problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Given the function $f(x) = x^2 + 4x + 6$, determine the range of $f(x)$ on the interval $[-3, 0)$. | [2, 6) | math | 38 |
Given $|a|=3$, $|b|=4$, and $(a+b)\cdot(a+3b)=33$, find the angle between $a$ and $b$. | 120^{\circ} | math | 38 |
Let $a$ and $b$ be real numbers such that $a^2 + b^2 \geq 1$. Find the maximum value of $a \sin \theta + b \cos \theta$, for $\theta \in [0, 2\pi]$. | \sqrt{a^2 + b^2} | math | 59 |
If \(3 \sin^2 \alpha + 2 \sin^2 \beta - 2 \sin \alpha = 0\), calculate the range of values for \(\cos^2 \alpha + \cos^2 \beta\). | \left[ \frac{14}{9}, 2 \right] | math | 52 |
The number of integers whose absolute value is less than 4, their sum, and their product are. | 0 | math | 21 |
Evaluate the expression
\[
\left(d^d - d(d-2)^d + (d-1)!\right)^2
\]
when $d = 4$. | 39204 | math | 38 |
What is the lowest prime number that is thirteen more than a cube? | 229 | math | 14 |
Given that the sum of the interior numbers in the sixth row of Pascal's Triangle is 30, calculate the sum of the interior numbers in the eighth row. | 126 | math | 33 |
A regular tetrahedron $ABCD$ has its centroid marked as point $P$. Compute the ratio $\frac{s}{S}$ where:
- $s$ is the sum of the distances from $P$ to the planes containing $DAB$, $DBC$, and $DCA$.
- $S$ is the sum of the distances from $P$ to the lines containing $AB$, $BC$, and $CA$.
A) $\frac{\sqrt{2}}{3}$
B) $1$
C... | \frac{\sqrt{2}}{2} | math | 134 |
Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$ . Find $100m+n$ .
[i]Proposed by Michael Tang | 4306 | math | 75 |
Given triangle $\triangle ABC$, $P$ and $Q$ are points on sides $AB$ and $BC$ respectively, such that $\overrightarrow{AP}=\frac{1}{3}\overrightarrow{AB}$ and $\overrightarrow{BQ}=\frac{1}{3}\overrightarrow{BC}$. If $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{AC}=\overrightarrow{b}$, find the vector $... | \frac{1}{3}\overrightarrow{a}+\frac{1}{3}\overrightarrow{b} | math | 105 |
The domain of the function $y=\dfrac{1}{{\log _{2}}(x-2)}$ | (2,3)\cup (3,+\infty ) | math | 26 |
The opening ceremony of the 2022 Beijing Winter Olympics was held at the National Stadium in China on the evening of February 4th. The TV broadcast audience reached 316 million people. The number written on the blank line is: ______ people. The closing ceremony was held on the evening of February 20th, with an audience... | 2.4 \text{ billion people} | math | 91 |
Shifting the graph of the function $f(x)=\sin (2x- \frac {\pi}{6})$ to the left by $\frac {\pi}{3}$ units to obtain the graph of the function $g(x)$, find the equation of one of the axes of symmetry of the graph of $g(x)$. | \frac {\pi}{2} | math | 68 |
\( T = \sqrt{(2021 + 2021) + (2021 - 2021) + (2021 \times 2021) + (2021 \div 2021)} \). What is the largest prime factor of \( T \)? | 337 | math | 71 |
In a fuel tank, there are 40 liters of oil. The oil flows out of the tank uniformly at a rate of 0.2 liters per minute. Determine the function relationship between the remaining oil quantity Q (in liters) in the tank and the time t (in minutes) of outflow. | Q=40-0.2t | math | 63 |
A square garden is completely covered with identical square tiles. Each diagonal of the garden has an equal number of tiles. If the total number of tiles that lie across both diagonals is 25, how many tiles cover the entire garden? | 169 | math | 48 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{(a^2 + 4a + 2)(b^2 + 4b + 2)(c^2 + 4c + 2)}{abc}.\] | 216 | math | 64 |
Given $p: 1 \leq x < 3$; $q: x^2 - ax \leq x - a$; If $\neg p$ is a sufficient condition for $\neg q$, find the range of the real number $a$. | [1, 3) | math | 55 |
Given a point Q(2, 0) on the Cartesian coordinate plane and a circle C: $x^2+y^2=1$, the length of the tangent from a moving point M to circle C equals the sum of the radius of circle C and the distance |MQ|. Find the equation of the trajectory of the moving point M. | 3x^2-y^2-8x+5=0 | math | 70 |
The equation of circle A is $x^2+y^2-2x-2y-7=0$, and the equation of circle B is $x^2+y^2+2x+2y-2=0$. Determine whether circle A and circle B intersect. If they intersect, find the equation of the line passing through the two intersection points and the distance between these two points; if they do not intersect, expla... | \frac{\sqrt{238}}{4} | math | 91 |
Find the number of odd digits in the base-4 representation of $523_{10}$. | 2 | math | 22 |
The side \( AB \) of the trapezoid \( ABCD \) is divided into five equal parts, and through the third division point, counting from point \( B \), a line is drawn parallel to the bases \( BC \) and \( AD \). Find the segment of this line that is enclosed between the sides of the trapezoid, given \( BC = a \) and \( AD ... | \frac{2a + 3b}{5} | math | 88 |
Given a box containing 3 screw base bulbs and 7 bayonet base bulbs, all of which have the same shape and power and are placed with their bases down, calculate the probability that the electrician gets a bayonet base bulb on his third try. | \frac{7}{120} | math | 52 |
Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in "random" mode, he can listen to the 11 tracks in a ... | 7/11 | math | 95 |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | 24 | math | 28 |
Suppose in February a city recorded a total rainfall of 280 inches. Calculate the average rainfall in inches per hour during that month. | \frac{5}{12} | math | 29 |
The graph of the function $f(x)=\sin (\omega x+\frac{π}{3})$ ($\omega\ \ \gt 0$) is shifted to the left by $\frac{π}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, then determine the minimum value of $\omega$. | \frac{1}{3} | math | 77 |
The function $g(x)$ satisfies
\[g(x - y) = g(x)g(y)\] for all real numbers $x$ and $y,$ and $g(x) \neq 0$ for all real numbers $x.$ Find $g(2).$ | 1 | math | 58 |
Both $a$ and $b$ are positive integers, and $b > 1$. When $a^b$ is the greatest possible value less than 500, and additionally, $a + b$ is even, what is the sum of $a$ and $b$? | 24 | math | 61 |
A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, calculate the total number of smaller cubes. | 20 | math | 48 |
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\textdollar10500$ for both taxes. How many dollars was his inheritance? | 37500 | math | 70 |
Given a positive integer \(N\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \(N\), including \(N\) itself. For example, the integer substrings of 3208 are \(3, 2, 0, 8, 32, 20, 320, 208\), and 3208. (The substring 08 is omitted from... | 88,888,888 | math | 162 |
Using the oblique projection method, we get:
① The intuitive diagram of a triangle must be a triangle;
② The intuitive diagram of a square must be a rhombus;
③ The intuitive diagram of an isosceles trapezoid can be a parallelogram;
④ The intuitive diagram of a rhombus must be a rhombus.
Among the above conclu... | ① | math | 92 |
A frog eats 80 pests every day. Based on this calculation, how many pests do 5 frogs eat in a week? | 2800 | math | 27 |
Given the set of numbers $\{ 1,2,3,4,5,6,7,8,9 \}$, and a target sum of $15$, determine the number of sets of three different numbers that contain $5$. | 4 | math | 51 |
Given the inequality in terms of x, $x^2 - 2ax + a^2 - ab + 4 \leq 0$ has exactly one solution, determine the minimum value of $a^2 + b^2$. | 8 | math | 50 |
Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points? | 1.5 | math | 76 |
If line $l: x+y-2=0$ intersects circle $C: x^2+y^2-2x-6y+2=0$ at points $A$ and $B$, then the area of $\triangle ABC$ is. | 2\sqrt{3} | math | 53 |
Each of the 5 brothers owns a piece of land. One day they pooled their money, bought their neighbor's garden, and divided the new land equally among themselves. As a result, Andrey's plot increased by $10\%$, Boris's by $\frac{1}{15}$, Vladimir's by $5\%$, Grigory's by $4\%$, and Dmitry's by $\frac{1}{30}$. By what per... | 5\% | math | 105 |
The value range of the function $f(x)=\sin x-\cos \left(x+ \frac {\pi}{6}\right)$, where $x\in[0,\pi]$, is ______. | \left[- \frac { \sqrt {3}}{2}, \sqrt {3}\right] | math | 43 |
Express 33 as the sum of \(n\) consecutive natural numbers. When \(n\) is maximized, replace all "+" signs in the resulting sum with "×". What is the product of the terms in this expression? | 20160 | math | 46 |
The 31st World University Summer Games will be held in Chengdu, Sichuan from July 28th to August 8th, 2023. A company decided to evaluate a certain product under its umbrella for bidding for related endorsement activities. The original selling price of the product was $25 per unit, with an annual sales volume of 80,000... | 30 \text{ dollars} | math | 301 |
Which of the numbers is greater: \(31^{11}\) or \(17^{14}\)? | 17^{14} | math | 24 |
Given \\(a > 1\\), when the graphs of \\(y=a^{x}\\) and \\(y=\log_{a}x\\) have exactly one common point, \\(\ln \ln a=\\) \_\_\_\_\_\_. | -1 | math | 55 |
Each bag of sugar has a standard weight of $500$ grams. The quality supervision department wants to understand the weight condition of a batch of sugar. They randomly selected $9$ bags and weighed each bag (in grams) as follows:
$490$ $495$ $493$ $498$ $499$ $500$ $503$ $507$ $506$
(I) Calculate the mea... | 1 | math | 152 |
Given in the geometric sequence $\{a_n\}$, it is always true that $a_{n+1} > a_n$ for $n\in\mathbb{N}^*$, and $a_1a_4=8$, $a_2+a_3=6$.
(I) Find the general formula for the sequence $\{a_n\}$.
(II) If the sequence $\{b_n\}$ satisfies $\dfrac{a_1}{b_1} + \dfrac{3a_2}{b_2} + \ldots + \dfrac{(2n-1)a_n}{b_n} = n$ ($n\in\m... | S_n=(2n-3)\cdot2^n+3 | math | 178 |
Given the hyperbola $C$: $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with asymptote equations $y = \pm \frac{1}{2}x$, calculate the eccentricity of the hyperbola $C$. | \sqrt{5} | math | 76 |
How many ordered triples of positive integers $(a, b, c)$ are there for which $a^{4} b^{2} c=54000$ ? | 16 | math | 36 |
A pet store has 15 puppies, 10 kittens, and 8 hamsters. Alice, Bob, and Charlie each want to buy a pet. For variety, they each prefer to buy a different type of pet. How many ways can Alice, Bob, and Charlie buy their preferred pets and leave the store satisfied assuming they all pick a different type of pet? | 7200 | math | 76 |
A solid cube of side length $5$ is removed from each corner of a solid cube of side length $10$. Calculate the total number of edges of the remaining solid. | 60 | math | 36 |
Bill can buy pans, pins, and pons for $\$3$, $\$5$ and $\$8$ each, respectively. What is the largest number of pons he can purchase if he must buy at least one of each item and will spend exactly $\$100$? | 11 | math | 60 |
Given a Ferris wheel with a radius of 30 feet, if it makes one complete revolution every 90 seconds, determine the time in seconds for a rider to move from the bottom of the wheel to a point 15 feet above the bottom. | 30 | math | 53 |
In triangle $ABC$ with $\angle ABC = 90^\circ$, $AB = 90$, and $AC = 150$. Point $D$ is the midpoint of $AC$. Calculate the area of $\triangle ABD$. Use rounded final values. | 3375 | math | 57 |
Assume we have a modified calendrical system in which leap years normally happen every four years, except every 100th year is not a leap year unless it is also divisible by 400. In a 200-year period, starting with year 1 which is a leap year, what is the maximum possible number of leap years? | 48 | math | 73 |
Nine cookies of radius $0.5$ inches are cut from a circle of cookie dough. The layout has eight cookies surrounding one centered cookie, all equally spaced and each outer cookie tangent to the central cookie as well as adjacent cookies. The diameter of the circle from which cookies are cut measures exactly twice as muc... | \sqrt{6.75} | math | 122 |
Using pennies, nickels, dimes, quarters, and half dollars, calculate the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar. | 6 | math | 38 |
A plank of wood has one end, \( A \), against a vertical wall. Its other end, \( B \), is on horizontal ground. When end \( A \) slips down \( 8 \text{ cm} \), end \( B \) moves \( 4 \text{ cm} \) further away from the wall. When end \( A \) slips down a further \( 9 \text{ cm} \), end \( B \) moves a further \( 3 \tex... | 65 \, \text{cm} | math | 120 |
The center of the upper base of a cube with edge length $a$ is connected to the midpoints of the sides of the lower base, which are also connected in a sequential order. Calculate the total surface area of the resulting pyramid. | 2a^2 | math | 48 |
Given the ellipse $C: \frac{x^2}{16} + \frac{y^2}{12} = 1$ with foci $F_1$ and $F_2$, find the largest angle in $\triangle F_1MF_2$ for a point $M$ that lies on $C$ and satisfies $|MF_1| - |MF_2| = 2$. | 90^{\circ} | math | 88 |
The sum of the first n terms of an arithmetic sequence {a_n} is S_n. If a_1=2 and S_3=12, calculate the value of a_5. | 10 | math | 41 |
A box contains 5 labels marked with the numbers $1$, $2$, $3$, $4$, $5$. If 3 labels are randomly drawn from the box, what is the probability that the average of the numbers on the drawn labels is $3$? | \frac{1}{5} | math | 55 |
Given that YZ and XZ form the legs of a right triangle XYZ, where YZ = 3 and XZ = d, and a new right triangle XYW is constructed externally sharing hypotenuse XY, with XW = 4, find the length of YW. | \sqrt{d^2 - 7} | math | 58 |
The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. Determine the number of such pairs. | 3 | math | 29 |
Find all real numbers \( x \) which satisfy \[\frac{x-2}{x-4} \ge 3.\] | (4, 5] | math | 27 |
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tre... | 140 | math | 121 |
Find the standard equation of the hyperbola with the same asymptotes as the hyperbola $x^2 - 2y^2 = 2$ and that passes through the point M(2, -2). | \frac{y^2}{2} - \frac{x^2}{4} = 1 | math | 46 |
For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ *Proposed by Melih Üçer* | 2 | math | 51 |
A right circular cone is inscribed in a right rectangular prism. The base of the cone is tangent to all four sides of the rectangular base of the prism. If the base of the rectangle has side lengths $a$ and $b$, and the height of both the prism and the cone is $h$, what is the ratio of the volume of the cone to the vol... | \frac{\pi}{12} | math | 92 |
Two circles are placed outside a square $ABCD$. The first circle is tangent to side $\overline{AB}$, and the second is tangent to side $\overline{CD}$. Both circles are tangent to the lines extended from $BC$ and $AD$. Calculate the ratio of the area of the first circle to that of the second circle. | 1 | math | 72 |
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$ . What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$ ?
*2018 CCA Math Bonanza Lightning Round #1.3* | 45^\circ | math | 81 |
Given a quadratic equation $ax^2+bx+c=0$ whose sum of roots is $p$, the sum of squares of the roots is $q$, and the sum of cubes of the roots is $r$, calculate the value of $ar+bq+cp$. | 0 | math | 56 |
Between 1000 and 9999, how many four-digit integers with all different digits have an absolute difference of 2 between the first and last digits? | 840 | math | 36 |
Alice and Bob play a similar game with a basketball. On each turn, if Alice has the basketball, there is a 5/8 chance that she will toss it to Bob and a 3/8 chance that she will keep the basketball. If Bob has the basketball, there is a 1/4 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he ke... | \frac{19}{64} | math | 106 |
Convert the binary number $11011100_{(2)}$ to a decimal number, and then convert it to an octal number. | 334 | math | 33 |
Given vectors in the plane $\overset{→}{OA}=(-1,-3)$, $\overset{→}{OB}=(5,3)$, and $\overset{→}{OM}=(2,2)$, point $P$ is on line $OM$, and $\overset{→}{PA} \cdot \overset{→}{PB}=-16$.
$(1)$ Find the coordinates of $\overset{→}{OP}$.
$(2)$ Find the cosine of $\angle APB$.
$(3)$ Let $t \in \mathbb{R}$, find the min... | \sqrt{2} | math | 152 |
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$ | 2148 | math | 114 |
In a square \(ABCD\), let \(P\) be a point on the side \(BC\) such that \(BP = 3PC\) and \(Q\) be the midpoint of \(CD\). If the area of the triangle \(PCQ\) is 5, what is the area of triangle \(QDA\)? | 20 | math | 67 |
Transform the expression:
$$
\frac{x^{3}+x^{2}-4 x-4}{(x-1)(x-3)+3 x-5}=\frac{(x+1)\left(x^{2}-4\right)}{x^{2}-x-2}=\frac{(x+1)(x-2)(x+2)}{(x-2)(x+1)}
$$
$$
\begin{gathered}
f(x)=\left|\frac{x^{3}+x^{2}-4 x-4}{(x-1)(x-3)+3 x-5}\right|=|x+2| \\
D(f): x \neq 2, x \neq -1;
\end{gathered}
$$
$$
p(x)=\sqrt{x^{2}}+a=|x|+a... | a \in (-2, 0) \cup (0, 2) | math | 462 |
There are $169$ lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly $19$ switches at once. (Every time you use this remote control, you can choose which $19$ switches are to be changed.)
(a) Given that at the beginning some lamps are on, can you turn all the la... | 9 | math | 119 |
Given $p$: $|x-a|<4$, $q$: $-x^2+5x-6>0$, and $q$ is a sufficient but not necessary condition for $p$, then the range of values for $a$ is \_\_\_\_\_\_. | [-1, 6] | math | 60 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus is point $F$, and point $A$ lies on the asymptote of the hyperbola. Triangle $OAF$ is an equilateral triangle with a side length of $2$ (where $O$ is the origin). Find the equation of the hyperbola. | x^2 - \frac{y^2}{3} = 1 | math | 103 |
Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | \frac{7\pi}{4} | math | 56 |
Simplify $8 \cdot \frac{15}{4} \cdot \frac{-28}{45}$. | -\frac{56}{3} | math | 26 |
Let $c_1, c_2, \ldots$ and $d_1, d_2, \ldots$ be arithmetic progressions such that $c_1 = 10, d_1 = 90$, and $c_{50} + d_{50} = 500$. Find the sum of the first fifty terms of the progression $c_1 + d_1, c_2 + d_2, \ldots$ | 15000 | math | 102 |
Given that $a_n$ is the coefficient of the $x^2$ term in the expansion of $(2+x)^n$ (where $n \in \mathbb{N}^*$, $n \geq 2$, and $x \in \mathbb{R}$), find the limit as $n$ approaches infinity:
$$
\lim_{n \to \infty}\left( \frac{2^{2}}{a_{2}} + \frac{2^{3}}{a_{3}} + \ldots + \frac{2^{n}}{a_{n}} \right).
$$ | 8 | math | 132 |
A novel begins on page 101 and ends on page 599. Each page number is printed once in the book. How many more times is the digit 5 printed compared to the digit 2 in the page numbers? | 0 | math | 49 |
Find the remainder when $3 \times 13 \times 23 \times ... \times 93$ and $7 \times 17 \times 27 \times ... \times 97$ multiplied together is divided by $8$. | 1 | math | 55 |
Let $O$ be the origin, and $(2a, 2b, 2c)$ be a fixed point. A plane passes through $(2a, 2b, 2c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A$, $B$, and $C$, respectively. Assume none of these points coincides with $O$ and let $(p, q, r)$ be the center of the sphere passing through $A$, $B$, $C$, $... | 1 | math | 145 |
The additive inverse of $a$ is itself, and the multiplicative inverse of $b$ is itself. Then, calculate the value of $a+b$. | 1 \text{ or }-1 | math | 32 |
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$ | (2, 4, 8) | math | 53 |
Given that $a$ and $b$ are positive numbers satisfying the equation $a + 2b = 1$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | 3 + 2\sqrt{2} | math | 46 |
For how many integers $n$ where $2 \le n \le 100$ is $\binom{n}{2}$ odd? | 50 | math | 30 |
$2.5$,$-0.5252252225\ldots$ (add a $2$ between every two $5$s),$-10^{2}$,$-5$,$0$,$\frac{1}{3}$,$|-4|$,$-3.6$,$\frac{π}{2}$.<br/>$(1)$Negative numbers:$\{ \_\_\_\_\_\_\ldots \}$;<br/>$(2)$Non-negative integers:$\{ \_\_\_\_\_\_\ldots \}$;<br/>$(3)$Fractions:$\{ \_\_\_\_\_\_\ldots \}$;<br/>$(4)$Irrational numbers:$\{ \_\... | -0.5252252225\ldots, \frac{π}{2} | math | 168 |
Let $p(x)$ be the polynomial of degree at most 8 such that
\[p(n) = \frac{1}{n}\]for $n = 1,$ 2, 3, $\dots,$ 9. Find $p(10).$ | \frac{1}{5} | math | 58 |
Let \( p, \) \( q, \) and \( r \) be positive real numbers. Find the minimum value of
\[
\frac{5r}{3p + q} + \frac{5p}{q + 3r} + \frac{2q}{p + r}.
\] | 4 | math | 66 |
Let \( F \) be the set consisting of all functions \( f \) such that \( f: P(S) \rightarrow \mathbb{R} \) and for all \( X, Y \subseteq P(S) \), \( f(X \cap Y) = \min (f(X), f(Y)) \), where \( S \) is a finite set and \( P(S) \) is the power set of \( S \). Find \( \max _{f \in F} \mid \operatorname{Im}(f) \mid \), whe... | n+1 | math | 138 |
Calculate:<br/>$(1)\frac{5}{7}+(-\frac{5}{6})-(-\frac{2}{7})+1\frac{1}{6}$;<br/>$(2)(\frac{1}{2}-1\frac{1}{3}+\frac{3}{8})÷(-\frac{1}{24})$;<br/>$(3)\left(-3\right)^{3}+\left(-5\right)^{2}-|-3|\times 4$;<br/>$(4)-{1}^{101}-[-0.5-(1-\frac{3}{5}×0.7)÷{(-\frac{1}{2})}^{2}]$. | \frac{91}{50} | math | 158 |
Given positive integers \( k, m, n \) such that \( 1 \leqslant k \leqslant m \leqslant n \), find the value of
$$
\sum_{i=0}^{n}(-1)^{i} \frac{1}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}
$$
and describe the calculation process. | 0 | math | 96 |
Three different numbers are randomly selected from the set $T = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$. The probability that their sum is 15 would be greater if the number $m$ had first been removed from set $T$. What is the value of $m$? | 5 | math | 94 |
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