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26,600 | Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. The answer can be expressed as $\frac{P\sqrt{Q}+R}{S}$, where $P$, $Q$, $R$, and $S$ are integers, $S$ is positive, and $Q$ is not divisible by the square of any prime. If the greatest common divisor of $P$, $R$, and $S$ is 1, find $P+Q+R+S$. | 23 | 2.34375 |
26,601 | Given the taxi fare of $3.50 for the first 0.5 mile, and an additional charge of $0.30 for each 0.1 mile thereafter, and a $3 tip, calculate the total number of miles that can be ridden with a $15 budget. | 3.333 | 0 |
26,602 | What is the greatest prime factor of $15! + 18!$? | 4897 | 0.78125 |
26,603 | Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$. | -670 | 10.9375 |
26,604 | Between 1000 and 9999, the number of four-digit integers with distinct digits where the absolute difference between the first and last digit is 2. | 840 | 1.5625 |
26,605 | Six six-sided dice are rolled. We are told there are no four-of-a-kind, but there are two different pairs of dice showing the same numbers. These four dice (two pairs) are set aside, and the other two dice are re-rolled. What is the probability that after re-rolling these two dice, at least three of the six dice show the same value? | \frac{2}{3} | 3.90625 |
26,606 | Given the real numbers \( x \) and \( y \) satisfy the equations:
\[ 2^x + 4x + 12 = \log_2{(y-1)^3} + 3y + 12 = 0 \]
find the value of \( x + y \). | -2 | 1.5625 |
26,607 | Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$. | 64 | 33.59375 |
26,608 |
The base of pyramid \( T ABCD \) is an isosceles trapezoid \( ABCD \) with the length of the shorter base \( BC \) equal to \( \sqrt{3} \). The ratio of the areas of the parts of the trapezoid \( ABCD \), divided by the median line, is \( 5:7 \). All the lateral faces of the pyramid \( T ABCD \) are inclined at an angle of \( 30^\circ \) with respect to the base. The plane \( AKN \), where points \( K \) and \( N \) are the midpoints of the edges \( TB \) and \( TC \) respectively, divides the pyramid into two parts. Find the volume of the larger part.
**(16 points)** | 0.875 | 0 |
26,609 | When fitting a set of data with the model $y=ce^{kx}$, in order to find the regression equation, let $z=\ln y$ and transform it to get the linear equation $z=0.3x+4$. Then, the values of $c$ and $k$ are respectively \_\_\_\_\_\_ and \_\_\_\_\_\_. | 0.3 | 35.9375 |
26,610 | Let $c$ and $d$ be real numbers such that
\[c^3 - 18c^2 + 25c - 75 = 0 \quad \text{and} \quad 9d^3 - 72d^2 - 345d + 3060 = 0.\]Compute $c + d.$ | 10 | 14.84375 |
26,611 | Suppose there are $160$ pigeons and $n$ holes. The $1$ st pigeon flies to the $1$ st hole, the $2$ nd pigeon flies to the $4$ th hole, and so on, such that the $i$ th pigeon flies to the $(i^2\text{ mod }n)$ th hole, where $k\text{ mod }n$ is the remainder when $k$ is divided by $n$ . What is minimum $n$ such that there is at most one pigeon per hole?
*Proposed by Christina Yao* | 326 | 64.84375 |
26,612 | Compute the integer $m > 3$ for which
\[\log_{10} (m - 3)! + \log_{10} (m - 1)! + 3 = 2 \log_{10} m!.\] | 10 | 4.6875 |
26,613 | A geometric sequence of positive integers is formed for which the first term is 3 and the fifth term is 375. What is the sixth term of the sequence? | 9375 | 24.21875 |
26,614 | A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | \frac{1}{8} | 6.25 |
26,615 | A train arrives randomly sometime between 3:00 and 4:00 PM and waits for 15 minutes before leaving. If John arrives randomly between 3:15 and 4:15 PM, what is the probability that the train will still be there when John arrives? | \frac{3}{16} | 1.5625 |
26,616 | Four vertices of a cube are given as \(A=(1, 2, 3)\), \(B=(1, 8, 3)\), \(C=(5, 2, 3)\), and \(D=(5, 8, 3)\). Calculate the surface area of the cube. | 96 | 11.71875 |
26,617 | Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 95 | 12.5 |
26,618 | How many 12-digit positive integers have all digits either 1 or 2, and have numbers ending in 12, but have no two consecutive 1's? | 89 | 7.03125 |
26,619 | In quadrilateral $EFGH$, $m\angle F = 100^\circ, m\angle G = 140^\circ$, $EF=6, FG=5,$ and $GH=7$. Calculate the area of $EFGH$. | 26.02 | 0.78125 |
26,620 | In the diagram, \(PRTY\) and \(WRSU\) are squares. Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, the length of \(ST\) is closest to | 5.5 | 0 |
26,621 | Let $M=123456789101112\dots5253$ be the $96$-digit number that is formed by writing integers from $1$ to $53$ in order. What is the remainder when $M$ is divided by $55$? | 53 | 10.9375 |
26,622 | Find the smallest natural number ending with the digit 2, which doubles if this digit is moved to the beginning. | 105263157894736842 | 92.1875 |
26,623 | In the numbers from 100 to 999, calculate how many numbers have digits in strictly increasing or strictly decreasing order. | 204 | 50 |
26,624 | A cheesecake is shaped like a $3 \times 3 \times 3$ cube and is decorated such that only the top and three of the vertical faces are covered with icing. The cake is cut into $27$ smaller cubes, each $1 \times 1 \times 1$ inch. Calculate the number of the smaller cubes that will have icing on exactly two sides. | 12 | 21.09375 |
26,625 | Given a bag with 1 red ball and 2 black balls of the same size, two balls are randomly drawn. Let $\xi$ represent the number of red balls drawn. Calculate $E\xi$ and $D\xi$. | \frac{2}{9} | 41.40625 |
26,626 | In the quadratic equation $3x^{2}-6x-7=0$, the coefficient of the quadratic term is ____ and the constant term is ____. | -7 | 60.15625 |
26,627 | Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$. | 11 | 0 |
26,628 | In a sequence of triangles, each successive triangle has its small triangles numbering as square numbers (1, 4, 9,...). Each triangle's smallest sub-triangles are shaded according to a pascal triangle arrangement. What fraction of the eighth triangle in the sequence will be shaded if colors alternate in levels of the pascal triangle by double layers unshaded followed by double layers shaded, and this pattern starts from the first (smallest) sub-triangle?
A) $\frac{1}{4}$
B) $\frac{1}{8}$
C) $\frac{1}{2}$
D) $\frac{3}{4}$
E) $\frac{1}{16}$ | \frac{1}{4} | 7.8125 |
26,629 | Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the holes caused by the nails are negligible. | 35\sqrt{2} | 0 |
26,630 | If a positive integer is equal to the sum of all its factors (including 1 but excluding the number itself), then this number is called a "perfect number". For example, 28 is a "perfect number" because $1 + 2 + 4 + 7 + 14 = 28$. If the sum of all factors of a positive integer (including 1 but excluding the number itself) is one less than the number, then this number is called an "almost perfect number". For example, 8 is an "almost perfect number" because $1 + 2 + 4 = 7$. The fifth "almost perfect number" in ascending order is . | 32 | 89.0625 |
26,631 | Given an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where $a>b>0$) that passes through the point $E(\sqrt 3, 1)$, with an eccentricity of $\frac{\sqrt{6}}{3}$, and O as the origin of the coordinate system.
(Ⅰ) Find the equation of the ellipse C.
(Ⅱ) If point P is a moving point on ellipse C, and the perpendicular bisector of AP, with A(3, 0), intersects the y-axis at point B, find the minimum value of |OB|. | \sqrt{6} | 68.75 |
26,632 | An organization initially consists of 10 leaders and a certain number of regular members. Each year, all leaders are replaced, and each regular member recruits two new members. After this, 10 new leaders are elected from outside the organization. Initially, there are 30 people in total in the organization. How many people will be in the organization after 10 years? | 1180990 | 0 |
26,633 | 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 $1386$, then what is the sum of the numbers written on the faces? | 38 | 10.15625 |
26,634 | In triangle \( \triangle ABC \), the lengths of sides opposite to angles A, B, C are denoted by a, b, c respectively. If \( c = \sqrt{3} \), \( b = 1 \), and \( B = 30^\circ \),
(1) find angles A and C;
(2) find the area of \( \triangle ABC \). | \frac{\sqrt{3}}{2} | 42.96875 |
26,635 | Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the minimum value of $\frac{1}{2a} + \frac{1}{b}$. | \sqrt{2} + \frac{3}{2} | 8.59375 |
26,636 | Let $a$, $b$, and $c$ be the sides of a triangle with angles $\alpha$, $\beta$, and $\gamma$ opposite them respectively. Suppose $a^2 + b^2 = 9c^2$. Find the value of
\[\frac{\tan \gamma}{\tan \alpha + \tan \beta}.\] | -1 | 0 |
26,637 | The circumradius R of triangle △ABC is $\sqrt{3}$. The sides opposite to angles A, B, and C are a, b, c respectively, and it is given that $\frac{2\sin A-\sin C}{\sin B} = \frac{\cos C}{\cos B}$.
(1) Find the angle B and the side length b.
(2) Find the maximum value of the area $S_{\triangle ABC}$ and the values of a and c when this maximum area is achieved, and determine the shape of the triangle at that time. | \frac{9\sqrt{3}}{4} | 12.5 |
26,638 | Let point $P$ be a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\triangle PF_1F_2$. If $S_{\triangle IF_1P} + S_{\triangle IPF_2} = 2S_{\triangle IF_1F_2}$, determine the eccentricity of the ellipse. | \frac{1}{2} | 57.03125 |
26,639 | Simplify: $$\sqrt[3]{9112500}$$ | 209 | 21.09375 |
26,640 | In a store, there are four types of nuts: hazelnuts, almonds, cashews, and pistachios. Stepan wants to buy 1 kilogram of nuts of one type and 1 kilogram of nuts of another type. He has calculated the cost of such a purchase depending on which two types of nuts he chooses. Five of Stepan's six possible purchases would cost 1900, 2070, 2110, 2330, and 2500 rubles. How many rubles is the cost of the sixth possible purchase? | 2290 | 0.78125 |
26,641 | Given a circle C that passes through points A(1, 4) and B(3, -2), and the distance from the center of the circle C to line AB is $\sqrt{10}$, find the equation of circle C. | 20 | 0.78125 |
26,642 | We randomly throw a 10-filér coin onto a $40 \times 40$ cm chessboard. What is the probability that the coin:
a) will be entirely inside one square?
b) will partially cover one and only one square edge?
c) will partially cover two square edges but not a vertex?
d) will cover a vertex? (The diameter of the 10-filér coin is $19 \mathrm{~mm}$). | 0.1134 | 0 |
26,643 | Let $n \in \mathbb{N}^*$, $a_n$ be the sum of the coefficients of the expanded form of $(x+4)^n - (x+1)^n$, $c=\frac{3}{4}t-2$, $t \in \mathbb{R}$, and $b_n = \left[\frac{a_1}{5}\right] + \left[\frac{2a_2}{5^2}\right] + ... + \left[\frac{na_n}{5^n}\right]$ (where $[x]$ represents the largest integer not greater than the real number $x$). Find the minimum value of $(n-t)^2 + (b_n + c)^2$. | \frac{4}{25} | 0 |
26,644 | Given the function $f(x)=\sin^2x+\sin x\cos x$, when $x=\theta$ the function $f(x)$ attains its minimum value, find the value of $\dfrac{\sin 2\theta+2\cos \theta}{\sin 2\theta -2\cos 2\theta}$. | -\dfrac{1}{3} | 1.5625 |
26,645 | Inside a square with side length 12, two congruent equilateral triangles are drawn such that each has one vertex touching two adjacent vertices of the square and they share one side. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles? | 12 - 4\sqrt{3} | 0 |
26,646 | How many positive four-digit integers of the form $\_\_35$ are divisible by 35? | 13 | 77.34375 |
26,647 | Given the function $$f(x)=\cos\omega x\cdot \sin(\omega x- \frac {\pi}{3})+ \sqrt {3}\cos^{2}\omega x- \frac { \sqrt {3}}{4}(\omega＞0，x\in\mathbb{R})$$, and the distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $$\frac {\pi}{4}$$.
(Ⅰ) Find the value of $\omega$ and the equation of the axis of symmetry for $f(x)$;
(Ⅱ) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $$f(A)= \frac { \sqrt {3}}{4}, \sin C= \frac {1}{3}, a= \sqrt {3}$$, find the value of $b$. | \frac {3+2 \sqrt {6}}{3} | 0 |
26,648 | A four-dimensional rectangular hyper-box has side lengths $W$, $X$, $Y$, and $Z$. It has "faces" (three-dimensional volumes) whose measures are $60$, $80$, $120$, $60$, $80$, $120$ cubic units. What is $W$ + $X$ + $Y$ + $Z$?
**A)** 200
**B)** 250
**C)** 300
**D)** 318.5
**E)** 400 | 318.5 | 1.5625 |
26,649 | In the sequence of positive integers \(1, 2, 3, \ldots\), all perfect squares are deleted. The 2003rd term of this new sequence is ____ . | 2047 | 0 |
26,650 | Trapezoid $EFGH$ has sides $EF=100$, $FG=55$, $GH=23$, and $HE=75$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the length of $EQ$ if it is expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. | \frac{750}{13} | 0 |
26,651 | Given a triangular pyramid where two of the three lateral faces are isosceles right triangles and the third face is an equilateral triangle with a side length of 1, calculate the volume of this triangular pyramid. | \frac{\sqrt{3}}{12} | 1.5625 |
26,652 | The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$. | 555 | 0 |
26,653 | From the $8$ vertices of a cube, choose any $4$ vertices, the probability that these $4$ points lie in the same plane is ______. | \frac{6}{35} | 25 |
26,654 | A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \_\_\_\_\_\_. | 72 | 0.78125 |
26,655 | The calculator's keyboard has digits from 0 to 9 and symbols of two operations. Initially, the display shows the number 0. Any keys can be pressed. The calculator performs operations in the sequence of key presses. If an operation symbol is pressed several times in a row, the calculator will remember only the last press. The absent-minded Scientist pressed very many buttons in a random sequence. Find the approximate probability that the result of the resulting sequence of operations is an odd number. | 1/3 | 0 |
26,656 | Let $p$, $q$, and $r$ be the roots of the polynomial $x^3 - x - 1 = 0$. Find the value of $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$. | \frac{11}{7} | 0 |
26,657 | Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^3}{9^3 - 1} + \frac{3^4}{9^4 - 1} + \cdots.$$ | \frac{1}{2} | 25.78125 |
26,658 |
In the trapezoid \( ABCD \) with bases \( AD \) and \( BC \), the side \( AB \) is equal to 2. The angle bisector of \( \angle BAD \) intersects the line \( BC \) at point \( E \). A circle is inscribed in triangle \( ABE \), touching side \( AB \) at point \( M \) and side \( BE \) at point \( H \). Given that \( MH = 1 \), find the angle \( \angle BAD \). | 120 | 5.46875 |
26,659 | Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the minimum value of $n$ .
*Leng Gangsong* | 11 | 5.46875 |
26,660 | Given that point $P$ is a moving point on the parabola $x^{2}=2y$, and the focus is $F$. If the fixed point is $M(1,2)$, calculate the minimum value of $(|PM|+|PF|)$ when point $P$ moves on the parabola. | \frac{5}{2} | 16.40625 |
26,661 | For a pyramid S-ABCD, each vertex is colored with one color, and the two ends of the same edge are colored differently. If there are exactly 5 colors available, calculate the number of different coloring methods. | 420 | 3.90625 |
26,662 | Let's modify the problem slightly. Sara writes down four integers $a > b > c > d$ whose sum is $52$. The pairwise positive differences of these numbers are $2, 3, 5, 6, 8,$ and $11$. What is the sum of the possible values for $a$? | 19 | 8.59375 |
26,663 | Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$ , respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt3 PS$ and the area of $PQRS$ is $28\sqrt3$ , what is the length of $PC$ ? | 2\sqrt{7} | 16.40625 |
26,664 | Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}}\right\}$. | -\frac{4}{35} | 1.5625 |
26,665 | Write the process of using the Horner's algorithm to find the value of the function $\_(f)\_()=1+\_x+0.5x^2+0.16667x^3+0.04167x^4+0.00833x^5$ at $x=-0.2$. | 0.81873 | 0 |
26,666 | <u>Set 1</u>**p1.** Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?**p2.** Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?**p3.** Twain is trying to crack a $4$ -digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules?
PS. You should use hide for answers. | 1715 | 11.71875 |
26,667 | Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $50$. Given that $q$ is an even number, determine the number of values of $n$ for which $q - r$ is divisible by $7$. | 7200 | 0.78125 |
26,668 | If a non-negative integer \( m \) and the sum of its digits are both multiples of 6, then \( m \) is called a "Liuhe number." Find the number of Liuhe numbers less than 2012. | 168 | 99.21875 |
26,669 | Among the three-digit numbers composed of the digits $0$ to $9$, the number of numbers where the digits are arranged in strictly increasing or strictly decreasing order, calculate the total. | 204 | 29.6875 |
26,670 | Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits? | 360 | 0.78125 |
26,671 | Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the perimeter of $\triangle ADE$. | 13 | 14.84375 |
26,672 | In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations for line l and curve C respectively.
2. Point A is a moving point on curve C in the first quadrant, and point B is a moving point on line l in the second quadrant. If ∠AOB=$$\frac {π}{4}$$, find the maximum value of $$\frac {|OA|}{|OB|}$$. | \frac { \sqrt {2}}{2} | 0 |
26,673 | Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 74 | 0 |
26,674 | If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0 |
26,675 | Let $AB$ be a segment of length $2$ . The locus of points $P$ such that the $P$ -median of triangle $ABP$ and its reflection over the $P$ -angle bisector of triangle $ABP$ are perpendicular determines some region $R$ . Find the area of $R$ . | 2\pi | 12.5 |
26,676 | In the book "The Nine Chapters on the Mathematical Art," a right square pyramid, whose base is a rectangle and has a side edge perpendicular to the base, is called a "yangma." Given a right square pyramid $M-ABCD$, where side edge $MA$ is perpendicular to the base $ABCD$, and $MA = BC = AB = 2$, calculate the sum of the surface areas of the inscribed and circumscribed spheres. | 36\pi - 16\sqrt{2}\pi | 0 |
26,677 | A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} | 1.5625 |
26,678 | If
\[ 1 \cdot 1995 + 2 \cdot 1994 + 3 \cdot 1993 + \dots + 1994 \cdot 2 + 1995 \cdot 1 = 1995 \cdot 997 \cdot y, \]
compute the integer \( y \). | 665 | 14.84375 |
26,679 | Through points \(A(0, 14)\) and \(B(0, 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).
What is the value of \(\frac{AL - AK}{BN - BM}\)? | 3.5 | 3.125 |
26,680 | In a square, points $R$ and $S$ are midpoints of two adjacent sides. A line segment is drawn from the bottom left vertex to point $S$, and another from the top right vertex to point $R$. What fraction of the interior of the square is shaded?
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--gray,linewidth(1));
filldraw((0,1)--(1,2)--(2,1)--(1,0)--(0,1)--cycle,white,linewidth(1));
label("R",(0,1),W);
label("S",(1,2),N);
[/asy] | \frac{3}{4} | 7.03125 |
26,681 | Let $2005 = c_1 \cdot 3^{a_1} + c_2 \cdot 3^{a_2} + \ldots + c_n \cdot 3^{a_n}$, where $n$ is a positive integer, $a_1, a_2, \ldots, a_n$ are distinct natural numbers (including 0, with the convention that $3^0 = 1$), and each of $c_1, c_2, \ldots, c_n$ is equal to 1 or -1. Find the sum $a_1 + a_2 + \ldots + a_n$. | 22 | 12.5 |
26,682 | There are two boxes, A and B, each containing four cards labeled with the numbers 1, 2, 3, and 4. One card is drawn from each box, and each card is equally likely to be chosen;
(I) Find the probability that the product of the numbers on the two cards drawn is divisible by 3;
(II) Suppose that Xiao Wang and Xiao Li draw two cards, and the person whose sum of the numbers on the two cards is greater wins. If Xiao Wang goes first and draws cards numbered 3 and 4, and the cards drawn by Xiao Wang are not returned to the boxes, Xiao Li draws next; find the probability that Xiao Wang wins. | \frac{8}{9} | 3.90625 |
26,683 | Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ . | 329 | 1.5625 |
26,684 | Given a triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $A= \frac{\pi}{3}$, and $\frac{\sqrt{3}}{2} - \sin(B-C) = \sin 2B$, find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 28.125 |
26,685 | In the diagram, \(PR, PS, QS, QT,\) and \(RT\) are straight line segments. \(QT\) intersects \(PR\) and \(PS\) at \(U\) and \(V\), respectively. If \(PU = PV\), \(\angle UPV = 24^\circ\), \(\angle PSQ = x^\circ\), and \(\angle TQS = y^\circ\), what is the value of \(x + y\)? | 78 | 10.15625 |
26,686 | Given 6 parking spaces in a row and 3 cars that need to be parked such that no two cars are next to each other, calculate the number of different parking methods. | 24 | 89.0625 |
26,687 | Distribute 4 college students to three factories A, B, and C for internship activities. Factory A can only arrange for 1 college student, the other factories must arrange for at least 1 student each, and student A cannot be assigned to factory C. The number of different distribution schemes is ______. | 12 | 8.59375 |
26,688 | Roll three dice once each, and let event A be "the three numbers are all different," and event B be "at least one 1 is rolled." Then the conditional probabilities P(A|B) and P(B|A) are respectively ( ). | \frac{1}{2} | 21.09375 |
26,689 | Given vectors $\overrightarrow {OA} = (1, -2)$, $\overrightarrow {OB} = (4, -1)$, $\overrightarrow {OC} = (m, m+1)$.
(1) If $\overrightarrow {AB} \parallel \overrightarrow {OC}$, find the value of the real number $m$;
(2) If $\triangle ABC$ is a right-angled triangle, find the value of the real number $m$. | \frac{5}{2} | 42.1875 |
26,690 | The sum of an infinite geometric series is $81$ times the series that results if the first five terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{3} | 67.1875 |
26,691 | The lengths of the six edges of a tetrahedron \(ABCD\) are 7, 13, 18, 27, 36, and 41, respectively. If \(AB = 41\), what is the length of \(CD\)? | 13 | 73.4375 |
26,692 | Let $S$ be the set of all non-zero real numbers. Define a function $f : S \to S$ such that for all $x, y \in S$ with $x + y \neq 0$, the following holds:
\[f(x) + f(y) = f\left(\frac{xy}{x+y}f(x+y)\right).\]
Determine the number of possible values of $f(3)$, denoted as $n$, and the sum of all possible values of $f(3)$, denoted as $s$. Finally, compute $n \times s$. | \frac{1}{3} | 15.625 |
26,693 |
Points \( A \) and \( B \) are located on a straight highway running from west to east. Point \( B \) is 9 km east of \( A \). A car leaves point \( A \) heading east at a speed of 40 km/h. At the same time, a motorcycle leaves point \( B \) in the same direction with a constant acceleration of 32 km/h\(^2\). Determine the maximum distance between the car and the motorcycle during the first two hours of their journey. | 25 | 3.90625 |
26,694 | Fill in the blanks with numbers $1 \sim 3$, so that each row and each column contains exactly one number appearing twice. The numbers outside the table indicate how many numbers are visible from that direction. A number can block equal or smaller numbers. What is the four-digit number $\overline{\mathrm{ABCD}}$? | 2213 | 0 |
26,695 | Let $P$ be a point not on line $XZ$ and $Q$ a point on line $XZ$ such that $PQ \perp XZ.$ Meanwhile, $R$ is a point on line $PZ$ such that $SR \perp PZ.$ If $SR = 5,$ $PQ = 6,$ and $XZ = 7,$ then what is the length of $PZ?$ | 8.4 | 5.46875 |
26,696 | From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows:
75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, 75, 80, 80. The number of students in this school who spend more than or equal to one and a half hours on homework outside of class is $\boxed{\text{\_\_\_\_\_\_\_\_}}$. | 630 | 77.34375 |
26,697 | Compute the value of $\left(81\right)^{0.25} \cdot \left(81\right)^{0.2}$. | 3 \cdot \sqrt[5]{3^4} | 0 |
26,698 | Given that $F(1,0)$ is the focus of the ellipse $\frac{x^2}{9} + \frac{y^2}{m} = 1$, $P$ is a moving point on the ellipse, and $A(1,1)$, find the minimum value of $|PA| + |PF|$. | 6 - \sqrt{5} | 41.40625 |
26,699 | Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step. | 10 | 14.0625 |
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