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|---|---|---|---|
27,800 | For a finite sequence $B = (b_1, b_2, \dots, b_{50})$ of numbers, the Cesaro sum of $B$ is defined as
\[\frac{T_1 + \cdots + T_{50}}{50},\]
where $T_k = b_1 + \cdots + b_k$ for $1 \leq k \leq 50$.
If the Cesaro sum of the 50-term sequence $(b_1, \dots, b_{50})$ is 200, what is the Cesaro sum of the 51-term sequence $(2, b_1, \dots, b_{50})$? | 198.078431372549 | 0 |
27,801 | In the diagram, $\triangle ABF$, $\triangle BCF$, and $\triangle CDF$ are right-angled, with $\angle ABF=\angle BCF = 90^\circ$ and $\angle CDF = 45^\circ$, and $AF=36$. Find the length of $CF$.
[asy]
pair A, B, C, D, F;
A=(0,25);
B=(0,0);
C=(0,-12);
D=(12, -12);
F=(24,0);
draw(A--B--C--D--F--A);
draw(B--F);
draw(C--F);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, SE);
label("F", F, NE);
[/asy] | 36 | 7.03125 |
27,802 | A certain scenic area has two attractions that require tickets for visiting. The three ticket purchase options presented at the ticket office are as follows:
Option 1: Visit attraction A only, $30$ yuan per person;
Option 2: Visit attraction B only, $50$ yuan per person;
Option 3: Combined ticket for attractions A and B, $70$ yuan per person.
It is predicted that in April, $20,000$ people will choose option 1, $10,000$ people will choose option 2, and $10,000$ people will choose option 3. In order to increase revenue, the ticket prices are adjusted. It is found that when the prices of options 1 and 2 remain unchanged, for every $1$ yuan decrease in the price of the combined ticket (option 3), $400$ people who originally planned to buy tickets for attraction A only and $600$ people who originally planned to buy tickets for attraction B only will switch to buying the combined ticket.
$(1)$ If the price of the combined ticket decreases by $5$ yuan, the number of people buying tickets for option 1 will be _______ thousand people, the number of people buying tickets for option 2 will be _______ thousand people, the number of people buying tickets for option 3 will be _______ thousand people; and calculate how many tens of thousands of yuan the total ticket revenue will be?
$(2)$ When the price of the combined ticket decreases by $x$ (yuan), find the functional relationship between the total ticket revenue $w$ (in tens of thousands of yuan) in April and $x$ (yuan), and determine at what price the combined ticket should be to maximize the total ticket revenue in April. What is the maximum value in tens of thousands of yuan? | 188.1 | 3.125 |
27,803 | Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$. | 4045 | 2.34375 |
27,804 |
Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers, while this week 15 chicks were sitting with their beaks open and 11 were growing feathers.
a) How many chicks were sitting with their beaks open two weeks ago?
b) How many chicks will be growing feathers next week?
Record the product of these numbers as the answer. | 165 | 9.375 |
27,805 | Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the length of the minor axis being $2$ and the eccentricity being $\frac{\sqrt{2}}{2}$, the line $l: y = kx + m$ intersects the ellipse $C$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ passes through the point $(0, -\frac{1}{2})$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Find the maximum area of $\triangle AOB$ ($O$ is the origin). | \frac{\sqrt{2}}{2} | 21.875 |
27,806 | Two cells in a \(20 \times 20\) board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a \(20 \times 20\) board such that every cell is adjacent to at most one marked cell? | 100 | 3.125 |
27,807 | James borrows $2000$ dollars from Alice, who charges an interest of $3\%$ per month (which compounds monthly). What is the least integer number of months after which James will owe more than three times as much as he borrowed? | 37 | 17.96875 |
27,808 | If the community center has 8 cans of soup and 2 loaves of bread, with each can of soup feeding 4 adults or 7 children and each loaf of bread feeding 3 adults or 4 children, and the center needs to feed 24 children, calculate the number of adults that can be fed with the remaining resources. | 22 | 3.125 |
27,809 | In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \). | 6\pi | 2.34375 |
27,810 | What is the total number of digits used when the first 4500 positive even integers are written? | 19444 | 2.34375 |
27,811 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = 1$, $| \overrightarrow{b}| = 6$, and $\overrightarrow{a} \cdot (\overrightarrow{b} - \overrightarrow{a}) = 2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi}{3} | 98.4375 |
27,812 | Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$ , find the modulo $2007$ residue of $m$ . | 2004 | 9.375 |
27,813 | The pentagon \( A B C D E \) is inscribed around a circle.
The angles \( \angle A B C \), \( \angle B A E \), and \( \angle C D E \) each measure \( 104^\circ \). Find \( \angle A D B \). Provide the answer in degrees (only the number, without units). | 38 | 28.90625 |
27,814 | A spinner with seven congruent sectors numbered from 1 to 7 is used. If Jane and her brother each spin the spinner once, and Jane wins if the absolute difference of their numbers is less than 4, what is the probability that Jane wins? Express your answer as a common fraction. | \frac{37}{49} | 46.09375 |
27,815 | In $\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \sqrt{2}$, $b=2$, and $\sin B - \cos B = \sqrt{2}$, find the measure of angle $A$. | \frac{\pi}{6} | 57.03125 |
27,816 | Suppose $a$, $b$, and $c$ are real numbers, and the roots of the equation \[x^4 - 10x^3 + ax^2 + bx + c = 0\] are four distinct positive integers. Compute $a + b + c.$ | 109 | 0.78125 |
27,817 | How many multiples of 4 are between 100 and 350? | 62 | 0 |
27,818 | A circle with a radius of 6 is inscribed around the trapezoid \(ABCD\). The center of this circle lies on the base \(AD\), and \(BC = 4\). Find the area of the trapezoid. | 24\sqrt{2} | 0 |
27,819 | The Intermediate Maths Challenge has 25 questions with the following scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks.
Where no answer is given 0 marks are scored.
Fiona scored 80 marks in total. What possible answers are there to the number of questions Fiona answered correctly? | 16 | 3.125 |
27,820 | 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 by at least 2? Express your answer as a common fraction. | \frac{4}{9} | 1.5625 |
27,821 | What is the area of the circle defined by \(x^2 - 8x + y^2 - 16y + 48 = 0\) that lies above the line \(y = 4\)? | 24\pi | 1.5625 |
27,822 | Given $\cos(α-β)= \frac{3}{5}$, $\sin β= -\frac{5}{13}$, and $α∈(0, \frac{π}{2})$, $β∈(-\frac{π}{2},0)$, find $\sin α$. | \frac{33}{65} | 26.5625 |
27,823 | In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | \sqrt{3} | 0 |
27,824 | Let $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$. | 47 | 5.46875 |
27,825 | The union of sets \( A \) and \( B \) is \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), \((A, B)\) and \((B, A)\) are considered different pairs. How many such pairs \((A, B)\) exist? | 27 | 6.25 |
27,826 | 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. | 63 | 15.625 |
27,827 | Given that \(A, B, C,\) and \(D\) are points on a circle with radius 1, \(\overrightarrow{AB} + 2 \overrightarrow{AC} = \overrightarrow{AD}\), and \(|AC| = 1\). Find the area of the quadrilateral \(ABDC\). | \frac{3 \sqrt{3}}{4} | 14.0625 |
27,828 | Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers. | 110880 | 67.1875 |
27,829 | Let \(b = 8\) and \(S_n\) be the sum of the reciprocals of the non-zero digits of the integers from \(1\) to \(8^n\) inclusive. Find the smallest positive integer \(n\) for which \(S_n\) is an integer. | 105 | 1.5625 |
27,830 | In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$ ?
[center]<see attached>[/center] | \frac{20}{3} | 14.0625 |
27,831 | On the side \( CD \) of the trapezoid \( ABCD \) with \( AD \parallel BC \), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped to the segment \( BM \). It is known that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \). | 18 | 12.5 |
27,832 | The circumcircle of acute $\triangle ABC$ has center $O$ . The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$ , respectively. Also $AB=5$ , $BC=4$ , $BQ=4.5$ , and $BP=\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 23 | 39.0625 |
27,833 | Find the sum of the values of \( x \) such that \( \cos^2 3x + \cos^2 7x = 6 \cos^2 4x \cos^2 2x \), where \( x \) is measured in degrees and \( 150 < x < 250. \) | 420 | 0 |
27,834 | Add $7A3_{16} + 1F4_{16}$. Express your answer in base 16, using A for 10, B for 11, ..., F for 15. | 997_{16} | 34.375 |
27,835 | In triangle \( A B C \), angle \( B \) equals \( 45^\circ \) and angle \( C \) equals \( 30^\circ \). Circles are constructed on the medians \( B M \) and \( C N \) as diameters, intersecting at points \( P \) and \( Q \). The chord \( P Q \) intersects side \( B C \) at point \( D \). Find the ratio of segments \( B D \) to \( D C \). | \frac{1}{\sqrt{3}} | 0 |
27,836 | Find the largest 5-digit number \( A \) that satisfies the following conditions:
1. Its 4th digit is greater than its 5th digit.
2. Its 3rd digit is greater than the sum of its 4th and 5th digits.
3. Its 2nd digit is greater than the sum of its 3rd, 4th, and 5th digits.
4. Its 1st digit is greater than the sum of all other digits.
(from the 43rd Moscow Mathematical Olympiad, 1980) | 95210 | 93.75 |
27,837 | (1) Evaluate the expression $$(\frac{\sqrt{121}}{2018} - 5)^0 + 2^{-2} \cdot (2\frac{1}{4})^{-\frac{1}{2}} - \log_4 3 \cdot \log_3 \sqrt{8}$$;
(2) The function $f(x) = x^{2-m}$ is an odd function defined on $[-3-m, m^2-m]$. Find the value of $f(m)$. | -1 | 25.78125 |
27,838 | Taylor is tiling his 12 feet by 16 feet living room floor. He plans to place 1 foot by 1 foot tiles along the edges to form a border, and then use 2 feet by 2 feet tiles to fill the remaining floor area. How many tiles will he use in total? | 87 | 67.1875 |
27,839 | The digits of the positive integer $N$ consist only of 1s and 0s, and $225$ divides $N$. What is the minimum value of $N$? | 111,111,100 | 0 |
27,840 | A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club have after 4 years? | 980 | 0.78125 |
27,841 | Given that Mary reversed the digits of a two-digit multiplier $b$, while multiplying correctly with a positive integer $a$, and her erroneous product was $180$, determine the correct product of $a$ and $b$. | 180 | 12.5 |
27,842 | Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $200$. For how many values of $n$ is $q+r$ divisible by $7$? | 13050 | 3.125 |
27,843 | A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they are then joined by a green line segment. In total, there are 52 red dots. There are 2 red dots at corners with an additional 16 red dots on the edges of the array. The remainder of the red dots are inside the array. There are 98 green line segments. The number of blue line segments is | 37 | 0 |
27,844 | (This question is worth 14 points.)
A newspaper stand in a city buys the "Evening News" from the newspaper office at a price of 0.20 yuan per copy and sells it at 0.30 yuan per copy. The unsold newspapers can be returned to the newspaper office at a price of 0.05 yuan per copy. In a month (calculated as 30 days), there are 20 days when 400 copies can be sold each day, and for the remaining 10 days, only 250 copies can be sold each day. However, the number of copies bought from the newspaper office each day must be the same. How many copies should the stand owner buy from the newspaper office each day to maximize the monthly profit? And calculate the maximum amount of money he can earn in a month? | 825 | 18.75 |
27,845 | A solid cube of side length \(4 \mathrm{~cm}\) is cut into two pieces by a plane that passed through the midpoints of six edges. To the nearest square centimetre, the surface area of each half cube created is: | 69 | 0 |
27,846 | On an exam there are 5 questions, each with 4 possible answers. 2000 students went on the exam and each of them chose one answer to each of the questions. Find the least possible value of $n$ , for which it is possible for the answers that the students gave to have the following property: From every $n$ students there are 4, among each, every 2 of them have no more than 3 identical answers. | 25 | 0.78125 |
27,847 | Given vectors $\overrightarrow {a}=( \sqrt {3}\sin x, m+\cos x)$ and $\overrightarrow {b}=(\cos x, -m+\cos x)$, and a function $f(x)= \overrightarrow {a}\cdot \overrightarrow {b}$
(1) Find the analytical expression of function $f(x)$;
(2) When $x\in[- \frac {\pi}{6}, \frac {\pi}{3}]$, the minimum value of $f(x)$ is $-4$. Find the maximum value of the function $f(x)$ and the corresponding $x$ value. | \frac {\pi}{6} | 48.4375 |
27,848 | For how many primes \( p < 50 \) is \( p^{4} + 5p^{3} + 4 \) divisible by 5? | 13 | 9.375 |
27,849 | As shown in the diagram, in the tetrahedron \(A B C D\), the face \(A B C\) intersects the face \(B C D\) at a dihedral angle of \(60^{\circ}\). The projection of vertex \(A\) onto the plane \(B C D\) is \(H\), which is the orthocenter of \(\triangle B C D\). \(G\) is the centroid of \(\triangle A B C\). Given that \(A H = 4\) and \(A B = A C\), find \(G H\). | \frac{4\sqrt{21}}{9} | 0 |
27,850 | Under normal circumstances, for people aged between 18 and 38, the regression equation of weight $y$ (kg) to height $x$ (cm) is $\overset{\land }{y} = 0.72x - 58.2$. Zhang Hong, who is 20 years old and has a height of 178 cm, should have a weight of approximately \_\_\_\_\_ kg. | 69.96 | 0 |
27,851 | What is the sum of all integer solutions to \( |n| < |n-5| < 10 \)? | -12 | 1.5625 |
27,852 | A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find the value of $x$. | 20 | 10.15625 |
27,853 | Let $b = \pi/2010$. Find the smallest positive integer $m$ such that
\[2[\cos(b)\sin(b) + \cos(4b)\sin(2b) + \cos(9b)\sin(3b) + \cdots + \cos(m^2b)\sin(mb)]\]
is an integer. | 67 | 8.59375 |
27,854 | If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\overrightarrow{a}=\left(0,-2,-1\right)$ and $\overrightarrow{b}=\left(2,0,4\right)$, calculate the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$. | \frac{2}{5} | 0 |
27,855 | A cone has a volume of $2592\pi$ cubic inches and the vertex angle of the vertical cross section is 90 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. | 20.0 | 0 |
27,856 | There are five concentric circles \(\Gamma_{0}, \Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \Gamma_{4}\) whose radii form a geometric sequence with a common ratio \(q\). Find the maximum value of \(q\) such that a closed polyline \(A_{0} A_{1} A_{2} A_{3} A_{4}\) can be drawn, where each segment has equal length and the point \(A_{i} (i=0,1, \ldots, 4)\) is on the circle \(\Gamma_{i}\). | \frac{\sqrt{5} + 1}{2} | 2.34375 |
27,857 | How many possible distinct arrangements are there of the letters in the word SUCCESS? | 420 | 21.09375 |
27,858 | Consider a 9x9 chessboard where the squares are labelled from a starting square at the bottom left (1,1) increasing incrementally across each row to the top right (9,9). Each square at position $(i,j)$ is labelled with $\frac{1}{i+j-1}$. Nine squares are chosen such that there is exactly one chosen square in each row and each column. Find the minimum product of the labels of the nine chosen squares. | \frac{1}{362880} | 12.5 |
27,859 | Let $p$ be a polynomial with integer coefficients such that $p(15)=6$ , $p(22)=1196$ , and $p(35)=26$ . Find an integer $n$ such that $p(n)=n+82$ . | 28 | 10.15625 |
27,860 | Given $f(x) = \frac {\log_{2}x-1}{2\log_{2}x+1}$ (where $x > 2$), and $f(x_1) + f(2x_2) = \frac {1}{2}$, find the minimum value of $f(x_1x_2)$. | \frac {1}{3} | 12.5 |
27,861 | The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula. | 1333\frac{1}{3} | 0 |
27,862 | Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, and it passes through point $M(-2, 0)$.
(I) Find the standard equation of ellipse $C$;
(II) Let line $l$ with a slope of $1$ intersect ellipse $C$ at points $A(x\_1, y\_1)$ and $B(x\_2, y\_2)$. Connect $MA$ and $MB$, then extend them to intersect line $x = 4$ at points $P$ and $Q$. Let $y\_P$ and $y\_Q$ be the y-coordinates of points $P$ and $Q$ respectively, and $\frac{1}{y\_1} + \frac{1}{y\_2} = \frac{1}{y\_P} + \frac{1}{y\_Q}$. Find the area of triangle $ABM$. | S = \sqrt{10} | 5.46875 |
27,863 | Let \( x, y, z, u, v \in \mathbf{R}_{+} \). The maximum value of
\[
f = \frac{x y + y z + z u + u v}{2 x^{2} + y^{2} + 2 z^{2} + u^{2} + 2 v^{2}}
\]
is $\qquad$ . | \frac{\sqrt{6}}{4} | 0 |
27,864 | In an opaque bag, there are 2 red balls and 5 black balls, all identical in size and material. Balls are drawn one by one without replacement until all red balls are drawn. Calculate the expected number of draws. | \dfrac{16}{3} | 3.90625 |
27,865 | Two people, A and B, are collaborating to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they reach half of the total amount, A's speed triples while B takes a 5-minute break and then continues at the original speed. By the time the document is completed, A and B have typed an equal number of characters. How many characters are in the document in total? | 18000 | 22.65625 |
27,866 | How many values of $\theta$ in the interval $0 < \theta \leq 4\pi$ satisfy the equation $2 + 4\sin2\theta - 3\cos4\theta + 2\tan\theta = 0$? | 16 | 5.46875 |
27,867 | Given a point M$(x_0, y_0)$ moves on the circle $x^2+y^2=4$, and N$(4, 0)$, the point P$(x, y)$ is the midpoint of the line segment MN.
(1) Find the trajectory equation of point P$(x, y)$.
(2) Find the maximum and minimum distances from point P$(x, y)$ to the line $3x+4y-86=0$. | 15 | 80.46875 |
27,868 | Given the function $f(x) = 2\sin\omega x\cos\omega x + 2\sqrt{3}\sin^2\omega x - \sqrt{3}$ ($\omega > 0$) has the smallest positive period of $\pi$.
(1) Find the interval of monotonic increase for the function $f(x)$;
(2) The graph of $f(x)$ is obtained by translating the graph of $y=\sin x$ in what way;
(3) If the graph of the function $f(x)$ is translated to the left by $\frac{\pi}{6}$ units and then translated up by 1 unit to get the graph of the function $y=g(x)$, and if $y=g(x)$ has at least 10 zeros in the interval $[0, b]$ ($b>0$), find the minimum value of $b$. | \frac{59\pi}{12} | 22.65625 |
27,869 | In the diagram, $\angle PQR = 90^\circ$. A line PS bisects $\angle PQR$, and $\angle PQS = y^\circ$. If $\angle SQR = 2x^\circ$ and $\angle PQS = 2y^\circ$, what is the value of $x + y$?
[asy]
size(100);
draw((0,1)--(0,0)--(1,0));
draw((0,0)--(.9,.47));
draw((0,.1)--(.1,.1)--(.1,0));
label("$P$",(0,1),N); label("$Q$",(0,0),SW); label("$R$",(1,0),E); label("$S$",(.9,.47),NE);
label("$2y^\circ$",(0.15,.2)); label("$2x^\circ$",(.32,-.02),N);
[/asy] | 45 | 63.28125 |
27,870 | Each of the $25$ balls is tossed independently and at random into one of $5$ bins. Let $r$ be the probability that one bin ends up with $6$ balls, another with $7$ balls, and the other three with $4$ balls each. Let $s$ be the probability that one bin ends up with $5$ balls and the other four with $5$ balls each. Compute the ratio $\frac{r}{s}$.
**A)** 5
**B)** $\frac{10}{3}$
**C)** $\frac{10 \cdot \binom{25}{6}\binom{19}{7}\binom{12}{4}\binom{8}{4}\binom{4}{4}}{\binom{25}{5}\binom{20}{5}\binom{15}{5}\binom{10}{5}\binom{5}{5}}$
**D)** 15 | \frac{10 \cdot \binom{25}{6}\binom{19}{7}\binom{12}{4}\binom{8}{4}\binom{4}{4}}{\binom{25}{5}\binom{20}{5}\binom{15}{5}\binom{10}{5}\binom{5}{5}} | 0 |
27,871 | Let \( r(\theta) = \frac{1}{1-2\theta} \). Calculate \( r(r(r(r(r(r(10)))))) \) (where \( r \) is applied 6 times). | 10 | 0 |
27,872 | A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \(120^\circ\) and the other two angles being \(60^\circ\) each. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest side of 120 m. | \frac{1}{2} | 47.65625 |
27,873 | For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? | 212 | 0 |
27,874 | The sequence of integers $ a_1 $ , $ a_2 $ , $ \dots $ is defined as follows: $ a_1 = 1 $ and $ n> 1 $ , $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different.
Define $ a_ {2004} $ . | 3006 | 7.03125 |
27,875 | Let $S$ be a set. We say $S$ is $D^\ast$ *-finite* if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$ , there exists a $y \in Y$ such that $f(y) \notin Y$ . The function $f$ is called a *witness* of $S$ . How many witnesses does $\{0,1,\cdots,5\}$ have?
*Proposed by Evan Chen* | 120 | 14.84375 |
27,876 | Calculate the value of the following product as a common fraction:
\[ \left(2 \cdot \left(1-\frac{1}{2}\right)\right) \cdot \left(1-\frac{1}{3}\right) \cdot \left(1-\frac{1}{4}\right) \dotsm \left(1-\frac{1}{50}\right) \] | \frac{1}{50} | 18.75 |
27,877 | Given the hyperbola $$E: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ with left and right vertices A and B, respectively. Let M be a point on the hyperbola such that ∆ABM is an isosceles triangle, and the area of its circumcircle is 4πa², then the eccentricity of the hyperbola E is _____. | \sqrt{2} | 9.375 |
27,878 | The solutions to the equation $x^2 = x$ are $x=0$ and $x=1$. | 0 or 1 | 0 |
27,879 | Find $x+y+z$ when $$ a_1x+a_2y+a_3z= a $$ $$ b_1x+b_2y+b_3z=b $$ $$ c_1x+c_2y+c_3z=c $$ Given that $$ a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9 $$ $$ a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17 $$ $$ a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c-bc_1\right)=-8 $$ $$ a_1\left(b_2c-bc_2\right)-a_2\left(b_1c-bc_1\right)+a\left(b_1c_2-b_2c_1\right)=7. $$ *2017 CCA Math Bonanza Lightning Round #5.1* | 16/9 | 96.875 |
27,880 | Given circle $O$: $x^{2}+y^{2}=10$, a line $l$ passing through point $P(-3,-4)$ intersects with circle $O$ at points $A$ and $B$. If the area of triangle $AOB$ is $5$, find the slope of line $l$. | \frac{11}{2} | 0.78125 |
27,881 | Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determine the number of sweets Beth had originally. | 50 | 6.25 |
27,882 |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height, dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$.
$A_{1}(2, -1, -2)$
$A_{2}(1, 2, 1)$
$A_{3}(5, 0, -6)$
$A_{4}(-10, 9, -7)$ | 4\sqrt{14} | 3.125 |
27,883 | Given the function $f\left(x\right)=|2x-3|+|x-2|$.<br/>$(1)$ Find the solution set $M$ of the inequality $f\left(x\right)\leqslant 3$;<br/>$(2)$ Under the condition of (1), let the smallest number in $M$ be $m$, and let positive numbers $a$ and $b$ satisfy $a+b=3m$, find the minimum value of $\frac{{{b^2}+5}}{a}+\frac{{{a^2}}}{b}$. | \frac{13}{2} | 0.78125 |
27,884 | The maximum value and the minimum positive period of the function $f(x)=\cos 4x \cdot \cos 2x \cdot \cos x \cdot \sin x$ are to be determined. | \frac{\pi}{4} | 65.625 |
27,885 | Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
*Author: Anderson Wang* | 1025 | 91.40625 |
27,886 | A standard deck of 52 cards is divided into 4 suits, with each suit containing 13 cards. Two of these suits are red, and the other two are black. The deck is shuffled, placing the cards in random order. What is the probability that the first three cards drawn from the deck are all the same color? | \frac{40}{85} | 0 |
27,887 | Find the smallest constant $D$ so that
\[ 2x^2 + 3y^2 + z^2 + 3 \ge D(x + y + z) \]
for all real numbers $x$, $y$, and $z$. | -\sqrt{\frac{72}{11}} | 0 |
27,888 | A tower is $45 \mathrm{~m}$ away from the bank of a river. If the width of the river is seen at an angle of $20^{\circ}$ from a height of $18 \mathrm{~m}$ in the tower, how wide is the river? | 16.38 | 27.34375 |
27,889 | Find the values of $a$, $b$, and $c$ such that the equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 3$ can be reduced to an equivalent form involving $\cos ax \cos bx \cos cx = 0$ for some positive integers $a$, $b$, and $c$, and then find $a + b + c$. | 12 | 19.53125 |
27,890 | Given the ellipse $E$: $\frac{x^{2}}{2}+y^{2}=1$ with its right focus $F$, two perpendicular lines passing through $F$ intersect with $E$ at points $A$, $C$ and $B$, $D$.
1. Can the quadrilateral $ABCD$ form a parallelogram? Please explain the reason.
2. Find the minimum value of $|AC|+|BD|$. | \frac{8 \sqrt{2}}{3} | 5.46875 |
27,891 | In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find the length of $GH$ and $GI$. | 18.7 | 0 |
27,892 | Define $E(n)$ as the sum of the even digits of $n$ and $O(n)$ as the sum of the odd digits of $n$. Find the value of $E(1) + O(1) + E(2) + O(2) + \dots + E(150) + O(150)$.
A) 1200
B) 1300
C) 1350
D) 1400
E) 1450 | 1350 | 54.6875 |
27,893 | A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$ , where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$ , compute the area of the concave quadrilateral $ABHC$ . | \frac{\sqrt{3}}{2} | 3.90625 |
27,894 | A line parallel to leg \(AC\) of right triangle \(ABC\) intersects leg \(BC\) at point \(K\) and the hypotenuse \(AB\) at point \(N\). On leg \(AC\), a point \(M\) is chosen such that \(MK = MN\). Find the ratio \(\frac{AM}{MC}\) if \(\frac{BK}{BC} = 14\). | 27 | 1.5625 |
27,895 | The bases \( AB \) and \( CD \) of trapezoid \( ABCD \) are 65 and 31, respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \). | 2015 | 35.15625 |
27,896 | In the sequence $\{a_n\}$, $a_1= \sqrt{2}$, $a_n= \sqrt{a_{n-1}^2 + 2}\ (n\geqslant 2,\ n\in\mathbb{N}^*)$, let $b_n= \frac{n+1}{a_n^4(n+2)^2}$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. The value of $16S_n+ \frac{1}{(n+1)^2}+ \frac{1}{(n+2)^2}$ is ______. | \frac{5}{4} | 48.4375 |
27,897 | (1) Given $ \frac {\pi}{2} < \beta < \alpha < \frac {3\pi}{4}$, $\cos (\alpha-\beta)= \frac {12}{13}$, $\sin (\alpha+\beta)=- \frac {3}{5}$, find the value of $\sin 2\alpha$.
(2) Given $ \frac {\pi}{2} < \alpha < \pi$, $0 < \beta < \frac {\pi}{2}$, $\tan \alpha=- \frac {3}{4}$, $\cos (\beta-\alpha)= \frac {5}{13}$, find the value of $\sin \beta$. | \frac {63}{65} | 47.65625 |
27,898 | Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM. | 16 | 28.90625 |
27,899 | The first term of a sequence is $3107$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $614^{\text{th}}$ term of the sequence? | 20 | 0.78125 |
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