34375 |
27,470 | Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=|\vec{b}|=2$, and $\vec{b}$ is perpendicular to $(2\vec{a}+\vec{b})$, find the angle between vector $\vec{a}$ and $\vec{b}$. | \frac{2\pi}{3} | 97.65625 |
27,471 | Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 3 & 7 \\ 4 & x & y \\ 4 & y & x+1 \end{vmatrix}
= 0.\]Find $x + y.$ | 20 | 13.28125 |
27,472 | What is the smallest positive integer \(n\) such that \(\frac{n}{n+51}\) is equal to a terminating decimal? | 74 | 1.5625 |
27,473 | On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers that immediately precede it in a clockwise direction.
Determine the maximum possible number of positive numbers that can be among the 2018 numbers. | 1008 | 0 |
27,474 | In the quadratic equation $2x^{2}-1=6x$, the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______. | -1 | 25.78125 |
27,475 | Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy the equation $a\sin B = \sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$.
$(2)$ Choose one set of conditions from the following three sets to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.
Set 1: $a = \sqrt{19}$, $c = 5$;
Set 2: The altitude $h$ on side $AB$ is $\sqrt{3}$, $a = 3$;
Set 3: $\cos C = \frac{1}{3}$, $c = 4\sqrt{2}$. | 4\sqrt{3} + 3\sqrt{2} | 0 |
27,476 | Determine how many "super prime dates" occurred in 2007, where a "super prime date" is defined as a date where both the month and day are prime numbers, and additionally, the day is less than or equal to the typical maximum number of days in the respective prime month. | 50 | 2.34375 |
27,477 | Given the function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. | \frac{\pi}{2} | 72.65625 |
27,478 | A cinema is showing four animated movies: Toy Story, Ice Age, Shrek, and Monkey King. The ticket prices are 50 yuan, 55 yuan, 60 yuan, and 65 yuan, respectively. Each viewer watches at least one movie and at most two movies. However, due to time constraints, viewers cannot watch both Ice Age and Shrek. Given that there are exactly 200 people who spent the exact same amount of money on movie tickets today, what is the minimum total number of viewers the cinema received today? | 1792 | 1.5625 |
27,479 | Write the number in the form of a fraction (if possible):
$$
x=0.5123412341234123412341234123412341234 \ldots
$$
Can you generalize this method to all real numbers with a periodic decimal expansion? And conversely? | \frac{51229}{99990} | 7.03125 |
27,480 | Suppose that $\sec y - \tan y = \frac{15}{8}$ and that $\csc y - \cot y = \frac{p}{q},$ where $\frac{p}{q}$ is in lowest terms. Find $p+q.$ | 30 | 6.25 |
27,481 | Find the absolute value of the difference of single-digit integers \( C \) and \( D \) such that in base \( 5 \):
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_5 \\
& & & \mathbf{3} & \mathbf{2} & D_5 \\
& & + & C & \mathbf{2} & \mathbf{4_5} \\
\cline{2-6}
& & C & \mathbf{2} & \mathbf{3} & \mathbf{1_5} \\
\end{array} $$ | 1_5 | 0 |
27,482 | An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nine balls. What is the probability that there are exactly five red balls and four blue balls in the urn?
A) $\frac{1}{10}$
B) $\frac{2}{10}$
C) $\frac{3}{10}$
D) $\frac{4}{10}$
E) $\frac{5}{10}$ | \frac{3}{10} | 30.46875 |
27,483 | Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. | 127 | 2.34375 |
27,484 | Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ . Let $c$ be the coefficient of $x^{48}$ in $f(x)$ . When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.) | 60 | 77.34375 |
27,485 | The equation \(2008=1111+444+222+99+77+55\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.
i) Find a similar decomposition for the number 2009.
ii) Determine all possible such decompositions of the number 2009 that use the minimum number of terms (the order of terms does not matter). | 1111 + 777 + 66 + 55 | 0 |
27,486 | Calculate the largest prime factor of $18^4 + 12^5 - 6^6$. | 11 | 1.5625 |
27,487 | For a positive real number $a$ , let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$ , find $k$ . | 4/3 | 53.90625 |
27,488 | Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after this directional change will it take for Tom to catch up to Alice? | 60 | 3.125 |
27,489 | Point $D$ lies on side $AC$ of equilateral triangle $ABC$ such that the measure of angle $DBC$ is 30 degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$? | \frac{1}{3} | 0.78125 |
27,490 | Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$ , respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$ , and denote by $P$ the reflection of $B$ across $\ell$ . Compute the expected value of $OP^2$ .
*Proposed by Lewis Chen* | 10004 | 11.71875 |
27,491 | In the diagram, pentagon \( PQRST \) has \( PQ = 13 \), \( QR = 18 \), \( ST = 30 \), and a perimeter of 82. Also, \( \angle QRS = \angle RST = \angle STP = 90^\circ \). The area of the pentagon \( PQRST \) is: | 270 | 0 |
27,492 | Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \leqslant S_5$. The sum of the first $9$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$ is ______. | - \frac{1}{9} | 2.34375 |
27,493 | Starting with the display "1," calculate the fewest number of keystrokes needed to reach "400". | 10 | 5.46875 |
27,494 | To enhance and beautify the city, all seven streetlights on a road are to be changed to colored lights. If there are three colors available for the colored lights - red, yellow, and blue - and the installation requires that no two adjacent streetlights are of the same color, with at least two lights of each color, there are ____ different installation methods. | 114 | 0.78125 |
27,495 | Find the sum of $202_4 + 330_4 + 1000_4$. Express your answer first in base 4, then convert that sum to base 10. | 158 | 0.78125 |
27,496 | Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements. | 192 | 0.78125 |
27,497 | A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___. | 600 | 22.65625 |
27,498 | The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports. | 20\% | 3.90625 |
27,499 | Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 4.6875 |
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