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26,500 | Given the following conditions:①$asinB=bsin({A+\frac{π}{3}})$; ②$S=\frac{{\sqrt{3}}}{2}\overrightarrow{BA}•\overrightarrow{CA}$; ③$c\tan A=\left(2b-c\right)\tan C$. Choose one of the three conditions and fill in the blank below, then answer the following questions.<br/>In $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, the area of $\triangle ABC$ is $S$, and it satisfies _____.<br/>$(1)$ Find the value of angle $A$;<br/>$(2)$ Given that the area of $\triangle ABC$ is $2\sqrt{3}$, point $D$ lies on side $BC$ such that $BD=2DC$, find the minimum value of $AD$. | \frac{4\sqrt{3}}{3} | 7.03125 |
26,501 | Let n be the smallest positive integer such that n is divisible by 20, n^2 is a perfect square, and n^3 is a perfect fifth power. Find the value of n. | 3200000 | 24.21875 |
26,502 | A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$ . Therefore, each square has exactly $8$ neighbors)
What is the maximum possible number of colored squares if $:$ $a) k=6$ $b)k=1$ | 300 | 4.6875 |
26,503 | In the diagram, point $D$ is one-third of the way from $A$ to $B$, and point $E$ is the midpoint of $\overline{BC}$. Given $A(0,8)$, $B(0,0)$, and $C(10,0)$, find the sum of the slope and $y$-intercept of the line passing through points $C$ and $D$. | \frac{12}{5} | 40.625 |
26,504 | The two banners below:
中华少年 杯赛联谊 切磋勾股
炎黄子孙 惠州弘志 振兴中华
Each character represents a non-zero natural number less than 25. Different characters represent different numbers, and the same characters represent the same number. It is known that the average value of these 34 characters is 12. What is the maximum possible sum of the natural numbers represented by "中华" (China)? | 46 | 1.5625 |
26,505 | Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w}$, let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}$, and let $\mathbf{r}$ be the projection of $\mathbf{w}$ onto $\mathbf{p}$. If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4}$, find $\frac{\|\mathbf{r}\|}{\|\mathbf{w}\|}$. | \frac{3}{4} | 10.9375 |
26,506 | In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$ .
 | 1 + \sqrt{2} | 17.1875 |
26,507 | Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$ , $OG = 1$ and $OG \parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.) | 12 | 0.78125 |
26,508 | In a circle with radius 7, two intersecting chords $PQ$ and $RS$ are given such that $PQ=10$ and $RS$ bisects $PQ$ at point $T$. It is also given that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the sine of the central angle subtended by the minor arc $PR$ and express it as a fraction $\frac{a}{b}$ in lowest terms. Find the product $ab$. | 15 | 0 |
26,509 | Given $\sqrt{18} < \sqrt{n^2}$ and $\sqrt{n^2} < \sqrt{200}$, determine the number of integers n. | 10 | 21.09375 |
26,510 | Given $-π < x < 0$, $\sin x + \cos x = \frac{1}{5}$,
(1) Find the value of $\sin x - \cos x$;
(2) Find the value of $\frac{3\sin^2 \frac{x}{2} - 2\sin \frac{x}{2}\cos \frac{x}{2} + \cos^2 \frac{x}{2}}{\tan x + \frac{1}{\tan x}}$. | -\frac{132}{125} | 0.78125 |
26,511 | In a rectangular parallelepiped $\mathrm{ABCDA}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with $\mathrm{AB}=1$ cm, $\mathrm{AD}=2$, and $\mathrm{AA}_{1}=1$, find the minimum area of triangle $\mathrm{PA}_{1} \mathrm{C}$, where vertex $\mathrm{P}$ lies on the line $\mathrm{AB}_{1}$. | \frac{\sqrt{3}}{2} | 0.78125 |
26,512 | At a conference of $40$ people, there are $25$ people who each know each other, and among them, $5$ people do not know $3$ other specific individuals in their group. The remaining $15$ people do not know anyone at the conference. People who know each other hug, and people who do not know each other shake hands. Determine the total number of handshakes that occur within this group. | 495 | 4.6875 |
26,513 | Given that $\theta$ is an angle in the fourth quadrant, and $\sin\theta + \cos\theta = \frac{1}{5}$, find:
(1) $\sin\theta - \cos\theta$;
(2) $\tan\theta$. | -\frac{3}{4} | 96.09375 |
26,514 | Six teams play in a soccer tournament where each team faces every other team exactly once. Each match results in a win or a loss, with no ties allowed. Winners receive one point; losers receive none. In the first match, Team $A$ defeats Team $B$. Assuming each team has an equal chance to win each match, and all match outcomes are independent, find the probability that Team $A$ scores more points than Team $B$ throughout the tournament. Let this probability be expressed as a fraction $m/n$ in its simplest form. Find $m+n$. | 419 | 0.78125 |
26,515 | Find the number of cubic centimeters in the volume of the cylinder formed by rotating a rectangle with side lengths 8 cm and 16 cm about its longer side. Express your answer in terms of \(\pi\). | 256\pi | 0.78125 |
26,516 | each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P. | 1009 | 10.9375 |
26,517 | The equation \( \sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 6x = \frac{81}{32} \) can be reduced to the equivalent equation
\[ \cos ax \cos bx \cos cx = 0, \] for some positive integers \( a, \) \( b, \) and \( c. \) Find \( a + b + c. \) | 14 | 50.78125 |
26,518 | In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\). | 15 | 0.78125 |
26,519 | Consider a quadrilateral ABCD inscribed in a circle with radius 300 meters, where AB = BC = AD = 300 meters, and CD being the side of unknown length. Determine the length of side CD. | 300 | 20.3125 |
26,520 | What is the least six-digit positive integer which is congruent to 7 (mod 17)? | 100,008 | 0 |
26,521 | Given that $0 < α < \frac{π}{2}$ and $\frac{π}{2} < β < π$, with $\cos(α + \frac{π}{4}) = \frac{1}{3}$ and $\cos(\frac{π}{4} - \frac{β}{2}) = \frac{\sqrt{3}}{3}$,
1. Find the value of $\cos β$;
2. Find the value of $\cos(2α + β)$. | -1 | 0 |
26,522 | How many multiples of 15 are between 25 and 225? | 14 | 79.6875 |
26,523 | A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ? | 4:26.8 | 0 |
26,524 | Find the radius of the circle inscribed in triangle $PQR$ if $PQ = 26$, $PR=10$, and $QR=18$. Express your answer in simplest radical form. | \sqrt{17} | 0 |
26,525 | Given that $\sqrt {3}\sin x+\cos x= \frac {2}{3}$, find the value of $\tan (x+ \frac {7\pi}{6})$. | \frac{\sqrt{2}}{4} | 31.25 |
26,526 | Given that a website publishes pictures of five celebrities along with five photos of the celebrities when they were teenagers, but only three of the teen photos are correctly labeled, and two are unlabeled, determine the probability that a visitor guessing at random will match both celebrities with the correct unlabeled teen photos. | \frac{1}{20} | 44.53125 |
26,527 | Compute
\[
\sum_{n=1}^{\infty} \frac{4n + 1}{(4n - 1)^3 (4n + 3)^3}.
\] | \frac{1}{432} | 0 |
26,528 | Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$ . In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play? | 999 | 5.46875 |
26,529 | Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 1.5625 |
26,530 | A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$ . How many bricks remain in the end? | 240 | 97.65625 |
26,531 | The sum of the first n terms of the sequence {a_n} is S_n = n^2 + n + 1, and b_n = (-1)^n a_n (n ∈ N^*). Determine the sum of the first 50 terms of the sequence {b_n}. | 49 | 32.8125 |
26,532 | In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by Factory A? | 0.665 | 6.25 |
26,533 | Let the base of the rectangular prism $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a rhombus with an area of $2 \sqrt{3}$ and $\angle ABC = 60^\circ$. Points $E$ and $F$ lie on edges $CC'$ and $BB'$, respectively, such that $EC = BC = 2FB$. What is the volume of the pyramid $A-BCFE$? | $\sqrt{3}$ | 0 |
26,534 | Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A, B, C \) such that \( AB = BC \). Perpendiculars \( AD, BE, CF \) are drawn from points \( A, B, \) and \( C \) to line \( m \) respectively, with feet of perpendiculars at \( D, E, \) and \( F \). Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \). | \sqrt{6} | 0 |
26,535 | Given a circle $C$ passes through points $A(-1,0)$ and $B(3,0)$, and the center of the circle is on the line $x-y=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ If point $P(x,y)$ is any point on circle $C$, find the maximum and minimum distance from point $P$ to the line $x+2y+4=0$. | \frac{2}{5}\sqrt{5} | 0 |
26,536 | Right triangle $ABC$ has its right angle at $A$ . A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$ . Let $D$ be the point where the semicircle is tangent to $BC$ . If $AD=4$ and $CO=5$ , find $\cos\angle{ABC}$ .
[asy]
import olympiad;
pair A, B, C, D, O;
A = (1,0);
B = origin;
C = (1,1);
O = incenter(C, B, (1,-1));
draw(A--B--C--cycle);
dot(O);
draw(arc(O, 0.41421356237,0,180));
D = O+0.41421356237*dir(135);
label(" $A$ ",A,SE);
label(" $B$ ",B,SW);
label(" $C$ ",C,NE);
label(" $D$ ",D,NW);
label(" $O$ ",O,S);
[/asy] | $\frac{4}{5}$ | 0 |
26,537 | A soccer ball rolls at $4 \mathrm{~m} / \mathrm{s}$ towards Marcos in a direct line from Michael. The ball is $15 \mathrm{~m}$ ahead of Michael who is chasing it at $9 \mathrm{~m} / \mathrm{s}$. Marcos is $30 \mathrm{~m}$ away from the ball and is running towards it at $8 \mathrm{~m} / \mathrm{s}$. Calculate the distance between Michael and Marcos when the ball is touched for the first time by one of them. | 2.5 | 3.125 |
26,538 | Four carpenters were hired by a guest to build a yard. The first carpenter said: "If only I alone were to build the yard, I would complete it in one year." The second carpenter said: "If only I alone were to build the yard, I would complete it in two years." The third carpenter said: "If only I alone were to build the yard, I would complete it in three years." And the fourth carpenter said: "If only I alone were to build the yard, I would complete it in four years." All four carpenters started building the yard together. For how long will they work together to complete the yard? | 175.2 | 0.78125 |
26,539 | Given that $\sum_{k=1}^{40}\sin 4k=\tan \frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\frac{p}{q}<90,$ find $p+q.$ | 85 | 0 |
26,540 | What is the largest $n$ for which the numbers $1,2, \ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$? | 11 | 50 |
26,541 | Three people are sitting in a row of eight seats. If there must be empty seats on both sides of each person, then the number of different seating arrangements is. | 24 | 22.65625 |
26,542 | Mr. Morgan G. Bloomgarten wants to distribute 1,000,000 dollars among his friends. He has two specific rules for distributing the money:
1. Each gift must be either 1 dollar or a power of 7 (7, 49, 343, 2401, etc.).
2. No more than six people can receive the same amount.
How can he distribute the 1,000,000 dollars under these conditions? | 1,000,000 | 1.5625 |
26,543 | The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$ . What is the angle between the diagonals $D_1D_3$ and $D_2D_5$ ? | 90 | 0 |
26,544 | Find the number of six-digit palindromes. | 900 | 93.75 |
26,545 | Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\triangle ABC$, in cm $^2$, is | 2535 | 4.6875 |
26,546 | A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price of each shirt is reduced by $1, the store can sell an additional 2 shirts per day on average. If the store makes an average daily profit of $2000, what should be the selling price of each shirt? | 120 | 14.0625 |
26,547 | Consider the following data from a new season graph, showing the number of home runs hit in April by the top hitters in the baseball league:
- 5 players hit 6 home runs each.
- 6 players hit 8 home runs each.
- 4 players hit 10 home runs each.
Calculate the mean number of home runs hit by these players. | \frac{118}{15} | 3.90625 |
26,548 | $12 \cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\cos ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}=$ | \frac{3}{2} | 0 |
26,549 | In triangle $PQR$, the sides $PQ$, $QR$, and $RP$ measure 17, 15, and 8 units, respectively. Let $J$ be the incenter of triangle $PQR$. The incircle of triangle $PQR$ touches side $QR$, $RP$, and $PQ$ at points $K$, $L$, and $M$, respectively. Determine the length of $PJ$. | \sqrt{34} | 22.65625 |
26,550 | A cross, consisting of two identical large squares and two identical small squares, was placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is 810 cm². | 36 | 2.34375 |
26,551 | Six small circles, each of radius 4 units, are tangent to a large circle. Each small circle is also tangent to its two neighboring small circles. Additionally, all small circles are tangent to a horizontal line that bisects the large circle. What is the diameter of the large circle in units? | 20 | 0 |
26,552 | If \( a^{2} = 1000 \times 1001 \times 1002 \times 1003 + 1 \), find the value of \( a \). | 1002001 | 7.03125 |
26,553 | Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$.
Point $X$ is the midpoint of $AD$ and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find $p+q$ if the ratio is $p:q$. | 12 | 3.125 |
26,554 | How many different lines pass through at least two points in this 4-by-4 grid of lattice points? | 20 | 11.71875 |
26,555 | \[\frac{\tan 96^{\circ} - \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)}{1 + \tan 96^{\circ} \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)} =\] | \frac{\sqrt{3}}{3} | 1.5625 |
26,556 | Let \( X = \{1, 2, \ldots, 2001\} \). Find the smallest positive integer \( m \) such that in any \( m \)-element subset \( W \) of \( X \), there exist \( u, v \in W \) (where \( u \) and \( v \) are allowed to be the same) such that \( u + v \) is a power of 2. | 1000 | 0 |
26,557 | Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers with safe vegetables, a rural cooperative invests 2 million yuan each year to build two pollution-free vegetable greenhouses, A and B. Each greenhouse requires an investment of at least 200,000 yuan. Greenhouse A grows tomatoes, and Greenhouse B grows cucumbers. Based on past gardening experience, it has been found that the annual income $P$ from growing tomatoes and the annual income $Q$ from growing cucumbers with an investment of $a$ (unit: 10,000 yuan) satisfy $P=80+4\sqrt{2a}, Q=\frac{1}{4}a+120$. Let the investment in Greenhouse A be $x$ (unit: 10,000 yuan), and the total annual income from the two greenhouses be $f(x)$ (unit: 10,000 yuan).
$(I)$ Calculate the value of $f(50)$;
$(II)$ How should the investments in Greenhouses A and B be arranged to maximize the total income $f(x)$? | 282 | 17.1875 |
26,558 | Given the function $f(x)=\sin (3x+ \frac {\pi}{3})+\cos (3x+ \frac {\pi}{6})+m\sin 3x$ ($m\in\mathbb{R}$), and $f( \frac {17\pi}{18})=-1$
$(1)$ Find the value of $m$;
$(2)$ In triangle $ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, if $f( \frac {B}{3})= \sqrt {3}$, and $a^{2}=2c^{2}+b^{2}$, find $\tan A$． | -3 \sqrt {3} | 0 |
26,559 | Given that the volume of a regular triangular pyramid $P-ABC$ is $\frac{1}{12}$, the center of its circumscribed sphere is $O$, and it satisfies $\vec{OA} + \vec{OB} + \vec{OC} = \vec{0}$, find the radius of the circumscribed sphere of the regular triangular pyramid $P-ABC$. | \frac{\sqrt{3}}{3} | 2.34375 |
26,560 | In a particular sequence, the first term is $a_1=1007$ and the second term is $a_2=1008$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{500}$. | 1339 | 0.78125 |
26,561 | A radio system consisting of 1000 components (each with a failure rate of $\lambda_{i} = 10^{-6}$ failures/hour) has been tested and accepted by the customer. Determine the probability of the system operating without failure over the interval $t_{1} < (t = t_{1} + \Delta t) < t_{2}$, where $\Delta t = 1000$ hours. | 0.367879 | 0 |
26,562 | The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $20$ with center $O$ on line $\overline{AP}$ is drawn such that it is tangent to $\overline{AM}$ and $\overline{PM}$. Let $\overline{AM} = 2\overline{PM}$. Find the length of $OP$. | 20 | 3.125 |
26,563 | Real numbers \( x, y, z, w \) satisfy \( x + y + z + w = 1 \). Determine the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \). | \frac{3}{2} | 50 |
26,564 | Given $S = \{1, 2, 3, 4\}$. Let $a_{1}, a_{2}, \cdots, a_{k}$ be a sequence composed of numbers from $S$, which includes all permutations of $(1, 2, 3, 4)$ that do not end with 1. That is, if $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is a permutation of $(1, 2, 3, 4)$ and $b_{4} \neq 1$, then there exist indices $1 \leq i_{1} < i_{2} < i_{3} < i_{4} \leq k$ such that $\left(a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}}\right)=\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$. Find the minimum value of $k$.
| 11 | 0 |
26,565 | Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), how many sets of solutions satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\) ? | 10 | 25 |
26,566 | Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | \frac{3}{64} | 25.78125 |
26,567 | Rectangle $PQRS$ is inscribed in triangle $XYZ$, with side $PS$ of the rectangle on side $XZ$. Triangle $XYZ$ has an altitude from $Y$ to side $XZ$ of 8 inches, and the side $XZ$ measures 15 inches. The length of segment $PQ$ is one-third the length of segment $PS$. Calculate the area of rectangle $PQRS$, and express your answer as a common fraction. | \frac{4800}{169} | 30.46875 |
26,568 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 240 | 0 |
26,569 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. The radius of the circumcircle is $1$, and it is given that $\frac{\tan A}{\tan B} = \frac{2c-b}{b}$. Find the maximum value of the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} | 2.34375 |
26,570 | A cube with side length 2 has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and for $1\le i\le 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. The vertices on the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, and $\overline{Q_1Q_4}$ are $\frac{2}{3}$ from $Q_1$. Find the side length of the octahedron. | \frac{4\sqrt{2}}{3} | 48.4375 |
26,571 | Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$ . Find $$ \sum_{(a,b,c)\in S} abc. $$ | \binom{2015}{5} | 0 |
26,572 | If a four-digit number is called a "good number" when its unit digit is 1 and it has exactly three identical digits, then how many "good numbers" are there among the four-digit numbers formed by the digits 1, 2, 3, and 4 with repetitions? | 12 | 23.4375 |
26,573 | An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? | 221 | 0.78125 |
26,574 | In triangle $ABC$, $AB=15$, $AC=20$, and $BC=25$. A rectangle $PQRS$ is embedded inside triangle $ABC$ such that $PQ$ is parallel to $BC$ and $RS$ is parallel to $AB$. If $PQ=12$, find the area of rectangle $PQRS$. | 115.2 | 0 |
26,575 | There are 9 digits: 0, 1, 2, …, 8. Using five cards, with the two sides respectively marked as 0/8, 1/7, 2/5, 3/4, 6/9; and 6 can be used as 9. How many different four-digit numbers can be formed with these five cards? | 1728 | 25.78125 |
26,576 | Some of the digits in the following correct addition have been replaced by the letters \( P, Q, R, \) and \( S \), as shown.
What is the value of \( P+Q+R+S \)?
\[
\begin{array}{r}
P 45 \\
+\quad Q R \quad S \\
\hline
654
\end{array}
\] | 15 | 73.4375 |
26,577 | Given that the unit vectors $\overrightarrow{e\_1}$ and $\overrightarrow{e\_2}$ satisfy the equation $|2\overrightarrow{e\_1} + \overrightarrow{e\_2}| = |\overrightarrow{e\_1}|$, find the projection of $\overrightarrow{e\_1}$ onto the direction of $\overrightarrow{e\_2}$. | -1 | 29.6875 |
26,578 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 32.8125 |
26,579 | Given that a multiple-choice math question has incorrect options that cannot be adjacent, calculate the total number of arrangements of $4$ different options that meet the requirements. | 36 | 0 |
26,580 | Suppose you have 3 red candies, 2 green candies, and 4 blue candies. A flavor is defined by the ratio of red to green to blue candies, and different flavors are identified by distinct ratios (flavors are considered the same if they can be reduced to the same ratio). How many different flavors can be created if a flavor must include at least one candy of each color? | 21 | 16.40625 |
26,581 | (1) Calculate $\dfrac{2A_{8}^{5}+7A_{8}^{4}}{A_{8}^{8}-A_{9}^{5}}$,
(2) Calculate $C_{200}^{198}+C_{200}^{196}+2C_{200}^{197}$. | 67331650 | 0 |
26,582 | What is the area enclosed by the graph of $|x| + |3y| + |x - y| = 20$? | \frac{200}{3} | 0 |
26,583 | Knowing that segment \( CD \) has a length of 6 and its midpoint is \( M \), two triangles \( \triangle ACD \) and \( \triangle BCD \) are constructed on the same side with \( CD \) as a common side and both having a perimeter of 16, such that \( \angle AMB = 90^{\circ} \). Find the minimum area of \( \triangle AMB \). | 400/41 | 0 |
26,584 | In Zuminglish, words consist only of the letters $M, O,$ and $P$. $O$ is a vowel, while $M$ and $P$ are consonants. A valid Zuminglish word must have at least two consonants between any two $O's$. Determine the number of valid 7-letter words in Zuminglish and find the remainder when this number is divided by $1000$. | 912 | 3.125 |
26,585 | Among 6 internists and 4 surgeons, there is one chief internist and one chief surgeon. Now, a 5-person medical team is to be formed to provide medical services in rural areas. How many ways are there to select the team under the following conditions?
(1) The team includes 3 internists and 2 surgeons;
(2) The team includes both internists and surgeons;
(3) The team includes at least one chief;
(4) The team includes both a chief and surgeons. | 191 | 0 |
26,586 | Given that the terminal side of angle $\alpha$ passes through point $P(-4a, 3a) (a \neq 0)$, find the value of $\sin \alpha + \cos \alpha - \tan \alpha$. | \frac{19}{20} | 4.6875 |
26,587 | Given Madeline has 80 fair coins. She flips all the coins. Any coin that lands on tails is tossed again. Additionally, any coin that lands on heads in the first two tosses is also tossed again, but only once. What is the expected number of coins that are heads after these conditions? | 40 | 9.375 |
26,588 | The vertices of a $3 \times 1 \times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L F=1$, points $M$ and $N$ are on $C G$ so that $C M=M N=N G=1$, and points $O$ and $P$ are on $D H$ so that $D O=O P=P H=1$. For every pair of the 16 points $A$ through $P$, Maria computes the distance between them and lists the 120 distances. How many of these 120 distances are equal to $\sqrt{2}$? | 32 | 0 |
26,589 | Given that the function $f(x)$ is monotonic on $(-1, +\infty)$, and the graph of the function $y = f(x - 2)$ is symmetrical about the line $x = 1$, if the sequence $\{a_n\}$ is an arithmetic sequence with a nonzero common difference and $f(a_{50}) = f(a_{51})$, determine the sum of the first 100 terms of $\{a_n\}$. | -100 | 55.46875 |
26,590 | Given that YQZC is a rectangle with YC = 8 and CZ = 15, and equilateral triangles ABC and PQR each with a side length of 9, where R and B are on sides YQ and CZ, respectively, determine the length of AP. | 10 | 4.6875 |
26,591 | Given the approximate values $\lg 2 = 0.301$ and $\lg 3 = 0.477$, find the best approximation for $\log_{5} 10$. | $\frac{10}{7}$ | 0 |
26,592 | In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, three edges with vertex $A$ as an endpoint are all of length $2$, and their angles with each other are all $60^{\circ}$. Determine the cosine value of the angle between the line $BD_{1}$ and the line $AC$. | \frac{\sqrt{6}}{6} | 3.125 |
26,593 | In the arithmetic sequence $\{a_n\}$, it is given that $a_{15}+a_{16}+a_{17}=-45$ and $a_{9}=-36$. Let $S_n$ be the sum of the first $n$ terms.
$(1)$ Find the minimum value of $S_n$ and the corresponding value of $n$;
$(2)$ Calculate $T_n=|a_1|+|a_2|+\cdots+|a_n|$. | -630 | 12.5 |
26,594 | There are 21 nonzero numbers. For each pair of these numbers, their sum and product are calculated. It turns out that half of all the sums are positive and half are negative. What is the maximum possible number of positive products? | 120 | 7.03125 |
26,595 | Compute
\[
\sin^2 0^\circ + \sin^2 10^\circ + \sin^2 20^\circ + \dots + \sin^2 180^\circ.
\] | 10 | 3.90625 |
26,596 | The perimeter of triangle $BQN$ is $180$, and the angle $QBN$ is a right angle. A circle of radius $15$ with center $O$ on $\overline{BQ}$ is drawn such that it is tangent to $\overline{BN}$ and $\overline{QN}$. Given that $OQ=p/q$ where $p$ and $q$ are relatively prime positive integers, find $p+q$. | 79 | 0 |
26,597 | Through a point $P$ inside the $\triangle ABC$, a line is drawn parallel to the base $AB$, dividing the triangle into two regions where the area of the region containing the vertex $C$ is three times the area of the region adjacent to $AB$. If the altitude to $AB$ has a length of $2$, calculate the distance from $P$ to $AB$. | \frac{1}{2} | 1.5625 |
26,598 |
A right triangle \(ABC\) has a perimeter of 54, with the leg \(AC\) being greater than 10. A circle with a radius of 6, whose center lies on leg \(BC\), is tangent to lines \(AB\) and \(AC\). Find the area of triangle \(ABC\). | 243/2 | 0 |
26,599 | Given the convex pentagon $ABCDE$, where each pair of neighboring vertices must have different colors and vertices at the ends of each diagonal must not share the same color, determine the number of possible colorings using 5 available colors. | 240 | 1.5625 |
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