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25,600 | The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \sin t$, where $t \in [0, 12]$. The relationship between the storage capacity of reservoir B (in hundred tons) and time $t$ (in hours) is: $g(t) = 5 - |t - 6|$, where $t \in [0, 12]$. The question is: When do the combined storage capacities of reservoirs A and B reach their maximum value? And what is this maximum value?
(Reference data: $\sin 6 \approx -0.279$). | 6.721 | 39.0625 |
25,601 | In trapezoid \( KLMN \), diagonal \( KM \) is equal to 1 and is also its height. From points \( K \) and \( M \), perpendiculars \( KP \) and \( MQ \) are drawn to sides \( MN \) and \( KL \), respectively. Find \( LM \) if \( KN = MQ \) and \( LM = MP \). | \sqrt{2} | 10.15625 |
25,602 | In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees? | 146 | 2.34375 |
25,603 | Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn. | 55 | 85.15625 |
25,604 | Let $x$ and $y$ be real numbers, where $y > x > 0$, such that
\[
\frac{x}{y} + \frac{y}{x} = 4.
\]
Find the value of
\[
\frac{x + y}{x - y}.
\] | \sqrt{3} | 8.59375 |
25,605 | A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers. | 432 | 8.59375 |
25,606 | Calculate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 + n - 1}{(n+3)!}
\] | \frac{2}{3} | 0.78125 |
25,607 | How many irreducible fractions with numerator 2015 exist that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | 1440 | 50.78125 |
25,608 | A student research group at a school found that the attention index of students during class changes with the listening time. At the beginning of the lecture, students' interest surges; then, their interest remains in a relatively ideal state for a while, after which students' attention begins to disperse. Let $f(x)$ represent the student attention index, which changes with time $x$ (minutes) (the larger $f(x)$, the more concentrated the students' attention). The group discovered the following rule for $f(x)$ as time $x$ changes:
$$f(x)= \begin{cases} 100a^{ \frac {x}{10}}-60, & (0\leqslant x\leqslant 10) \\ 340, & (10 < x\leqslant 20) \\ 640-15x, & (20 < x\leqslant 40)\end{cases}$$
where $a > 0, a\neq 1$.
If the attention index at the 5th minute after class starts is 140, answer the following questions:
(Ⅰ) Find the value of $a$;
(Ⅱ) Compare the concentration of attention at the 5th minute after class starts and 5 minutes before class ends, and explain the reason.
(Ⅲ) During a class, how long can the student's attention index remain at least 140? | \dfrac {85}{3} | 37.5 |
25,609 | Let $ABC$ be a triangle in which $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$ . | 3/4 | 0 |
25,610 | Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes. | 34 | 0.78125 |
25,611 | Let $f_1(x)=x^2-1$ , and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial $f_{2004}$ have? | 2005 | 3.90625 |
25,612 | 22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$ , f satisfies $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$ If $f(2016) = 6102$ and $f(6102) = 2016$ , what is $f(1)$ ?
23. Let $d$ be a randomly chosen divisor of $2016$ . Find the expected value of $$ \frac{d^2}{d^2 + 2016} $$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$ ? | 8117 | 18.75 |
25,613 | For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | 121 | 10.15625 |
25,614 | Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. | 21 | 51.5625 |
25,615 | From 1000 to 9999, a four-digit number is randomly chosen. The probability that all the digits in this number are different and the units digit is an odd number is ( ). | $\frac{56}{225}$ | 0 |
25,616 | Find the positive solution to
\[\sqrt{x + 2 + \sqrt{x + 2 + \dotsb}} = \sqrt{x \sqrt{x \dotsm}}.\] | 1 + \sqrt{3} | 87.5 |
25,617 | A factory produced an original calculator that performs two operations:
(a) the usual addition, denoted by \( + \)
(b) an operation denoted by \( \circledast \).
We know that, for any natural number \( a \), the following hold:
\[
(i) \quad a \circledast a = a \quad \text{ and } \quad (ii) \quad a \circledast 0 = 2a
\]
and, for any four natural numbers \( a, b, c, \) and \( d \), the following holds:
\[
(iii) \quad (a \circledast b) + (c \circledast d) = (a+c) \circledast(b+d)
\]
What are the results of the operations \( (2+3) \circledast (0+3) \) and \( 1024 \circledast 48 \)? | 2000 | 30.46875 |
25,618 | Suppose that $x$ is an integer that satisfies the following congruences:
\[
4 + x \equiv 3^2 \pmod{2^3}, \\
6 + x \equiv 2^3 \pmod{3^3}, \\
8 + x \equiv 7^2 \pmod{5^3}.
\]
What is the remainder when $x$ is divided by $30$? | 17 | 17.96875 |
25,619 | Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\). | \frac{9\pi}{16} | 3.90625 |
25,620 | Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$ ). What part of the area of the triangle is painted over? | \frac{1}{2} | 18.75 |
25,621 | The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes. | 428 | 33.59375 |
25,622 | Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$. | \frac{9}{8} | 3.90625 |
25,623 | For any 2016 complex numbers \( z_1, z_2, \ldots, z_{2016} \), it holds that
\[
\sum_{k=1}^{2016} |z_k|^2 \geq \lambda \min_{1 \leq k \leq 2016} \{ |z_{k+1} - z_k|^2 \},
\]
where \( z_{2017} = z_1 \). Find the maximum value of \( \lambda \). | 504 | 7.8125 |
25,624 | Estimate the population of the island of Thalassa in the year 2050, knowing that its population doubles every 20 years and increases by an additional 500 people every decade thereafter, given that the population in the year 2000 was 250. | 1500 | 5.46875 |
25,625 | $A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards.
A) 600
B) 720
C) 840
D) 960
E) 1080 | 720 | 21.09375 |
25,626 | Given the sequence $\{a\_n\}$, where $a\_n= \sqrt {5n-1}$, $n\in\mathbb{N}^*$, arrange the integer terms of the sequence $\{a\_n\}$ in their original order to form a new sequence $\{b\_n\}$. Find the value of $b_{2015}$. | 5037 | 30.46875 |
25,627 | Given real numbers $x$ and $y$ satisfy the equation $x^2+y^2-4x+1=0$.
(1) Find the maximum and minimum value of $\frac {y}{x}$.
(2) Find the maximum and minimum value of $y-x$.
(3) Find the maximum and minimum value of $x^2+y^2$. | 7-4\sqrt{3} | 54.6875 |
25,628 | Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty? | 126 | 16.40625 |
25,629 | For what smallest natural $k$ is the number \( 2016 \cdot 20162016 \cdot 201620162016 \cdot \ldots \cdot 20162016\ldots2016 \) (with $k$ factors) divisible by \(3^{67}\)? | 34 | 72.65625 |
25,630 | Given that $x$ and $y$ satisfy the equation $(x-1)^{2}+(y+2)^{2}=4$, find the maximum and minimum values of $S=3x-y$. | 5 - 2\sqrt{10} | 85.15625 |
25,631 | Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\frac{1}{2}$ is $\frac{a - b \pi}{c}\left(a, b, c \in \mathbf{Z}_{+}, (a, b, c)=1\right)$. Find the value of $a + b + c$. | 59 | 0.78125 |
25,632 | Given a finite sequence $S=(2, 2x, 2x^2,\ldots ,2x^{200})$ of $n=201$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{200}+a_{201}}{2}\right)$ of $n-1=200$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le 150$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(2, 2x, 2x^2,\ldots ,2x^{200})$. If $A^{150}(S)=(2 \cdot 2^{-75})$, then what is $x$?
A) $1 - \frac{\sqrt{2}}{2}$
B) $2^{3/8} - 1$
C) $\sqrt{2} - 2$
D) $2^{1/5} - 1$ | 2^{3/8} - 1 | 15.625 |
25,633 | An organization has a structure where there is one president, two vice-presidents (VP1 and VP2), and each vice-president supervises two managers. If the organization currently has 12 members, in how many different ways can the leadership (president, vice-presidents, and managers) be chosen? | 554400 | 0 |
25,634 | A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | \frac{3}{128} | 11.71875 |
25,635 | Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 0.78125 |
25,636 | The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a "lucky coin." Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is "Yes," she needs to pay a consultation fee of 20 yuan. If the answer is "No," she needs to pay a consultation fee of 10 yuan. She can also choose not to ask any questions and directly buy some coins. What is the minimum amount of money (in yuan) Feifei needs to pay to guarantee she gets the lucky coin? | 130 | 1.5625 |
25,637 | Square $ABCD$ has sides of length 4. Set $T$ is the set of all line segments that have length 4 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $T$ enclose a region whose area to the nearest hundredth is $m$. Find $100m$. | 343 | 7.8125 |
25,638 | Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.
$(1)$ Find the equation of ellipse $C$ and its eccentricity;
$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why. | \dfrac{8}{5} | 0 |
25,639 | Inside a square $R_1$ with area 81, an equilateral triangle $T_1$ is inscribed such that each vertex of $T_1$ touches one side of $R_1$. Each midpoint of $T_1’s$ sides is connected to form a smaller triangle $T_2$. The process is repeated with $T_2$ to form $T_3$. Find the area of triangle $T_3$. | \frac{81\sqrt{3}}{256} | 0 |
25,640 | In the diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$. | 13:1 | 0 |
25,641 | Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \leqslant a \leqslant 4$ and $a \in R)$ units of the nutrient solution are released, its concentration $y (\text{g}/\text{L})$ changes over time $x (\text{days})$ according to the function $y = af(x)$, where $f(x)=\begin{cases} \frac{4+x}{4-x} & 0\leqslant x\leqslant 2 \\ 5-x & 2\prec x\leqslant 5 \end{cases}$. If the nutrient solution is released multiple times, the concentration at a given moment is the sum of the concentrations released at the corresponding times. According to experience, the nutrient solution is effective only when its concentration is not less than $4(\text{g}/\text{L})$.
(1) If $4$ units of the nutrient solution are released only once, how many days can it be effective?
(2) If $2$ units of the nutrient solution are released first, and then $b$ units are released after $3$ days. In order to keep the nutrient solution continuously effective in the next $2$ days, find the minimum value of $b$. | 24-16\sqrt{2} | 0 |
25,642 | A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral? | 1/2 | 3.90625 |
25,643 | Two rectangles, one $8 \times 10$ and the other $12 \times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. | 65 | 0 |
25,644 | A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \(\frac{10}{3}\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of 1? | 56 | 0 |
25,645 | Let \( T = 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \cdots + 2023 + 2024 - 2025 - 2026 \). What is the residue of \( T \), modulo 2027? | 2026 | 14.84375 |
25,646 | Given that $x$ and $y$ are real numbers, and they satisfy $xy + x + y = 17$, $x^2y + xy^2 = 66$, find the value of $x^4 + x^3y + x^2y^2 + xy^3 + y^4$. | 12499 | 48.4375 |
25,647 | Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces? | 40 | 4.6875 |
25,648 | Let $a^2 = \frac{9}{25}$ and $b^2 = \frac{(3+\sqrt{7})^2}{14}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a-b)^2$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | 22 | 0 |
25,649 | In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.
[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,linewidth(1));
draw((0,0)--(1,1)--(1,2)--(0,0)--cycle,linewidth(1));
dot((1,2));
label("A",(0,0),SW);
label("B",(2,0),SE);
label("C",(1,2),N);
label("D",(1,1),NE);
label("E",(1,0),S);
[/asy] | \frac{1}{2} | 18.75 |
25,650 | How many pairs of positive integers \((x, y)\) are there such that \(x < y\) and \(\frac{x^{2}+y^{2}}{x+y}\) is a divisor of 2835? | 20 | 1.5625 |
25,651 | Let \(x,\) \(y,\) \(z\) be real numbers such that \(9x^2 + 4y^2 + 25z^2 = 1.\) Find the maximum value of
\[8x + 3y + 10z.\] | \sqrt{173} | 1.5625 |
25,652 | A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round. | 11 | 42.96875 |
25,653 | Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an eight-sided die (numbered 1 to 8) is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
A) $\frac{1}{12}$
B) $\frac{1}{8}$
C) $\frac{1}{6}$
D) $\frac{1}{4}$
E) $\frac{1}{3}$ | \frac{1}{6} | 74.21875 |
25,654 | Let the function $f(x)=2\tan \frac{x}{4}\cdot \cos^2 \frac{x}{4}-2\cos^2\left(\frac{x}{4}+\frac{\pi }{12}\right)+1$.
(Ⅰ) Find the smallest positive period and the domain of $f(x)$;
(Ⅱ) Find the intervals of monotonicity and the extremum of $f(x)$ in the interval $[-\pi,0]$; | -\frac{\sqrt{3}}{2} | 7.8125 |
25,655 | Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$ . $X$ , $Y$ , and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$ ). Compute the square of the smallest possible area of $XYZ$ .
*Proposed by Michael Ren* | 7500 | 4.6875 |
25,656 | Given that there are 5 people standing in a row, calculate the number of ways for person A and person B to stand such that there is exactly one person between them. | 36 | 22.65625 |
25,657 | A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in terms of $\pi$. | 9\pi\sqrt{15} | 80.46875 |
25,658 | Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$ . Find the maximum possible value of $n$ . | 504 | 0 |
25,659 | In a flood control emergency, a large oil tank drifting downstream from upstream needs to be exploded by shooting. It is known that there are only $5$ bullets. The first hit can only cause the oil to flow out, and the second hit can cause an explosion. Each shot is independent, and the probability of hitting each time is $\frac{2}{3}$.
$(1)$ Calculate the probability that the oil tank will explode;
$(2)$ If the oil tank explodes or the bullets run out, the shooting will stop. Let $X$ be the number of shots. Calculate the probability that $X$ is not less than $4$. | \frac{7}{27} | 9.375 |
25,660 | 9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are. | 16 | 2.34375 |
25,661 | Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability $1/2$ and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die $5$ times and gets a $1, 2, 3, 4$ and $5$ in some unspecified
order. Compute the probability that his next roll is a $6$ . | 3/26 | 1.5625 |
25,662 | If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of "quasi-geometric progression." Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these $4$ numbers have the property of "quasi-geometric progression" is ____. | \frac{13}{35} | 0 |
25,663 |
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 | 11.71875 |
25,664 | Let the function be $$f(x)=\sin(2\omega x+ \frac {\pi}{3})+ \frac { \sqrt {3}}{2}+a(\omega＞0)$$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $$\frac {\pi}{6}$$.
(1) Find the value of $\omega$;
(2) If the minimum value of $f(x)$ in the interval $$[- \frac {\pi}{3}, \frac {5\pi}{6}]$$ is $$\sqrt {3}$$, find the value of $a$;
(3) If $g(x)=f(x)-a$, what transformations are applied to the graph of $y=\sin x$ ($x\in\mathbb{R}$) to obtain the graph of $g(x)$? Also, write down the axis of symmetry and the center of symmetry for $g(x)$. | \frac { \sqrt {3}+1}{2} | 0 |
25,665 | There are 6 balls of each of the four colors: red, blue, yellow, and green. Each set of 6 balls of the same color is numbered from 1 to 6. If 3 balls with different numbers are randomly selected, and these 3 balls have different colors and their numbers are not consecutive, the number of ways to do this is ______. | 96 | 38.28125 |
25,666 | Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\). | \frac{\ln 5}{2 \ln 2 - 3 \ln 5} | 1.5625 |
25,667 | In a school's mentoring program, several first-grade students can befriend one sixth-grade student, while one sixth-grade student cannot befriend multiple first-grade students. It is known that $\frac{1}{3}$ of the sixth-grade students and $\frac{2}{5}$ of the first-grade students have become friends. What fraction of the total number of students in the first and sixth grades are these friends? | $\frac{4}{11}$ | 0 |
25,668 | Remove all perfect squares from the sequence of positive integers \(1, 2, 3, \cdots\) to get a new sequence, and calculate the 2003rd term of this new sequence. | 2047 | 0 |
25,669 | The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:
$\frac{4}{3}\pi r^3 = 50,000$
Simplifying, we get $r^3 = \frac{50,000 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, we want to find the volume of the top portion of Logan’s model. It is given that this sphere should have a volume of 0.2 liters. Using the same formula, we can solve for the radius:
$\frac{4}{3}\pi r^3 = 0.2$
Simplifying, we get $r^3 = \frac{0.2 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.
Now, since the sphere is a model of the top portion of the water tower, the radius of the model is proportional to the radius of the real tower. Therefore, the ratio of the radius of the model to the radius of the real tower is equal to the ratio of the volume of the model to the volume of the real tower.
$\frac{r_{model}}{r_{real}} = \frac{0.2}{50000}$
We can now equate the two expressions for the radius and solve for the height of the model:
$\frac{r_{model}}{60} = \frac{0.2}{50000}$ | 0.95 | 0.78125 |
25,670 | Find the 20th term in the sequence: $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \ldots, \frac{1}{m+1}, \frac{2}{m+1}, \ldots, \frac{m}{m+1}, \ldots$ | \frac{6}{7} | 8.59375 |
25,671 | The area of square \(ABCD\) is 64. The midpoints of its sides are joined to form the square \(EFGH\). The midpoints of its sides are \(J, K, L,\) and \(M\). The area of the shaded region is: | 24 | 1.5625 |
25,672 | For environmental protection, Wuyang Mineral Water recycles empty bottles. Consumers can exchange 4 empty bottles for 1 bottle of mineral water (if there are fewer than 4 empty bottles, they cannot be exchanged). Huacheng Middle School bought 1999 bottles of Wuyang brand mineral water. If they exchange the empty bottles for new bottles of water as much as possible, then the teachers and students of Huacheng Middle School can drink a total of bottles of mineral water; conversely, if they can drink a total of 3126 bottles of mineral water, then they originally bought bottles of mineral water. | 2345 | 53.125 |
25,673 | One angle of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and a circle can be circumscribed around this trapezoid. | 1:3 | 1.5625 |
25,674 | Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day before. If he is late for work, he changes his mode of transportation the next day. Given all this, how likely is it that Maurice will be late for work on his 467th trip? | 2/3 | 7.8125 |
25,675 | The lighting power increased by
\[ \Delta N = N_{\text {after}} - N_{\text {before}} = 300\, \text{BT} - 240\, \text{BT} = 60\, \text{BT} \] | 60 | 35.15625 |
25,676 | If \(x + \frac{1}{y} = 3\) and \(y + \frac{1}{z} = 3\), what is the value of the product \(xyz\)? | -1 | 0.78125 |
25,677 | Given $\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\sin A= \sqrt {3}a\cos C$.
(Ⅰ) Find angle $C$;
(Ⅱ) If $c= \sqrt {21}$, and $\sin C+\sin (B-A)=5\sin 2A$, find the area of $\triangle ABC$. | \frac {5 \sqrt {3}}{4} | 0 |
25,678 | Four teams, including Quixajuba, are competing in a volleyball tournament where:
- Each team plays against every other team exactly once;
- Any match ends with one team winning;
- In any match, the teams have an equal probability of winning;
- At the end of the tournament, the teams are ranked by the number of victories.
a) Is it possible that, at the end of the tournament, all teams have the same number of victories? Why?
b) What is the probability that the tournament ends with Quixajuba alone in first place?
c) What is the probability that the tournament ends with three teams tied for first place? | \frac{1}{8} | 2.34375 |
25,679 | Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that
- each member in $S$ is a positive integer not exceeding $2002$ ,
- if $a,b\in S$ (not necessarily different), then $ab\not\in S$ .
| 1958 | 7.8125 |
25,680 | There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? | 90 | 0 |
25,681 | Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | \sqrt{3} | 18.75 |
25,682 | Express the quotient $2213_4 \div 13_4$ in base 4. | 53_4 | 0 |
25,683 | A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$? | 0.8 | 0.78125 |
25,684 | Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \). | 2/3 | 0.78125 |
25,685 | Given the set $A=\{(x,y) \,|\, |x| \leq 1, |y| \leq 1, x, y \in \mathbb{R}\}$, and $B=\{(x,y) \,|\, (x-a)^2+(y-b)^2 \leq 1, x, y \in \mathbb{R}, (a,b) \in A\}$, then the area represented by set $B$ is \_\_\_\_\_\_. | 12 + \pi | 0.78125 |
25,686 | What is the least positive integer with exactly $12$ positive factors? | 108 | 0 |
25,687 | Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$. | 26 | 21.09375 |
25,688 | The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. | 36\sqrt{3} | 39.84375 |
25,689 | The probability that a car driving on this road does not stop at point A and the probability that it does not stop at point B and the probability that it does not stop at point C are $\left(1-\frac{25}{60}\right)$, $\left(1-\frac{35}{60}\right)$, and $\left(1-\frac{45}{60}\right)$, respectively. | \frac{35}{192} | 0 |
25,690 | In the right trapezoid \(ABCD\), it is known that \(AB \perp BC\), \(BC \parallel AD\), \(AB = 12\), \(BC = 10\), and \(AD = 6\). Point \(F\) is a movable point on a circle centered at point \(C\) with radius 8, and point \(E\) is a point on \(AB\). When the value of \(DE + EF\) is minimized, what is the length of \(AE\)? | 4.5 | 1.5625 |
25,691 | Given $X \sim N(\mu, \sigma^2)$, $P(\mu-\sigma < X \leq \mu+\sigma) = 0.68$, $P(\mu-2\sigma < X \leq \mu+2\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scored above 120? | 500 | 0.78125 |
25,692 | In the set of all three-digit numbers composed of the digits 0, 1, 2, 3, 4, 5, without any repeating digits, how many such numbers have a digit-sum of 9? | 12 | 0 |
25,693 | Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$. | \frac{3\sqrt{6}}{2} + 1 | 0.78125 |
25,694 | Given that the line $x - 2y + 2k = 0$ encloses a triangle with an area of $1$ together with the two coordinate axes, find the value of the real number $k$. | -1 | 61.71875 |
25,695 | Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits. | 252 | 0.78125 |
25,696 |
Each segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of the squares of which is equal to 1, and the segments carry the products of the numbers at their ends. Then the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the maximum number that could be obtained? | -1 | 0 |
25,697 | The perpendicular to the side $AB$ of the trapezoid $ABCD$, passing through its midpoint $K$, intersects the side $CD$ at point $L$. It is known that the area of quadrilateral $AKLD$ is five times greater than the area of quadrilateral $BKLC$. Given $CL=3$, $DL=15$, and $KC=4$, find the length of segment $KD$. | 20 | 11.71875 |
25,698 | Albert now decides to extend his list to the 2000th digit. He writes down positive integers in increasing order with a first digit of 1, such as $1, 10, 11, 12, \ldots$. Determine the three-digit number formed by the 1998th, 1999th, and 2000th digits. | 141 | 0.78125 |
25,699 | What is ${-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}$? | -199 | 71.875 |
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