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31,300 | Let $S=\{1,2,\ldots ,98\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group. | 50 | 22.65625 |
31,301 | A council consists of nine women and three men. During their meetings, they sit around a round table with the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways can the nine chairs and three stools be arranged around the round table for a meeting? | 55 | 23.4375 |
31,302 | What is the smallest positive integer that is neither prime nor a cube and that has an even number of prime factors, all greater than 60? | 3721 | 17.96875 |
31,303 |
For what values of the parameter \( a \in \mathbb{R} \) does the maximum distance between the roots of the equation
\[ a \tan^{3} x + \left(2-a-a^{2}\right) \tan^{2} x + \left(a^{2}-2a-2\right) \tan x + 2a = 0 \]
belonging to the interval \(\left(-\frac{\pi}{2} , \frac{\pi}{2}\right)\), take the smallest value? Find this smallest value. | \frac{\pi}{4} | 3.90625 |
31,304 | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $\overrightarrow{m} = \overrightarrow{a} + t \overrightarrow{b} (t \in \mathbb{R})$, $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, the minimum value of $|\overrightarrow{m}|$ is obtained only when $t = \frac{1}{4}$. Determine the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 98.4375 |
31,305 | Compute the largest integer \( k \) such that \( 2010^k \) divides \( 2010! \). | 30 | 97.65625 |
31,306 | Given that point P is on the curve $y=\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$. | \frac{\sqrt{2}}{2}(1 - \ln 2) | 0.78125 |
31,307 | Cara is in a group photo with her seven friends. If Cara must stand between two of her friends, how many different possible pairs of friends could she be standing between? | 21 | 83.59375 |
31,308 | Given that all four roots of the equation $2x^4 + mx^2 + 8 = 0$ are integers, then $m = \ $, and the polynomial $2x^4 + mx^2 + 8$ can be factored into $\ $. | -10 | 1.5625 |
31,309 | Given the function $f(x)=-3x^2+6x$, let ${S_n}$ be the sum of the first $n$ terms of the sequence ${{a_n}}$. The points $(n, {S_n})$ (where $n \in \mathbb{N}^*$) lie on the curve $y=f(x)$.
(I) Find the general formula for the terms of the sequence ${{a_n}}$.
(II) If ${b_n}={(\frac{1}{2})^{n-1}}$ and ${c_n}=\frac{{a_n} \bullet {b_n}}{6}$, let ${T_n}$ be the sum of the first $n$ terms of the sequence ${c_n}$. Determine whether a maximum value of ${T_n}$ exists. If so, find the maximum value; if not, explain the reason. | \frac{1}{2} | 38.28125 |
31,310 | There were five teams entered in a competition. Each team consisted of either only boys or only girls. The number of team members was $9, 15, 17, 19$, and $21$. After one team of girls had been knocked out of the competition, the number of girls still competing was three times the number of boys. How many girls were in the team that was eliminated? | 21 | 21.09375 |
31,311 | Given the parabola $y^{2}=4x$, a line with a slope of $\frac{\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$. | 2\sqrt{2} | 0 |
31,312 | In a rectangle of size $3 \times 4$, 4 points are chosen. Find the smallest number $C$ such that the distance between some two of these points does not exceed $C$. | 2.5 | 53.125 |
31,313 | For how many integer values of $m$ ,
(i) $1\le m \le 5000$ (ii) $[\sqrt{m}] =[\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function | 72 | 1.5625 |
31,314 | Calculate the following expression:
\[\left( 1 - \frac{1}{\cos 30^\circ} \right) \left( 1 + \frac{1}{\sin 60^\circ} \right) \left( 1 - \frac{1}{\sin 30^\circ} \right) \left( 1 + \frac{1}{\cos 60^\circ} \right).\] | -1 | 0.78125 |
31,315 | Given the function $f(x) = \sin(\omega x + \frac{\pi}{6}) (\omega > 0)$, where the graph is symmetric about the point $(x_0, 0)$ and the distance between two adjacent symmetry axes of the function's graph is $\frac{\pi}{2}$, determine the value of $x_0$. | \frac{5\pi}{12} | 7.8125 |
31,316 | A bear is in the center of the left down corner of a $100*100$ square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum $k$ so that in any bear cycle we can remove a row or column so that the maximum length of the remaining pathes is at most $k$ . | 5000 | 19.53125 |
31,317 | A spinner is divided into six congruent sectors, numbered from 1 to 6. Jane and her brother each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? | \frac{5}{6} | 64.0625 |
31,318 | Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained by all the teams was 61. What is the maximum number of points that the tournament's winning team could have obtained? | 17 | 20.3125 |
31,319 | Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $(E)$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\_1NF\_2$ is $4$, let line $(l)$ pass through $F\_1$ intersecting $(E)$ at points $A$ and $B$ with $|AB|=\frac{4}{3}$.
1. Find the maximum value of $|AF\_2| \cdot |BF\_2|$.
2. If the slope of line $(l)$ is $45^{\circ}$, find the area of $\triangle ABF\_2$. | \frac{2}{3} | 7.03125 |
31,320 | Given that $x$ is a multiple of $54896$, find the greatest common divisor of $f(x)=(5x+4)(9x+7)(11x+3)(x+12)$ and $x$. | 112 | 11.71875 |
31,321 | Calculate the circulation of the vector field:
a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;
b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$. | -3 | 59.375 |
31,322 | Given an arc length of 50cm and the central angle corresponding to the arc is 200°, (1) find the radius of the circle containing this arc, (2) find the area of the sector formed by this arc and the radius. | \frac{1125}{\pi} | 55.46875 |
31,323 | In our province, the new college entrance examination adopts a "choose 3 out of 7" model, which means choosing 3 subjects from politics, history, geography, physics, chemistry, biology, and technology as elective subjects. How many possible combinations of elective subjects are there? If person A must choose physics and politics, and person B does not choose technology, how many combinations are there such that both persons have at least one subject in common? (Answer mathematically) | 92 | 0 |
31,324 | A regular decagon is formed by connecting three sequentially adjacent vertices of the decagon. Find the probability that all three sides of the triangle are also sides of the decagon. | \frac{1}{12} | 62.5 |
31,325 | Given the function $f(x)=\cos (2x- \frac {π}{6})\sin 2x- \frac{1}{4}(x∈R)$
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) Find the maximum and minimum values of the function $f(x)$ on $\[- \frac {π}{4},0\]$. | -\frac {1}{2} | 67.96875 |
31,326 | In the Cartesian coordinate system, given points A(1, -3), B(4, -1), P(a, 0), and N(a+1, 0), if the perimeter of the quadrilateral PABN is minimal, then find the value of a. | a = \frac{5}{2} | 41.40625 |
31,327 | Calculate the value of the expression \( \sqrt{\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \). | \frac{3\sqrt{2}}{4} | 56.25 |
31,328 | In triangle \(ABC\), the angle at vertex \(B\) is \(\frac{\pi}{3}\), and the line segments connecting the incenter to vertices \(A\) and \(C\) are 4 and 6, respectively. Find the radius of the circle inscribed in triangle \(ABC\). | \frac{6 \sqrt{3}}{\sqrt{19}} | 0 |
31,329 | The sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket contains one number, the sixth bracket contains two numbers, and so on in a cycle. What is the sum of the numbers in the 104th bracket? | 2072 | 17.1875 |
31,330 | Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. | 12 | 3.90625 |
31,331 | Given the hyperbola $C$: $\frac{x^2}{a^2} - y^2 = 1$ $(a > 0)$ and the line $l$: $x + y = 1$, which intersect at two distinct points $A$ and $B$.
1. Find the range of values for $a$.
2. Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\overrightarrow{PA} = \frac{5}{12}\overrightarrow{PB}$. Find the value of $a$. | \frac{17}{13} | 25.78125 |
31,332 | The traditional Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art" records: "There are 5 cows and 2 sheep, worth 19 taels of silver; 2 cows and 5 sheep, worth 16 taels of silver. How much is each cow and each sheep worth in silver?" According to the translation above, answer the following two questions:
$(1)$ Find out how much each cow and each sheep are worth in silver.
$(2)$ If the cost of raising a cow is 2 taels of silver and the cost of raising a sheep is 1.5 taels of silver, villager Li wants to raise a total of 10 cows and sheep (the number of cows does not exceed the number of sheep). When he sells them all, how many cows and sheep should Li raise to earn the most silver? | 7.5 | 7.8125 |
31,333 | Given a sequence where each term is either 1 or 2, starting with 1, and where between the \(k\)-th 1 and the \((k+1)\)-th 1 there are \(2^{k-1}\) 2's (i.e., the sequence is 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, ...), determine the sum of the first 1998 terms of this sequence. | 3986 | 5.46875 |
31,334 | Given the following arrays, each composed of three numbers: $(1,2,3)$, $(2,4,6)$, $(3,8,11)$, $(4,16,20)$, $(5,32,37)$, ..., $(a\_n,b\_n,c\_n)$. If the sum of the first $n$ terms of the sequence $\{c\_n\}$ is denoted as $M\_n$, find the value of $M\_{10}$. | 2101 | 77.34375 |
31,335 | For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a "naive number." For example, the four-digit number $7311$ is a "naive number" because $7-1=6$ and $3-1=2$. On the other hand, the four-digit number $8421$ is not a "naive number" because $8-1\neq 6$. Find the smallest "naive number" which is ______.
Let the digit in the thousands place of a "naive number" $M$ be $a$, the digit in the hundreds place be $b$, the digit in the tens place be $c$, and the digit in the units place be $d$. Define $P(M)=3(a+b)+c+d$ and $Q(M)=a-5$. If $\frac{{P(M)}}{{Q(M)}}$ is divisible by $10$, then the maximum value of $M$ that satisfies this condition is ______. | 9313 | 0.78125 |
31,336 | Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) and a line $l: y = -\frac { \sqrt {3}}{3}x + b$ intersect at two distinct points P and Q. The distance from the origin to line $l$ is $\frac { \sqrt {3}}{2}$, and the eccentricity of the ellipse is $\frac { \sqrt {6}}{3}$.
(Ⅰ) Find the equation of the ellipse;
(Ⅱ) Determine whether there exists a real number $k$ such that the line $y = kx + 2$ intersects the ellipse at points P and Q, and the circle with diameter PQ passes through point D(1, 0). If it exists, find the value of $k$; if not, explain why. | -\frac {7}{6} | 0.78125 |
31,337 | Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage. | 48 | 56.25 |
31,338 | Let $ABC$ be a right triangle, right at $B$ , and let $M$ be the midpoint of the side $BC$ . Let $P$ be the point in
bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$ ). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$ . | 120 | 1.5625 |
31,339 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 14.84375 |
31,340 | Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collected in any order before the fourth click. | 18 | 80.46875 |
31,341 | Given a right triangle where one leg has length $4x + 2$ feet and the other leg has length $(x-3)^2$ feet, find the value of $x$ if the hypotenuse is $5x + 1$ feet. | \sqrt{\frac{3}{2}} | 3.90625 |
31,342 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?
(Two rectangles are different if they do not share all four vertices.) | 100 | 23.4375 |
31,343 | The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$ | 671 | 78.125 |
31,344 | Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\frac{m}{n}$ where $m$ and $n$ are coprime integers and find $\log_2 n$. | 409 | 20.3125 |
31,345 | In 1980, the per capita income in our country was $255; by 2000, the standard of living had reached a moderately prosperous level, meaning the per capita income had reached $817. What was the annual average growth rate? | 6\% | 0.78125 |
31,346 | Determine the real value of $t$ that minimizes the expression
\[
\sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}.
\] | 7/2 | 0 |
31,347 | How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges. | 11 | 15.625 |
31,348 | Let $ABC$ be a triangle with $\angle BAC=117^\circ$ . The angle bisector of $\angle ABC$ intersects side $AC$ at $D$ . Suppose $\triangle ABD\sim\triangle ACB$ . Compute the measure of $\angle ABC$ , in degrees. | 42 | 69.53125 |
31,349 | In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once? | 1/2 | 4.6875 |
31,350 | Calculate the following product: $12 \times 0.5 \times 3 \times 0.2 =$
A) $\frac{20}{5}$
B) $\frac{22}{5}$
C) $\frac{16}{5}$
D) $\frac{18}{5}$
E) $\frac{14}{5}$ | \frac{18}{5} | 93.75 |
31,351 | In a press conference before a championship game, ten players from four teams will be taking questions. The teams are as follows: three Celtics, three Lakers, two Warriors, and two Nuggets. If teammates insist on sitting together and one specific Warrior must sit at the end of the row on the left, how many ways can the ten players be seated in a row? | 432 | 18.75 |
31,352 | Let $a$ and $b$ be real numbers bigger than $1$ . Find maximal value of $c \in \mathbb{R}$ such that $$ \frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c $$ | \frac{1}{3} | 10.15625 |
31,353 | A Gareth sequence is a sequence of numbers where each number after the second is the non-negative difference between the two previous numbers. For example, if a Gareth sequence begins 15, 12, then:
- The third number in the sequence is \(15 - 12 = 3\),
- The fourth number is \(12 - 3 = 9\),
- The fifth number is \(9 - 3 = 6\),
resulting in the sequence \(15, 12, 3, 9, 6, \ldots\).
If a Gareth sequence begins 10, 8, what is the sum of the first 30 numbers in the sequence? | 64 | 34.375 |
31,354 | In a regular 15-gon, three distinct segments are chosen at random among the segments whose end-points are the vertices. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
A) $\frac{345}{455}$
B) $\frac{100}{455}$
C) $\frac{310}{455}$
D) $\frac{305}{455}$
E) $\frac{450}{455}$ | \frac{345}{455} | 27.34375 |
31,355 | Given that a recipe calls for \( 4 \frac{1}{2} \) cups of flour, calculate the amount of flour needed if only half of the recipe is made. | 2 \frac{1}{4} | 92.1875 |
31,356 | Given that $α$ and $β$ are both acute angles, and $\cos (α+β)=\dfrac{\sin α}{\sin β}$, the maximum value of $\tan α$ is \_\_\_\_. | \dfrac{ \sqrt{2}}{4} | 12.5 |
31,357 | In quadrilateral $ABCD$, $\angle A = 120^\circ$, and $\angle B$ and $\angle D$ are right angles. Given $AB = 13$ and $AD = 46$, find the length of $AC$. | 62 | 0 |
31,358 | The equations
\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\]and
\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\]have a common rational root $r$ which is not an integer, and which is positive. What is $r?$ | \frac{1}{2} | 75 |
31,359 | What is the fifth-largest divisor of 3,640,350,000? | 227,521,875 | 0 |
31,360 | The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? | 17.67 | 42.96875 |
31,361 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{3}$, and $(3\overrightarrow{a} - 2\overrightarrow{b}) \perp \overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{6} | 98.4375 |
31,362 | Given an ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{\sqrt{3}}{2}$, and the length of its minor axis is $2$.
(I) Find the standard equation of the ellipse $C$.
(II) Suppose a tangent line $l$ of the circle $O: x^2 + y^2 = 1$ intersects curve $C$ at points $A$ and $B$. The midpoint of segment $AB$ is $M$. Find the maximum value of $|OM|$. | \dfrac{5}{4} | 7.03125 |
31,363 | Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?
*Roland Hablutzel, Venezuela*
<details><summary>Remark</summary>The original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?</details> | 2005 * 2004 * 2005 | 0 |
31,364 | Find all values of \( a \) such that the roots \( x_1, x_2, x_3 \) of the polynomial
\[ x^3 - 6x^2 + ax + a \]
satisfy
\[ \left(x_1 - 3\right)^3 + \left(x_2 - 3\right)^3 + \left(x_3 - 3\right)^3 = 0. \] | -9 | 35.15625 |
31,365 | There are eight envelopes numbered 1 to 8. Find the number of ways in which 4 identical red buttons and 4 identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red buttons is more than the sum of the numbers on the envelopes containing the blue buttons. | 31 | 82.8125 |
31,366 | In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 2.5$, then what is $BC$? | 2\sqrt{261} | 0 |
31,367 | A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them? | 144 | 10.15625 |
31,368 | In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the program list? (Answer with a number). | 1716 | 5.46875 |
31,369 | A line segment starts at $(2, 4)$ and ends at the point $(7, y)$ with $y > 0$. The segment is 6 units long. Find the value of $y$. | 4 + \sqrt{11} | 65.625 |
31,370 | Let planes \( \alpha \) and \( \beta \) be parallel to each other. Four points are selected on plane \( \alpha \) and five points are selected on plane \( \beta \). What is the maximum number of planes that can be determined by these points? | 72 | 5.46875 |
31,371 | Suppose we want to divide 12 puppies into three groups where one group has 4 puppies, one has 6 puppies, and one has 2 puppies. Determine how many ways we can form the groups such that Coco is in the 4-puppy group and Rocky is in the 6-puppy group. | 2520 | 90.625 |
31,372 | At a school cafeteria, Jenny wants to buy a meal consisting of one main dish, one drink, one dessert, and one side dish. The list below contains Jenny's preferred choices available:
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Main Dishes} & \textbf{Drinks} & \textbf{Desserts} & \textbf{Side Dishes} \\
\hline
Spaghetti & Water & Cookie & Salad \\
\hline
Turkey Sandwich & Juice & Cake & Fruit Cup \\
\hline
Veggie Burger & & & Chips \\
\hline
Mac and Cheese & & & \\
\hline
\end{tabular}
How many distinct possible meals can Jenny arrange from these options? | 48 | 62.5 |
31,373 | Let \( M(x, y, z) \) represent the minimum of the three numbers \( x, y, z \). If the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c > 0 \)) has a zero, determine the maximum value of \( M \left( \frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c} \right) \). | \frac{5}{4} | 3.125 |
31,374 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 96.875 |
31,375 | Let the function $f(x)=a\ln x-x- \frac {1}{2}x^{2}$.
(I) For $a=2$, find the extreme values of the function $f(x)$.
(II) Discuss the monotonicity of the function $f(x)$. | -\frac{3}{2} | 40.625 |
31,376 | Given the function $y=\cos(2x+ \frac {\pi}{3})$, determine the horizontal shift required to obtain this function from the graph of $y=\sin 2x$. | \frac{5\pi}{12} | 28.90625 |
31,377 | The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$. | \frac{2}{5} | 4.6875 |
31,378 | The numbers from 1 to 9 are placed in the cells of a 3x3 table so that the sum of the numbers on one diagonal is 7 and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? | 25 | 47.65625 |
31,379 | A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 16 | 46.875 |
31,380 | A plane flies between two cities. With the tailwind, it takes 5 hours and 30 minutes, and against the wind, it takes 6 hours. Given that the wind speed is 24 kilometers per hour, and assuming the plane's flying speed is $x$ kilometers per hour, then the speed of the plane with the tailwind is kilometers per hour, and the speed of the plane against the wind is kilometers per hour. | 528 | 61.71875 |
31,381 | Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$ | 27/4 | 0.78125 |
31,382 | Given $\cos \alpha= \frac {1}{3}$, $\cos (\alpha+\beta)=- \frac {1}{3}$, and $\alpha$, $\beta\in(0, \frac {\pi}{2})$, find the value of $\cos (\alpha-\beta)$. | \frac{23}{27} | 57.8125 |
31,383 | Segments $BD$ and $AE$ intersect at $C$, with $AB = BC$ and $CD = DE = EC$. Additionally, $\angle A = 4 \angle B$. Determine the degree measure of $\angle D$.
A) 45
B) 50
C) 52.5
D) 55
E) 60 | 52.5 | 8.59375 |
31,384 | A factory conducted a survey on the defective parts produced by a team. The number of defective parts produced by the team each day in 7 days is as follows (unit: pieces): 3, 3, 0, 2, 3, 0, 3. What is the value of the variance of the number of defective parts produced by the team in 7 days? | \frac{12}{7} | 31.25 |
31,385 | If there are exactly $3$ integer solutions for the inequality system about $x$: $\left\{\begin{array}{c}6x-5≥m\\ \frac{x}{2}-\frac{x-1}{3}＜1\end{array}\right.$, and the solution to the equation about $y$: $\frac{y-2}{3}=\frac{m-2}{3}+1$ is a non-negative number, find the sum of all integers $m$ that satisfy the conditions. | -5 | 62.5 |
31,386 | Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\frac{a}{b}$ is ______. | -\frac{15}{4} | 5.46875 |
31,387 | Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$. | \frac{5}{2} | 2.34375 |
31,388 | Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged. | 144 | 13.28125 |
31,389 | The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS. | 24 | 5.46875 |
31,390 | Given that there are 10 streetlights numbered from 1 to 10, two of which will be turned off under the conditions that two adjacent lights cannot be turned off at the same time and the lights at both ends cannot be turned off either, calculate the number of ways to turn off the lights. | 21 | 81.25 |
31,391 | Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$ . | 111 | 0 |
31,392 | Increase Grisha's yield by 40% and Vasya's yield by 20%.
Grisha, the most astute among them, calculated that in the first case their total yield would increase by 1 kg; in the second case, it would decrease by 0.5 kg; in the third case, it would increase by 4 kg. What was the total yield of the friends (in kilograms) before their encounter with Hottabych? | 15 | 8.59375 |
31,393 | A certain real estate property is holding a lottery for homebuyers, with the following rules: For homeowners who purchase the property, they can randomly draw 2 balls from box $A$, which contains 2 red balls and 2 white balls, and 2 balls from box $B$, which contains 3 red balls and 2 white balls. If all 4 balls drawn are red, the homeowner wins the first prize and receives $10,000 in renovation funds; if exactly 3 balls are red, the homeowner wins the second prize and receives $5,000 in renovation funds; if exactly 2 balls are red, the homeowner wins the third prize and receives $3,000 in renovation funds; any other outcome is considered a consolation prize, awarding $1,500 in renovation funds.
$(Ⅰ)$ Three homeowners participate in the lottery. Find the probability that exactly one homeowner wins the second prize.
$(Ⅱ)$ Let $X$ denote the amount of renovation funds won by a homeowner in the lottery. Find the probability distribution and the expected value of $X$. | 3,675 | 0 |
31,394 | A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain? | 20 | 17.96875 |
31,395 | Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point?
(Alexey Panasenko) | 2.4 | 18.75 |
31,396 | Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors. | 162000 | 6.25 |
31,397 | Given that $x$ and $y$ are positive integers, and $x^2 - y^2 = 53$, find the value of $x^3 - y^3 - 2(x + y) + 10$. | 2011 | 63.28125 |
31,398 | In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 15$, $\tan Z = \frac{4}{3}$, and $\tan X = \frac{3}{2}$, what is the length of $XY$? | \frac{20\sqrt{13}}{3} | 21.875 |
31,399 | Solve the following equations:
(1) $x^{2}-3x=4$;
(2) $x(x-2)+x-2=0$. | -1 | 0 |
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