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29,200 | There are 6 seats in a row and 3 people taking their seats, find the number of different ways of seating them such that there are exactly two adjacent empty seats. | 72 | 0.78125 |
29,201 | Among the nine fractions
$$
\frac{5}{4}, \frac{17}{6}, \frac{-5}{4}, \frac{10}{7}, \frac{2}{3}, \frac{14}{8}, \frac{-1}{3}, \frac{5}{3} \text { and } \frac{-3}{2}
$$
we have eight with the following properties:
- 2 fractions whose sum is $\frac{2}{5}$
- 2 fractions whose difference is $\frac{2}{5}$
- 2 fractions whose product is $\frac{2}{5}$
- 2 fractions whose quotient is $\frac{2}{5}$
Find the fraction that is left over. | -\frac{3}{2} | 15.625 |
29,202 | The area of the shaded region BEDC in parallelogram ABCD is to be found, where BC = 15, ED = 9, and the total area of ABCD is 150. If BE is the height of parallelogram ABCD from base BC and is shared with ABE, both of which overlap over BE, calculate the area of the shaded region BEDC. | 120 | 18.75 |
29,203 | The base of an oblique prism is a parallelogram with sides 3 and 6 and an acute angle of $45^{\circ}$. The lateral edge of the prism is 4 and is inclined at an angle of $30^{\circ}$ to the base plane. Find the volume of the prism. | 18\sqrt{6} | 0 |
29,204 | A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$ , but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$ ). Find the maximum number of contestants. | 64 | 24.21875 |
29,205 | Given that six students are to be seated in three rows of two seats each, with one seat reserved for a student council member who is Abby, calculate the probability that Abby and Bridget are seated next to each other in any row. | \frac{1}{5} | 36.71875 |
29,206 | Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | 530816 | 0 |
29,207 | Given that the graph of the function $f(x)=ax^3+bx^2+c$ passes through the point $(0,1)$, and the equation of the tangent line at $x=1$ is $y=x$.
(1) Find the analytic expression of $y=f(x)$;
(2) Find the extreme values of $y=f(x)$. | \frac{23}{27} | 50.78125 |
29,208 | For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots? | 625 | 10.9375 |
29,209 | Determine the maximum difference between the \(y\)-coordinates of the intersection points of the graphs \(y=5-x^2+2x^3\) and \(y=3+2x^2+2x^3\). | \frac{8\sqrt{6}}{9} | 56.25 |
29,210 | Person A and Person B start from points $A$ and $B$ simultaneously and move towards each other. It is known that the speed ratio of Person A to Person B is 6:5. When they meet, they are 5 kilometers from the midpoint between $A$ and $B$. How many kilometers away is Person B from point $A$ when Person A reaches point $B$? | 5/3 | 0 |
29,211 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 64.0625 |
29,212 | A regular pentagon \(Q_1 Q_2 \dotsb Q_5\) is drawn in the coordinate plane with \(Q_1\) at \((1,0)\) and \(Q_3\) at \((5,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_5 + y_5 i).\] | 242 | 0 |
29,213 | There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .
Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased? | 100! | 7.03125 |
29,214 | Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$ . | 1026 | 29.6875 |
29,215 | Before the soccer match between the teams "North" and "South", five predictions were made:
a) There will be no draw;
b) Goals will be scored against "South";
c) "North" will win;
d) "North" will not lose;
e) Exactly 3 goals will be scored in the match.
After the match, it turned out that exactly three of the predictions were correct. What was the final score of the match? | 1:2 | 9.375 |
29,216 | In a table tennis match between player A and player B, the "best of five sets" rule is applied, which means the first player to win three sets wins the match. If the probability of player A winning a set is $\dfrac{2}{3}$, and the probability of player B winning a set is $\dfrac{1}{3}$, then the probability of the match ending with player A winning three sets and losing one set is ______. | \dfrac{8}{27} | 7.8125 |
29,217 | Consider all 4-digit palindromes that can be written as $\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes. | 36 | 2.34375 |
29,218 | Given the function $f(x) = 2\cos(x)(\sin(x) + \cos(x))$ where $x \in \mathbb{R}$,
(Ⅰ) Find the smallest positive period of function $f(x)$;
(Ⅱ) Find the intervals on which function $f(x)$ is monotonically increasing;
(Ⅲ) Find the minimum and maximum values of function $f(x)$ on the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. | \sqrt{2} + 1 | 39.84375 |
29,219 | Alexio has 120 cards numbered 1-120, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 3, 4, or 7? Express your answer as a common fraction. | \dfrac{69}{120} | 0 |
29,220 | A sphere with radius $r$ is inside a cone, the cross section of which is an equilateral triangle inscribed in a circle. Find the ratio of the total surface area of the cone to the surface area of the sphere. | 9:4 | 0 |
29,221 | Let $r_1$ , $r_2$ , $\ldots$ , $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$ . If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$ . | 240 | 0 |
29,222 | What is the distance from Boguli to Bolifoyn? | 10 | 0 |
29,223 | How many four-digit positive integers are multiples of 7? | 1286 | 98.4375 |
29,224 | A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$ . For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$ , then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$ . | 15 | 14.84375 |
29,225 | Given a real number $a$ satisfying ${a}^{\frac{1}{2}}\leqslant 3$ and $\log _{a}3\leqslant \frac{1}{2}$.
$(1)$ Find the range of real number $a$;
$(2)$ If $a \gt 1$, $f\left(x\right)=mx^{a}+\ln \left(1+x\right)^{a}-a\ln \left(1-x\right)-2\left(m\in R\right)$, and $f(\frac{1}{2})=a$, find the value of $f(-\frac{1}{2})$. | -13 | 88.28125 |
29,226 | Let set $M=\{-1, 0, 1\}$, and set $N=\{a, a^2\}$. Find the real number $a$ such that $M \cap N = N$. | -1 | 1.5625 |
29,227 | As shown in the diagram, a square is divided into 4 identical rectangles, each of which has a perimeter of 20 centimeters. What is the area of this square? | 64 | 82.03125 |
29,228 | Eight spheres of radius 2 are each tangent to the faces of an inner cube centered at the origin, with each sphere located in one of the octants, and the side length of the cube is 6. Find the radius of the smallest sphere, centered at the origin, that encloses these eight spheres. | 3\sqrt{3} + 2 | 3.90625 |
29,229 | A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \mathrm{~cm}^{2}$. | 36 | 2.34375 |
29,230 | Alice conducted a survey among a group of students regarding their understanding of snakes. She found that $92.3\%$ of the students surveyed believed that snakes are venomous. Of the students who believed this, $38.4\%$ erroneously thought that all snakes are venomous. Knowing that only 31 students held this incorrect belief, calculate the total number of students Alice surveyed. | 88 | 21.09375 |
29,231 | Let $ a,b$ be integers greater than $ 1$ . What is the largest $ n$ which cannot be written in the form $ n \equal{} 7a \plus{} 5b$ ? | 47 | 2.34375 |
29,232 | In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$? | 180 | 71.09375 |
29,233 | The sequence $3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, ...$ consists of $3$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. Calculate the sum of the first $1024$ terms of this sequence.
A) $4166$
B) $4248$
C) $4303$
D) $4401$ | 4248 | 3.90625 |
29,234 | Given the equation in terms of \( x \)
$$
x^{4}-16 x^{3}+(81-2a) x^{2}+(16a-142) x + a^{2} - 21a + 68 = 0
$$
where all roots are integers, find the value of \( a \) and solve the equation. | -4 | 1.5625 |
29,235 | How many integers between $2$ and $100$ inclusive *cannot* be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$ ? Note that there are $25$ primes less than $100.$ | 35 | 1.5625 |
29,236 | A line that always passes through a fixed point is given by the equation $mx - ny - m = 0$, and it intersects with the parabola $y^2 = 4x$ at points $A$ and $B$. Find the number of different selections of distinct elements $m$ and $n$ from the set ${-3, -2, -1, 0, 1, 2, 3}$ such that $|AB| < 8$. | 18 | 8.59375 |
29,237 | Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____. | 128 | 3.90625 |
29,238 | A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if they studied for 5 hours and the test has a difficulty factor of 2? | 300 | 75 |
29,239 | A rectangle and a regular pentagon have the same perimeter. Let $A$ be the area of the circle circumscribed about the rectangle, with the rectangle having side lengths in the ratio 3:1, and $B$ the area of the circle circumscribed around the pentagon. Find $A/B$.
A) $\frac{125(\sqrt{5}-1)}{256}$
B) $\frac{5}{3} \cdot \frac{5-\sqrt{5}}{8}$
C) $\frac{500(5 - \sqrt{5})}{1024}$
D) $\frac{5(5-\sqrt{5})}{64}$
E) $\frac{250(5 - \sqrt{5})}{256}$ | \frac{5(5-\sqrt{5})}{64} | 1.5625 |
29,240 | The units of length include , and the conversion rate between two adjacent units is . | 10 | 25.78125 |
29,241 | Given a square grid with the letters AMC9 arranged as described, starting at the 'A' in the middle, determine the number of different paths that allow one to spell AMC9 without revisiting any cells. | 24 | 4.6875 |
29,242 | Given a sequence ${a_n}$ with the sum of its first $n$ terms denoted as $T_n$, where $a_1 = 1$ and $a_1 + 2a_2 + 4a_3 + ... + 2^{n-1}a_n = 2n - 1$, find the value of $T_8 - 2$. | \frac{63}{64} | 60.15625 |
29,243 | A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$ , then $a_m$ is a divisor of $a_n$ and $a_m<a_n$ . Find the least possible value of $a_{2000}$ . | 128 | 0 |
29,244 | Given 15 points in space, 5 of which are collinear, what is the maximum possible number of unique planes that can be determined by these points? | 445 | 46.875 |
29,245 | Elective 4-4: Coordinate System and Parametric Equations
Given the parametric equations of curve \\(C\\) as \\(\begin{cases}x=2\cos \left(\theta\right) \\ y= \sqrt{3}\sin \left(\theta\right)\end{cases} \\), in the same plane Cartesian coordinate system, the points on curve \\(C\\) are transformed by \\(\begin{cases} {x'}=\dfrac{1}{2}x \\ {y'}=\dfrac{1}{\sqrt{3}}y \\ \end{cases}\\) to obtain curve \\({C'}\\), with the origin as the pole and the positive half-axis of \\(x\\) as the polar axis, establishing a polar coordinate system.
\\((\\)Ⅰ\\()\\) Find the polar equation of curve \\({C'}\\);
\\((\\)Ⅱ\\()\\) If a line \\(l\\) passing through point \\(A\left(\dfrac{3}{2},\pi \right)\\) (in polar coordinates) with a slope angle of \\(\dfrac{\pi }{6}\\) intersects curve \\({C'}\\) at points \\(M\\) and \\(N\\), and the midpoint of chord \\(MN\\) is \\(P\\), find the value of \\(\dfrac{|AP|}{|AM|\cdot |AN|}\\). | \dfrac{3\sqrt{3}}{5} | 27.34375 |
29,246 | Find the angle of inclination of the tangent line to the curve $y= \frac {1}{2}x^{2}-2x$ at the point $(1,- \frac {3}{2})$. | \frac{3\pi}{4} | 0.78125 |
29,247 | Given that $\alpha \in \left( 0, \pi \right)$ and $3\cos 2\alpha = \sin \left( \frac{\pi}{4} - \alpha \right)$, find the value of $\sin 2\alpha$. | -\frac{17}{18} | 0.78125 |
29,248 | A positive integer $n$ is called*bad*if it cannot be expressed as the product of two distinct positive integers greater than $1$ . Find the number of bad positive integers less than $100. $ *Proposed by Michael Ren* | 30 | 1.5625 |
29,249 | A teacher intends to give the children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients and a leading coefficient of 1. Then he will provide them with \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately provide the value of the expression \( P(n_{1}) \cdot P(n_{2}) \cdot \ldots \cdot P(n_{k}) \). From this data, the children must determine the polynomial that the teacher had in mind. What is the minimum value of \( k \) such that the polynomial found by the children will necessarily be the same as the one conceived by the teacher? | 2017 | 1.5625 |
29,250 | The function $g(x)$ satisfies
\[g(x + g(x)) = 5g(x)\] for all \(x\), and \(g(1) = 5\). Find \(g(26)\). | 125 | 30.46875 |
29,251 | Hydras consist of heads and necks (each neck connects exactly two heads). With one sword strike, all the necks emanating from a head \( A \) of the hydra can be cut off. However, a new neck immediately grows from head \( A \) to every head that \( A \) was not previously connected to. Hercules wins if he manages to split the hydra into two disconnected parts. Find the smallest \( N \) for which Hercules can defeat any 100-headed hydra by making no more than \( N \) strikes. | 10 | 0 |
29,252 | The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$ , where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$ \dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}} $$ | 14 | 0 |
29,253 | What is the smallest positive integer with exactly 16 positive divisors? | 384 | 0 |
29,254 | Six identical rectangles are arranged to form a larger rectangle \( ABCD \). The area of \( ABCD \) is 6000 square units. What is the length \( z \), rounded off to the nearest integer? | 32 | 2.34375 |
29,255 | An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $20$, and one of the base angles is $\arcsin(0.6)$. Find the area of the trapezoid given that the height of the trapezoid is $9$. | 100 | 28.125 |
29,256 | A positive integer is called a perfect power if it can be written in the form \(a^b\), where \(a\) and \(b\) are positive integers with \(b \geq 2\). The increasing sequence \(2, 3, 5, 6, 7, 10, \ldots\) consists of all positive integers which are not perfect powers. Calculate the sum of the squares of the digits of the 1000th number in this sequence. | 21 | 57.8125 |
29,257 | Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to $90^\circ$ ?
*Proposed by Anton Trygub* | 2,042,220 | 0 |
29,258 | 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} | 0 |
29,259 | In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 6. $PQ$ and $RS$ intersect perpendicularly at the center $O$. The line segments $PR$ and $QS$ subtend central angles of 60° and 120° respectively at $O$. What is the area of the shaded region formed by $\triangle POR$, $\triangle SOQ$, sector $POS$, and sector $ROQ$? | 36 + 18\pi | 2.34375 |
29,260 | At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. | 12 | 3.125 |
29,261 | There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$ , and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. | 3/455 | 6.25 |
29,262 | Let $D$ be the circle with the equation $2x^2 - 8y - 6 = -2y^2 - 8x$. Determine the center $(c,d)$ of $D$ and its radius $s$, and calculate the sum $c + d + s$. | \sqrt{7} | 0 |
29,263 | Find the minimum point of the function $f(x)=x+2\cos x$ on the interval $[0, \pi]$. | \dfrac{5\pi}{6} | 0.78125 |
29,264 | Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. Determine the number of women in the luxury compartment if the train carries 300 passengers. | 32 | 94.53125 |
29,265 | In the rectangular coordinate system xOy, an ellipse C is given by the equation $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($$a>b>0$$), with left and right foci $$F_1$$ and $$F_2$$, respectively. The left vertex's coordinates are ($$-\sqrt {2}$$, 0), and point M lies on the ellipse C such that the perimeter of $$\triangle MF_1F_2$$ is $$2\sqrt {2}+2$$.
(1) Find the equation of the ellipse C;
(2) A line l passes through $$F_1$$ and intersects ellipse C at A and B, satisfying |$$\overrightarrow {OA}+2 \overrightarrow {OB}$$|=|$$\overrightarrow {BA}- \overrightarrow {OB}$$|, find the area of $$\triangle ABO$$. | \frac {2\sqrt {3}}{5} | 0 |
29,266 | Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$ , $AC = 1800$ , $BC = 2014$ . The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$ . Compute the length $XY$ .
*Proposed by Evan Chen* | 1186 | 3.90625 |
29,267 | The year 2000 is a leap year. The year 2100 is not a leap year. The following are the complete rules for determining a leap year:
(i) Year \(Y\) is not a leap year if \(Y\) is not divisible by 4.
(ii) Year \(Y\) is a leap year if \(Y\) is divisible by 4 but not by 100.
(iii) Year \(Y\) is not a leap year if \(Y\) is divisible by 100 but not by 400.
(iv) Year \(Y\) is a leap year if \(Y\) is divisible by 400.
How many leap years will there be from the years 2000 to 3000 inclusive? | 244 | 0 |
29,268 | Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. | \frac{5}{32} | 64.84375 |
29,269 | Given that the center of the ellipse $E$ is at the origin, the focus is on the $y$-axis with an eccentricity of $e=\frac{\sqrt{2}}{2}$, and it passes through the point $P\left( 1,\sqrt{2} \right)$:
(1) Find the equation of the ellipse $E$;
(2) Two pairs of perpendicular lines are drawn through the focus $F$ of ellipse $E$, intersecting the ellipse $E$ at $A,B$ and $C,D$ respectively. Find the minimum value of $| AB |+| CD |$. | \frac{16}{3} | 12.5 |
29,270 | How many positive odd integers greater than 1 and less than $200$ are square-free? | 80 | 96.875 |
29,271 | How many multiples of 5 are between 80 and 375? | 59 | 0.78125 |
29,272 | Given a bag contains 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls, determine the minimum number of balls that must be drawn to ensure that at least 15 balls of the same color are selected. | 75 | 90.625 |
29,273 | For how many pairs of consecutive integers in the set $\{1100, 1101, 1102, \ldots, 2200\}$ is no carrying required when the two integers are added? | 1100 | 0 |
29,274 | Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. What is the probability that the smallest positive difference between any two of those numbers is $3$ or greater? Express your answer as a common fraction. | \frac{1}{14} | 0 |
29,275 | **Q14.** Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$ . Suppose that $IH$ is perpendicular to $BC$ ( $H$ belongs to $BC$ ). If $HB=5 \text{cm}, \; HC=8 \text{cm}$ , compute the area of $\triangle ABC$ . | 40 | 0.78125 |
29,276 | A certain product costs $6$ per unit, sells for $x$ per unit $(x > 6)$, and has an annual sales volume of $u$ ten thousand units. It is known that $\frac{585}{8} - u$ is directly proportional to $(x - \frac{21}{4})^2$, and when the selling price is $10$ dollars, the annual sales volume is $28$ ten thousand units.
(1) Find the relationship between the annual sales profit $y$ and the selling price $x$.
(2) Find the selling price that maximizes the annual profit and determine the maximum annual profit. | 135 | 4.6875 |
29,277 | Given Mr. Thompson can choose between two routes to commute to his office: Route X, which is 8 miles long with an average speed of 35 miles per hour, and Route Y, which is 7 miles long with an average speed of 45 miles per hour excluding a 1-mile stretch with a reduced speed of 15 miles per hour. Calculate the time difference in minutes between Route Y and Route X. | 1.71 | 0.78125 |
29,278 | A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? | 12 | 0 |
29,279 | There are five students: A, B, C, D, and E;
(1) If these five students line up in a row, in how many ways can A not stand in the first position?
(2) If these five students line up in a row, and A and B must be next to each other while C and D must not be next to each other, in how many ways can they line up?
(3) If these five students participate in singing, dancing, chess, and drawing competitions, with at least one person in each competition, and each student must participate in exactly one competition, and A cannot participate in the dancing competition, how many participating arrangements are there? | 180 | 2.34375 |
29,280 | Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $BC$ intersects side $AB$ at point $E$ such that $BE = 6$. If $AE = 3$, find the length of $EC$. | 12 | 8.59375 |
29,281 | Let $\{a_n\}$ be a decreasing geometric sequence, where $q$ is the common ratio, and $S_n$ is the sum of the first $n$ terms. Given that $\{a_1, a_2, a_3\} \subseteq \{-4, -3, -2, 0, 1, 2, 3, 4\}$, find the value of $$\frac {S_{8}}{1-q^{4}}$$. | \frac {17}{2} | 35.15625 |
29,282 | If $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 14 | 0.78125 |
29,283 | For a pair of integers \((a, b)(0 < a < b < 1000)\), a set \(S \subseteq \{1, 2, \cdots, 2003\}\) is called a "jump set" for the pair \((a, b)\) if for any pair of elements \(\left(s_{1}, s_{2}\right)\) in \(S\), \(|s_{1}-s_{2}| \notin\{a, b\}\).
Let \(f(a, b)\) denote the maximum number of elements in a jump set for the pair \((a, b)\). Find the maximum and minimum values of \(f(a, b)\). | 668 | 14.0625 |
29,284 | There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely? | 361 | 2.34375 |
29,285 | If Alice is walking north at a speed of 4 miles per hour and Claire is walking south at a speed of 6 miles per hour, determine the time it will take for Claire to meet Alice, given that Claire is currently 5 miles north of Alice. | 30 | 3.90625 |
29,286 | A factory produces a certain product for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$; when the annual production is not less than 80 thousand units, $C(x)=51x+\frac{10000}{x}-1450$. The selling price per product is 0.05 ten thousand yuan. Through market analysis, it is determined that all the products produced by the factory can be sold.
(1) Write the analytical expression of the annual profit $L$ (in ten thousand yuan) as a function of the annual production $x$ (in thousand units);
(2) At what annual production volume (in thousand units) does the factory maximize its profit from this product? | 100 | 16.40625 |
29,287 | Express $7.\overline{123}$ as a common fraction in lowest terms. | \frac{2372}{333} | 75 |
29,288 | Given that Marie has 2500 coins consisting of pennies (1-cent coins), nickels (5-cent coins), and dimes (10-cent coins) with at least one of each type of coin, calculate the difference in cents between the greatest possible and least amounts of money that Marie can have. | 22473 | 63.28125 |
29,289 | Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 4$. Two of the roots of $g(x)$ are $r + 3$ and $r + 5$, and
\[ f(x) - g(x) = 2r + 1 \]
for all real numbers $x$. Find $r$. | \frac{1}{4} | 7.8125 |
29,290 | Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$ . | 2116 | 85.9375 |
29,291 | How many positive odd integers greater than 1 and less than 200 are square-free? | 81 | 0.78125 |
29,292 | Helena needs to save 40 files onto disks, each with 1.44 MB space. 5 of the files take up 1.2 MB, 15 of the files take up 0.6 MB, and the rest take up 0.3 MB. Determine the smallest number of disks needed to store all 40 files. | 16 | 5.46875 |
29,293 | In a configuration of two right triangles, $PQR$ and $PRS$, squares are constructed on three sides of the triangles. The areas of three of the squares are 25, 4, and 49 square units. Determine the area of the fourth square built on side $PS$.
[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,7)--(1,7.7)--(0,7.7));
draw((0,7)--(0,8.5)--(7,7));
draw((0,8.5)--(2.6,16)--(7,7));
draw((2.6,16)--(12.5,19.5)--(14,10.5)--(7,7));
draw((0,8.5)--(-5.6,11.4)--(-3.28,14.76)--(2.6,16));
draw((0,7)--(-2,7)--(-2,8.5)--(0,8.5));
draw((0.48,8.35)--(0.68,8.85)--(-0.15,9.08));
label("$P$",(7,7),SE);
label("$Q$",(0,7),SW);
label("$R$",(0,8.5),N);
label("$S$",(2.6,16),N);
label("25",(-2.8/2,7+1.5/2));
label("4",(-2.8/2+7,7+1.5/2));
label("49",(3.8,11.75));
[/asy] | 53 | 10.15625 |
29,294 | If $q(x) = x^4 - 2x^2 - 5x + 3$, find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$. | -125 | 1.5625 |
29,295 | Let $(a_1,a_2,a_3,\ldots,a_{15})$ be a permutation of $(1,2,3,\ldots,15)$ for which
$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$
Find the number of such permutations. | 3003 | 8.59375 |
29,296 | Given the function $f\left( x \right)=\frac{{{e}^{x}}-a}{{{e}^{x}}+1}\left( a\in R \right)$ defined on $R$ as an odd function.
(1) Find the range of the function $y=f\left( x \right)$;
(2) When ${{x}_{1}},{{x}_{2}}\in \left[ \ln \frac{1}{2},\ln 2 \right]$, the inequality $\left| \frac{f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)}{{{x}_{1}}+{{x}_{2}}} \right| < \lambda \left( \lambda \in R \right)$ always holds. Find the minimum value of the real number $\lambda$. | \frac{1}{2} | 1.5625 |
29,297 | Given that four A's, four B's, four C's, and four D's are to be placed in a 4 × 4 grid so that each row and column contains one of each letter, and A is placed in the upper right corner, calculate the number of possible arrangements. | 216 | 0.78125 |
29,298 | Some middle school students in a city participated in a mathematics invitational competition, which consisted of 6 problems. It is known that each problem was solved by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. What is the minimum number of middle school students who participated in this mathematics invitational competition? | 1000 | 17.1875 |
29,299 | Given a sequence $\{a_n\}$ satisfying $a_1=81$ and $a_n= \begin{cases} -1+\log_{3}a_{n-1}, & n=2k \\ 3^{a_{n-1}}, & n=2k+1 \end{cases}$ (where $k\in\mathbb{N}^*$), find the maximum value of the sum of the first $n$ terms of the sequence, $S_n$. | 127 | 25 |
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