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|---|---|---|---|
27,700 | A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. | 20 | 0.78125 |
27,701 | How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 50 | 19.53125 |
27,702 | The graph of the function in the form \( y=\frac{b}{|x|-a} \) (where \( a, b > 0 \)) resembles the Chinese character "唄". It is referred to as the "唄 function", and the point symmetric to its intersection with the y-axis about the origin is called the "目 point". A circle with its center at the 明 point that intersects the 唄 function is called the "唄 circle". For \( a=b=1 \), the minimum area of all 唄 circles is . | 3\pi | 0.78125 |
27,703 | Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Assume the shaded region is the smaller region formed inside the triangle including points $D$, $E$, $F$, $G$, and $H$. | \frac{1}{3} | 0.78125 |
27,704 | There are 21 different pairs of digits (a, b) such that $\overline{5a68} \times \overline{865b}$ is divisible by 824. | 19 | 0 |
27,705 | Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.
Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter. | 157 | 19.53125 |
27,706 | What is the product of the prime numbers less than 20? | 9699690 | 100 |
27,707 | Let \( a, b, c, d, e \) be natural numbers with \( a < b < c < d < e \), and \( a + 2b + 3c + 4d + 5e = 300 \). Determine the maximum value of \( a + b \). | 35 | 10.9375 |
27,708 | Given the parabola C: y² = 3x with focus F, and a line l with slope $\frac{3}{2}$ intersecting C at points A and B, and the x-axis at point P.
(1) If |AF| + |BF| = 4, find the equation of line l;
(2) If $\overrightarrow{AP}$ = 3$\overrightarrow{PB}$, find |AB|. | \frac{4\sqrt{13}}{3} | 5.46875 |
27,709 | The maximum point of the function $f(x)=\frac{1}{3}x^3+\frac{1}{2}x^2-2x+3$ is ______. | -2 | 0.78125 |
27,710 | The king called two wise men. He gave the first one 100 blank cards and ordered him to write a positive number on each (the numbers do not have to be different) without showing them to the second wise man. Then, the first wise man can communicate several different numbers to the second wise man, each of which is either written on one of the cards or is the sum of the numbers on some of the cards (without specifying how each number is obtained). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be beheaded; otherwise, each will lose as many beard hairs as the numbers the first wise man communicated to the second. How can the wise men, without colluding, stay alive and lose the minimum number of hairs? | 101 | 0 |
27,711 | The sequence is defined as \( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \) for \( k=1,2, \cdots, n-1 \). Determine the value of \( n \) for which \( a_{n}=0 \). | 201 | 0 |
27,712 | 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 | 49.21875 |
27,713 | Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$. | 3059 | 0 |
27,714 | In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. The value of \( x \) is | 80 | 14.84375 |
27,715 | Let the original number be expressed as $x$. When the decimal point of $x$ is moved one place to the right, the resulting number can be expressed as $100x$. According to the given information, we have the equation $100x = x + 34.65$. | 3.85 | 6.25 |
27,716 | $\frac{\text{华杯赛}}{\text{少} \times \text{俊} + \text{金坛} + \text{论} \times \text{数}} = 15$
In the above equation, different Chinese characters represent different digits between $1$ and $9$. When the three-digit number "华杯赛" reaches its maximum value, please write a solution where the equation holds. | 975 | 3.125 |
27,717 | We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number? | 606 | 40.625 |
27,718 | There are 8 seats in a row, and 3 people are sitting in the same row. If there are empty seats on both sides of each person, the number of different seating arrangements is \_\_\_\_\_\_\_\_\_. | 24 | 14.0625 |
27,719 | You have four textbooks for mandatory courses numbered 1 to 4. They are randomly placed on the same shelf.
1. Calculate the probability that Textbook 2 is to the left of Textbook 4.
2. Calculate the probability that Textbook 2 is to the left of Textbook 3, and Textbook 3 is to the left of Textbook 4. | \frac{1}{4} | 14.84375 |
27,720 | A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can travel to accomplish this?
A) $\sqrt{289} + 8$
B) $16 + \sqrt{185}$
C) $8 + \sqrt{545}$
D) $12 + \sqrt{400}$
E) $10 + \sqrt{365}$ | 8 + \sqrt{545} | 8.59375 |
27,721 | Determine the base seven product of the numbers $321_7$ and $13_7$. | 4503_7 | 44.53125 |
27,722 | 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{23}{36} | 0 |
27,723 | Determine the value of \( n \) if we know that
$$
\binom{n}{5}=\frac{n(n-1)(n-2)(n-3)(n-4)}{2 \cdot 3 \cdot 4 \cdot 5}
$$
(which, as we know, is an integer) in the decimal system is of the form \(\overline{ababa}\), where \( a \) and \( b \) represent digits. | 39 | 0 |
27,724 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, and $\overrightarrow{b} \perp (2\overrightarrow{a}+ \overrightarrow{b})$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{2\pi}{3} | 98.4375 |
27,725 | A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be filled? | 665,280 | 0 |
27,726 | Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by 50. Determine the number of values of $n$ for which $q+r$ is divisible by 7. | 12600 | 6.25 |
27,727 | If $\lceil{\sqrt{x}}\rceil=17$, how many possible integer values of $x$ are there? | 33 | 95.3125 |
27,728 | Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first 2 games, A and B each won 1 game.
(I) Calculate the probability of A winning the match;
(II) Let $\xi$ represent the number of games played from the 3rd game until the end of the match, calculate the distribution and the mathematical expectation of $\xi$. | 2.48 | 3.90625 |
27,729 | The diagram shows three touching semicircles with radius 1 inside an equilateral triangle, with each semicircle also touching the triangle. The diameter of each semicircle lies along a side of the triangle. What is the length of each side of the equilateral triangle? | $2 \sqrt{3}$ | 0 |
27,730 | Given a regular triangular pyramid \(P-ABC\), where points \(P\), \(A\), \(B\), and \(C\) all lie on the surface of a sphere with radius \(\sqrt{3}\), and \(PA\), \(PB\), and \(PC\) are mutually perpendicular, find the distance from the center of the sphere to the cross-section \(ABC\). | \frac{\sqrt{3}}{3} | 7.8125 |
27,731 | A boulevard has 25 houses on each side, for a total of 50 houses. The addresses on the east side of the boulevard follow an arithmetic sequence, as do the addresses on the west side. On the east side, the addresses start at 5 and increase by 7 (i.e., 5, 12, 19, etc.), while on the west side, they start at 2 and increase by 5 (i.e., 2, 7, 12, etc.). A sign painter charges $\$1$ per digit to paint house numbers. If he paints the house number on each of the 50 houses, how much will he earn? | 113 | 0 |
27,732 | In the diagram, points \( P_1, P_3, P_5, P_7 \) are on \( BA \) and points \( P_2, P_4, P_6, P_8 \) are on \( BC \) such that \( BP_1 = P_1P_2 = P_2P_3 = P_3P_4 = P_4P_5 = P_5P_6 = P_6P_7 = P_7P_8 \). If \(\angle ABC = 5^\circ\), what is the measure of \(\angle AP_7P_8\)? | 40 | 0.78125 |
27,733 | 20. Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ passes through point $M\left( 1,\frac{3}{2} \right)$, $F_1$ and $F_2$ are the two foci of ellipse $C$, and $\left| MF_1 \right|+\left| MF_2 \right|=4$, $O$ is the center of ellipse $C$.
(1) Find the equation of ellipse $C$;
(2) Suppose $P,Q$ are two different points on ellipse $C$, and $O$ is the centroid of $\Delta MPQ$, find the area of $\Delta MPQ$. | \frac{9}{2} | 0.78125 |
27,734 | Given the ellipse \(3x^{2} + y^{2} = 6\) and the point \(P\) with coordinates \((1, \sqrt{3})\). Find the maximum area of triangle \(PAB\) formed by point \(P\) and two points \(A\) and \(B\) on the ellipse. | \sqrt{3} | 3.90625 |
27,735 | Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed? | 48 | 21.875 |
27,736 | It is known that $F_1$ and $F_2$ are the upper and lower foci of the ellipse $C: \frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$ ($a>b>0$), where $F_1$ is also the focus of the parabola $C_1: x^{2}=4y$. Point $M$ is the intersection of $C_1$ and $C_2$ in the second quadrant, and $|MF_{1}|= \frac {5}{3}$.
(1) Find the equation of the ellipse $C_1$.
(2) Given $A(b,0)$, $B(0,a)$, the line $y=kx$ ($k>0$) intersects $AB$ at point $D$ and intersects the ellipse $C_1$ at points $E$ and $F$. Find the maximum area of the quadrilateral $AEBF$. | 2 \sqrt {6} | 0 |
27,737 | A point $P$ is randomly placed inside the right triangle $\triangle XYZ$ where $X$ is at $(0,6)$, $Y$ is at $(0,0)$, and $Z$ is at $(9,0)$. What is the probability that the area of triangle $PYZ$ is less than half of the area of triangle $XYZ$?
[asy]
size(7cm);
defaultpen(linewidth(0.7));
pair X=(0,6), Y=(0,0), Z=(9,0), P=(2,2);
draw(X--Y--Z--cycle);
draw(Y--P--Z);
label("$X$",X,NW);
label("$Y$",Y,SW);
label("$Z$",Z,E);
label("$P$",P,N);
draw((0,0.6)--(0.6,0.6)--(0.6,0));[/asy] | \frac{3}{4} | 1.5625 |
27,738 | It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$ . | 19 | 21.875 |
27,739 | In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles? | 10 | 10.15625 |
27,740 | The number of unordered pairs of edges of a given rectangular cuboid that determine a plane. | 66 | 5.46875 |
27,741 | A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling.
(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development team;
(2) After a month of learning and discussion, the research team decided to select two employees for an experiment. The method is to first select one employee from the team to conduct the experiment, and after that, select another employee from the remaining team members to conduct the experiment. Calculate the probability that exactly one female employee is among the two selected employees;
(3) After the experiment, the first employee to conduct the experiment obtained the data 68, 70, 71, 72, 74, and the second employee obtained the data 69, 70, 70, 72, 74. Which employee's experiment is more stable? Explain your reasoning. | \frac {1}{2} | 29.6875 |
27,742 | A metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into water that is initially at $80{ }^{\circ} \mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\circ} \mathrm{C}$. Without removing the first bar from the water, another metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into the water. What will the temperature of the water be after the new thermal equilibrium is reached? | 50 | 3.90625 |
27,743 | In the diagram below, trapezoid $ABCD$ with $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$, it is given that $CD = 15$, $\tan C = 1.2$, and $\tan B = 1.8$. What is the length of $BC$? | 2\sqrt{106} | 13.28125 |
27,744 | Let $g$ be a function defined for all real numbers that satisfies $g(3+x) = g(3-x)$ and $g(8+x) = g(8-x)$ for all $x$. If $g(0) = 0$, determine the least number of roots $g(x) = 0$ must have in the interval $-1000 \leq x \leq 1000$. | 402 | 0 |
27,745 | There are 7 students standing in a row. How many different arrangements are there in the following situations?
(1) A and B must stand together;
(2) A is not at the head of the line, and B is not at the end of the line;
(3) There must be exactly one person between A and B. | 1200 | 82.03125 |
27,746 | $A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After $n$ hours, B and C meet, and $n$ hours later, A catches up to D at location $C$. What is the distance between locations $B$ and $C$ in kilometers? | 30 | 17.1875 |
27,747 | In the plane, fixed points A, B, C, D satisfy $|\overrightarrow{DA}| = |\overrightarrow{DB}| = |\overrightarrow{DC}| = 2$, $\overrightarrow{DA} \cdot \overrightarrow{BC} = \overrightarrow{DB} \cdot \overrightarrow{AC} = \overrightarrow{DC} \cdot \overrightarrow{AB} = 0$. For moving points P and M satisfying $|\overrightarrow{AP}| = 1$, $\overrightarrow{PM} = \overrightarrow{MC}$, the maximum value of $|\overrightarrow{BM}|^2$ is \_\_\_\_\_\_. | \frac{49}{4} | 18.75 |
27,748 | Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$ | 245 | 54.6875 |
27,749 | The diagram shows the two squares \( BCDE \) and \( FGHI \) inside the triangle \( ABJ \), where \( E \) is the midpoint of \( AB \) and \( C \) is the midpoint of \( FG \). What is the ratio of the area of the square \( BCDE \) to the area of the triangle \( ABJ \)? | 1/3 | 2.34375 |
27,750 | Consider a square in the coordinate plane with vertices at $(2, 1)$, $(5, 1)$, $(5, 4)$, and $(2, 4)$. A line joining $(2, 3)$ and $(5, 1)$ divides the square shown into two parts. Determine the fraction of the area of the square that is above this line. | \frac{5}{6} | 3.90625 |
27,751 | Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$, respectively. Find the ratio $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$. | \frac{1}{16} | 6.25 |
27,752 | Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?
*Ray Li.* | 52 - 2\sqrt{235} | 3.90625 |
27,753 | A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.). | 27.1 | 3.90625 |
27,754 | What is the smallest possible sum of two consecutive integers whose product is greater than 420? | 43 | 76.5625 |
27,755 | In triangle $ABC$, $AB=AC$, and $D$ is the midpoint of both $\overline{AB}$ and $\overline{CE}$. If $\overline{BC}$ is 14 units long, determine the length of $\overline{CD}$. Express your answer as a decimal to the nearest tenth. | 14.0 | 0 |
27,756 | What is the smallest positive integer with exactly 12 positive integer divisors? | 150 | 0 |
27,757 | Given α ∈ (0,π), β ∈ (-π/2,π/2) satisfies sin(α + π/3) = 1/3, cos(β - π/6) = √6/6, determine sin(α + 2β). | \frac{2\sqrt{10}-2}{9} | 39.84375 |
27,758 | Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest point of the rotated square, to the line on which the bases of the original squares were placed. | \frac{2 + \sqrt{3}}{4} | 3.90625 |
27,759 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 12 + 8\sqrt{2} | 0 |
27,760 | What is the total number of digits used when the first 2500 positive even integers are written? | 9444 | 8.59375 |
27,761 | A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is at the back, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the truck; 5 minutes later, the car catches up with the bus; another $t$ minutes later, the truck catches up with the bus. Find the value of $t$. | 15 | 42.96875 |
27,762 | A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value) | \frac{3}{16} | 1.5625 |
27,763 | Given a line $l$ whose inclination angle $\alpha$ satisfies the condition $\sin \alpha +\cos \alpha = \frac{1}{5}$, determine the slope of $l$. | -\frac{4}{3} | 70.3125 |
27,764 | The perimeter of the triangles that make up rectangle \(ABCD\) is 180 cm. \(BK = KC = AE = ED\), \(AK = KD = 17 \) cm. Find the perimeter of a rectangle, one of whose sides is twice as long as \(AB\), and the other side is equal to \(BC\). | 112 | 25 |
27,765 | Point $P$ is located inside a square $ABCD$ of side length $10$ . Let $O_1$ , $O_2$ , $O_3$ , $O_4$ be the circumcenters of $P AB$ , $P BC$ , $P CD$ , and $P DA$ , respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$ , the second largest of the lengths $O_1O_2$ , $O_2O_3$ , $O_3O_4$ , $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$ . | 5001 | 57.03125 |
27,766 | Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm long. Calculate the time it takes Tia to reach the library in minutes. | 15.24 | 25.78125 |
27,767 | The value of the quadratic polynomial $a(x^3 - x^2 + 3x) + b(2x^2 + x) + x^3 - 5$ when $x = 2$ is $-17$. What is the value of this polynomial when $x = -2$? | -1 | 21.875 |
27,768 | Calculate the value of $\log_{2}9 \cdot \log_{3}5 \cdot \log_{\sqrt{5}}8 = \_\_\_\_\_\_.$ | 12 | 96.875 |
27,769 | Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$ , after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$ . For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$ , the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds? | 1/2 | 86.71875 |
27,770 | Let $x_1$, $x_2$, ..., $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. | 61 | 0 |
27,771 | Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How many pieces of string does Yuvraj obtain? | 16 | 10.9375 |
27,772 | Throw a dice twice to get the numbers $a$ and $b$, respectively. What is the probability that the line $ax-by=0$ intersects with the circle $(x-2)^2+y^2=2$? | \frac{5}{12} | 0 |
27,773 | Kelly is attempting to unlock her electronic device with a four-digit password. She remembers that she only used digits from 1 to 6, each digit possibly being repeated, and that each odd digit must be followed by an even digit, with no specific rule for the sequences following even digits. How many combinations might Kelly need to consider? | 648 | 11.71875 |
27,774 | A bug is on the edge of a ceiling of a circular room with a radius of 65 feet. The bug walks straight across the ceiling to the opposite edge, passing through the center of the circle. It next walks straight to another point on the edge of the circle but not back through the center. If the third part of its journey, back to the original starting point, was 100 feet long, how many total feet did the bug travel over the course of all three parts? | 313 | 0 |
27,775 | Given the function $f(x) = \sin(\omega x + \phi)$ ($\omega > 0$, $0 \leq \phi \leq \pi$) is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\sqrt{4+\pi^2}$.
(1) Find the analytical expression of the function $f(x)$.
(2) If $\sin\alpha + f(\alpha) = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan\alpha}$. | -\frac{5}{9} | 18.75 |
27,776 | The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$ , $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$ . Find $x_7$ . | 3456 | 46.875 |
27,777 | Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$ , and $\sigma(n)$ denotes the sum of divisors of $n$ ). As a hint, you are given that $641|2^{32}+1$ . | 31 | 52.34375 |
27,778 | Let \\(f(x)=3\sin (\omega x+ \frac {\pi}{6})\\), where \\(\omega > 0\\) and \\(x\in(-\infty,+\infty)\\), and the function has a minimum period of \\(\frac {\pi}{2}\\).
\\((1)\\) Find \\(f(0)\\).
\\((2)\\) Find the expression for \\(f(x)\\).
\\((3)\\) Given that \\(f( \frac {\alpha}{4}+ \frac {\pi}{12})= \frac {9}{5}\\), find the value of \\(\sin \alpha\\). | \frac {4}{5} | 12.5 |
27,779 | Homer started peeling a pile of 60 potatoes at a rate of 4 potatoes per minute. Five minutes later, Christen joined him peeling at a rate of 6 potatoes per minute. After working together for 3 minutes, Christen took a 2-minute break, then resumed peeling at a rate of 4 potatoes per minute. Calculate the total number of potatoes Christen peeled. | 23 | 2.34375 |
27,780 | Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(4,3)$, respectively. What is its area? | 37.5\sqrt{3} | 0 |
27,781 | Find all integer values of the parameter \(a\) for which the system
\[
\begin{cases}
x - 2y = y^2 + 2, \\
ax - 2y = y^2 + x^2 + 0.25a^2
\end{cases}
\]
has at least one solution. In the answer, indicate the sum of the found values of the parameter \(a\). | 10 | 1.5625 |
27,782 | Five people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Determine the probability that no two adjacent people stand up. | \frac{11}{32} | 3.125 |
27,783 | If we exchange a 10-dollar bill into dimes and quarters, what is the total number \( n \) of different ways to have two types of coins? | 20 | 0 |
27,784 | Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\div y$ is ______. | -2 | 74.21875 |
27,785 | Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? | 40 \sqrt{1925} | 0 |
27,786 | The sum of three numbers \( a \), \( b \), and \( c \) is 150. If we increase \( a \) by 10, decrease \( b \) by 5, and multiply \( c \) by 7, the three resulting numbers are equal. What is the value of \( b \)? | 77.\overline{3} | 0 |
27,787 | Given that the function $f(x)$ is defined on $\mathbb{R}$ and is not identically zero, and for any real numbers $x$, $y$, it satisfies: $f(2)=2$, $f(xy)=xf(y)+yf(x)$, $a_{n}= \dfrac {f(2^{n})}{2^{n}}(n\in\mathbb{N}^{*})$, $b_{n}= \dfrac {f(2^{n})}{n}(n\in\mathbb{N}^{*})$, consider the following statements:
$(1)f(1)=1$; $(2)f(x)$ is an odd function; $(3)$ The sequence $\{a_{n}\}$ is an arithmetic sequence; $(4)$ The sequence $\{b_{n}\}$ is a geometric sequence.
The correct statements are \_\_\_\_\_\_. | (2)(3)(4) | 7.8125 |
27,788 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 96.09375 |
27,789 | What is the smallest positive value of $x$ such that $x + 8901$ results in a palindrome? | 108 | 5.46875 |
27,790 | Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - 9 a b}{\lambda^3} \). | \frac{3\sqrt{3}}{2} | 0 |
27,791 | Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$ . Let $ r, r_1$ , and $r_2$ be the inradii of triangles $ABC, ABD$ , and $ACD$ , respectively. If $r, r_1$ , and $r_2$ are positive integers and one of them is $5$ , find the largest possible value of $r+r_1+ r_2$ . | 30 | 3.90625 |
27,792 | Given an infinite geometric sequence $\{a_n\}$, the product of its first $n$ terms is $T_n$, and $a_1 > 1$, $a_{2008}a_{2009} > 1$, $(a_{2008} - 1)(a_{2009} - 1) < 0$, determine the maximum positive integer $n$ for which $T_n > 1$. | 4016 | 11.71875 |
27,793 | 2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle? | 666 | 20.3125 |
27,794 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 6 + 4\sqrt{2} | 0 |
27,795 | For any positive integer $x$ , let $f(x)=x^x$ . Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of $n$ .
*Proposed by Luke Robitaille* | 13611 | 78.90625 |
27,796 | China's space station has entered the formal construction phase. The Tianhe core module, Wentian experimental module, and Mengtian experimental module will all dock in 2022, forming a "T" shaped structure. During the construction phase of the Chinese space station, there are 6 astronauts staying in the space station. It is expected that in a certain construction task, 6 astronauts need to work simultaneously in the Tianhe core module, Wentian experimental module, and Mengtian experimental module. Due to space limitations, each module must have at least 1 person and at most 3 people. The total number of different arrangement plans is ______. | 450 | 6.25 |
27,797 | Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and centroid $G$. If $2\sin A\overrightarrow{GA}+\sqrt{3}\sin B\overrightarrow{GB}+3\sin C\cdot \overrightarrow{GC}=\vec{0}$, then $\cos B=$_______. | \dfrac {1}{12} | 10.9375 |
27,798 | Given a set $T = \{a, b, c, d, e, f\}$, determine the number of ways to choose two subsets of $T$ such that their union is $T$ and their intersection contains exactly three elements. | 80 | 2.34375 |
27,799 | In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have? | 20 | 3.90625 |
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