problem stringlengths 14 4.09k | solution stringlengths 1 802k | task_type stringclasses 3
values | problem_tokens int64 5 895 |
|---|---|---|---|
Determine how many $4 \times 4$ arrays containing all the numbers from 1 to 16 can be formed such that each row and column contains numbers in strictly increasing order.
A) 24
B) 36
C) 48
D) 54
E) 60 | 36 | math | 68 |
A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number. | 178\% | math | 70 |
If the inequality $x + \frac{4}{x - a} \geq 5$ holds for all $x \in (a, +\infty)$, find the minimum value of the real number $a$. | 1 | math | 48 |
Given that the average score for the morning group is 90 and the evening group is 80, and the ratio of the number of students in the morning group to the evening group is $\frac{4}{5}$, calculate the overall average score of the students from both groups. | \frac{760}{9} | math | 59 |
Given a monotonically increasing geometric sequence $\{a_n\}$ satisfies: $a_2+a_3+a_4=28$, and $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) If $b_n=a_n\cdot \log_{\frac{1}{2}}a_n$, $S_n=b_1+b_2+\ldots+b_n$, find the smallest positive integer $n$ suc... | 5 | math | 138 |
If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ . | 1/9 | math | 68 |
(1) Given $f(x)=x^{3}-2f'(1)x$, then $f'(1)=$_______
(2) Given the function $f(x)=a^{2}x-2a+1$, "If $\forall x \in (0,1)$, $f(x) \neq 0$" is a false statement, then the range of $a$ is_______
(3) Given in $\Delta ABC$, $AC=\sqrt{2}$, $BC=\sqrt{6}$, the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$. If there exists a p... | (3,+\infty) | math | 259 |
A six-digit number 1234xy is divisible by both 8 and 9. Given that \( x + y = c \), find the value of \( c \). | 8 | math | 38 |
In triangle $ABC$ with side ratio $AB : AC = 4 : 3$, the angle bisector of $\angle BAC$ intersects side $BC$ at point $L$. Find the length of segment $AL$ if the length of vector $3 \cdot \overrightarrow{AB} + 4 \cdot \overrightarrow{AC}$ is equal to 2016. | 288 | math | 82 |
Given that one asymptote of the hyperbola $\frac{x^{2}}{a} - \frac{y^{2}}{2} = 1$ is $2x - y = 0$, find the real number $a$. | \frac{1}{2} | math | 51 |
If the price of a product increased from 5.00 reais to 5.55 reais, what was the percentage increase? | 11\% | math | 30 |
Given the function $f(x)=\sin(2x+\frac{5π}{6})-\cos^2x+1$.
$(1)$ Find the minimum value and the interval of monotonic increase of the function $f(x)$.
$(2)$ Let angles $A$, $B$, and $C$ be the three interior angles of $\triangle ABC$. If $\cos B=\frac{1}{3}$ and $f(\frac{C}{2})=-\frac{1}{4}$, find $\sin A$. | \frac{2\sqrt{2} + \sqrt{3}}{6} | math | 112 |
Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. What is the probability that at the end of t... | \frac{1}{4} | math | 125 |
At a conference with $30$ businessmen, fifteen businessmen drank coffee, thirteen drank tea, and eight drank soda. Six businessmen drank both coffee and tea, two drank both coffee and soda, and three drank both tea and soda. One businessman drank all three beverages. How many businessmen drank none of the beverages? | 4 | math | 63 |
Given the function $f(x) = ax + \cos x$ is monotonically decreasing on $\mathbb{R}$, determine the range of the real number $a$. | (-\infty,-1] | math | 37 |
Given a circle of radius $3$, determine the area of the region consisting of all line segments of length $6$ that are tangent to the circle at their midpoints. | 9\pi | math | 35 |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | 758 | math | 97 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, determine the measure of angle $B$. | \frac{3\pi}{10} | math | 70 |
Let $a$ be a real number, and the function $f(x) = (x-a)^2 + |x-a| - a(a-1)$.
1. If $f(0) \leq 1$, find the range of values for $a$;
2. Discuss the monotonicity of $f(x)$;
3. When $a > 2$, discuss the number of zeros of $f(x) + |x|$ on $\mathbb{R}$. | 2 | math | 102 |
A $\textit{palindrome}$ is a number that reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 52,712 to create a palindrome? | 113 | math | 61 |
A bag contains 13 balls of each of 4 different colors. How many balls must be taken out to ensure that among the balls taken out, there are at least 3 balls of different colors? | 27 | math | 42 |
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$ , there exist three elements $x,y,z$ in $A$ such that $x=a+b$ , $y=b+c$ , $z=c+a$ , where $a,b,c$ are in $S$ and are distinct integers.
*Proposed by Huawei Zhu* | 1008 | math | 117 |
A student who was "careless" during addition and subtraction operations mistakenly wrote "-5" as "+5" and performed the calculation. Calculate the difference between the incorrect answer and the correct answer. | 10 | math | 39 |
The expression \( x + \frac{1}{x} \) has a maximum for \( x < 0 \) and a minimum for \( x > 0 \). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \( x + \frac{1}{x} \). | 8 | math | 73 |
Given the set of numbers $\{-5, -4, -1, 3, 7, 9\}$, calculate the largest possible product when three different numbers from this set are multiplied together. | 189 | math | 42 |
Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time is 18 seconds. Given t... | 44 | math | 122 |
The real numbers $x, y, z$ satisfy $0 \leq x \leq y \leq z \leq 4$. If their squares form an arithmetic progression with common difference 2, determine the minimum possible value of $|x-y|+|y-z|$. | 4-2\sqrt{3} | math | 61 |
In the Cartesian coordinate system, $P\left(m,n\right)$ is an intersection point of the linear function $y=x-2022$ and the inverse proportion function $y=-\frac{{2022}}{x}$. Evaluate the algebraic expression $\frac{{2022}}{m}+\frac{{{m^2}-2022m}}{n}$. | 2022 | math | 85 |
Find the maximum real number \(\lambda\) such that for the real-coefficient polynomial
$$
f(x) = x^3 + ax^2 + bx + c
$$
with all roots being non-negative real numbers, the inequality
$$
f(x) \geqslant \lambda(x - a)^3 \quad \text{for all} \; x \geqslant 0
$$
holds. Also, determine when equality holds in this inequality... | -\frac{1}{27} | math | 100 |
Find the degree measure of the least positive angle $\theta$ for which
\[\cos 10^\circ = \sin 20^\circ + \sin \theta.\] | 40^\circ | math | 38 |
Given a hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \), where \( A_{1} \) and \( A_{2} \) are the left and right vertices of hyperbola \( C \), and \( F \) is a focus of \( C \). Let \( O \) be the origin, with a circle centered at \( O \) having a radius \( |OF| \). The circle intersects with one of the ... | \frac{\sqrt{21}}{3} | math | 154 |
Given the parabola C: x²=2py (p>0), where P is a point on the parabola, the tangent line at point P intersects the y-axis at point Q, and F is the focus of the parabola C. If |PF|=5, determine the value of |QF|. | 5 | math | 68 |
5 students stand in a row. If student A cannot stand at the far left and student B cannot stand at the far right, then calculate the total number of different arrangements. | 78 | math | 35 |
Little Ming has 10 math problems to be completed in ascending order of their serial numbers, and he must complete at least one problem per day. Find the total number of different ways Little Ming can complete the problems. | 512 | math | 44 |
Given that $f(x)$ is an even function that is increasing on $[0,+\infty)$, and $f(1)=0$, determine the range of $x$ that satisfies $f(\log_{\frac{1}{2}}x) > 0$. | (0, \frac{1}{2}) \cup (2, +\infty) | math | 58 |
If $2008=2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}}$, where $a_{1}, a_{2}, \cdots, a_{n}$ are distinct non-negative integers, then the relationship between $\sin \sum_{i=1}^{n} a_{i}$, $\cos \sum_{i=1}^{n} a_{i}$, and $\tan \sum_{i=1}^{n} a_{i}$ is ______. (Given that $\pi=3.141592 \cdots$) | \sin \sum_{i=1}^{n} a_{i} > \tan \sum_{i=1}^{n} a_{i} > \cos \sum_{i=1}^{n} a_{i} | math | 132 |
Find the equation of the line that passes through the point $(-1, 3)$ and is perpendicular to the line $x - 2y + 3 = 0$. | 2x + y - 1 = 0 | math | 37 |
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$. | 9999 | math | 41 |
On the door, BIOLOGY is spelled out with 7 magnets, one letter per magnet. Two vowels and two consonants fall off and are put away in a bag. If the O's are indistinguishable, how many distinct possible collections of letters could be put in the bag? | 12 | math | 58 |
A parallelepiped $PQRS TUVW$ is generated by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PT},$ as defined below. Compute the ratio:
\[\frac{PT^2 + QU^2 + RV^2 + SW^2}{PQ^2 + PR^2 + PT^2}.\] | 4 | math | 81 |
Given that $\alpha$ is an angle in the first quadrant, and $\cos\alpha = \frac{2\sqrt{5}}{5}$, calculate the value of $\cos 2 \alpha - \frac{\cos \alpha}{\sin \alpha}$. | -\frac{7}{5} | math | 56 |
Let \( p(x) \) be a polynomial of degree \( 3n \) such that \( p(0) = p(3) = \cdots = p(3n) = 2 \), \( p(1) = p(4) = \cdots = p(3n-2) = 1 \), \( p(2) = p(5) = \cdots = p(3n-1) = 0 \), and \( p(3n+1) = 730 \). Find \( n \). | 4 | math | 120 |
The standard equation of the parabola with its focus at the right focus of the ellipse $\frac{x^2}{3} + y^2 = 1$ is __________. | y^2 = 4\sqrt{2}x | math | 38 |
Among 5 lottery tickets with 1 winning ticket, 5 people take turns drawing one ticket each to determine who gets the winning ticket. Assuming that those who draw later do not know the results of the previous draws, do all individuals have an equal probability of drawing the winning ticket, regardless of their position ... | \frac{1}{5} | math | 65 |
Given the vertices of triangle ABC are A(-4, 0), B(4, 0), and the perimeter of triangle ABC is 18, determine the equation of the trajectory of vertex C. | \frac {x^{2}}{25}+ \frac {y^{2}}{9}=1 | math | 42 |
The lengths of the sides of an acute triangle form an arithmetic progression with a common difference of 5 cm. Find the largest number with the property that the length of the longest side of any triangle of this type is greater than this number. | 25 \, \text{cm} | math | 48 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, and that $a_1 - a_5 + a_9 - a_{13} + a_{17} = 117$, find the value of $a_3 + a_{15}$. | 234 | math | 62 |
Given a quadratic function $y=ax^2+bx+c$ (where $a$ is a positive integer) whose graph passes through points A(-1, 4) and B(2, 1), and it has two distinct intersections with the x-axis, the maximum value of $b+c$ is. | -4 | math | 64 |
Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x\plus{}b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1.$ | \frac{10}{3} \text{ and } -\frac{10}{3} | math | 53 |
Suppose the line $l$: $\begin{cases} x=1+\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and the curve $C_{1}$: $\begin{cases} x=\cos \theta \\ y=\sin \theta \end{cases} (\theta \text{ is the parameter})$.
(1) Let $l$ intersect $C_{1}$ at two points $A$ and $B$, find $|AB|$.
(2) If th... | \frac{\sqrt{6}}{4}(\sqrt{2}-1) | math | 214 |
One mole of an ideal monoatomic gas is first heated isobarically. In this process, it does 20 J of work. Then, it is heated isothermally, receiving the same amount of heat as in the first case. How much work (in joules) does the gas do in the second case? | 50 \text{ J} | math | 67 |
A ray emitted from the point $\left(-5,3\right)$, after reflecting off the $x$-axis, bisects the circumference of the circle $x^{2}+y^{2}-2x-2y-3=0$. Find the equation of the line where the reflected ray lies. | 2x-3y+1=0 | math | 66 |
Calculate:<br/>
$(1)12-\left(-8\right)+\left(-11)$;<br/>
$(2)-\frac{4}{9}-(-3)+(-\frac{5}{9})$;<br/>
$(3)(-\frac{7}{5})÷8×(-\frac{8}{5})×\frac{10}{7}$;<br/>
$(4)(-24)×(\frac{5}{6}-\frac{3}{8}+\frac{1}{4})$;<br/>
$(5)(-2)^3×(-\frac{3}{4})+30÷(-5)$;<br/>
$(6)-1^{2018}×(1-0.4)+\frac{1}{3}×[(-2)^2-6]$. | -\frac{19}{15} | math | 177 |
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be real numbers. Consider the \(2^{n}-1\) non-empty sums that can be formed from these numbers. How many of these sums can be positive? | 2^{n-1} | math | 52 |
A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is | 3 | math | 50 |
Given three numbers are turned up from three fair dice, determine the probability that these numbers can be arranged to form an arithmetic progression with common difference one. | \frac{1}{9} | math | 30 |
For any \( x_1, x_2 \) \((x_1 \neq x_2)\) in the domain of the function \( f(x) \), the following statements hold:
(1) \( f(x_1 + x_2) = f(x_1) \cdot f(x_2) \)
(2) \( f(x_1 \cdot x_2) = f(x_1) + f(x_2) \)
(3) \( \frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0 \)
(4) \( f\left(\frac{x_1 + x_2}{2}\right) > \frac{f(x_1) + f(x_2)}{2... | (1),(3) | math | 190 |
Consider the lines:
\[
y = 2x + 3, \quad 2y = 6x + 4, \quad 3y = 6x - 1, \quad 4y = 2x - 8, \quad 5y = 2x - 10.
\]
Determine how many pairs of these lines are either parallel or perpendicular to each other. | 1 | math | 87 |
The average of 12 numbers is 90. If the numbers 80 and 90 are added to this set of numbers, what is the average of the new set of numbers? | 89.2857142857 | math | 41 |
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$ , $$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016 $$ *(Macedonia)* | f(x) = x + 1 | math | 86 |
Snoopy saw a flash of lightning and heard the sound of thunder 10 seconds later. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, the distance from Snoopy to the flash of lightning. | 2 | math | 63 |
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose... | 17 | math | 91 |
Given that the remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$, express $R$ as a polynomial. | 2^{100}(x-1)-(x-2) | math | 46 |
Calculate: $1-2-3+4+5-6-7+8+\ldots+2005-2006-2007+2008$ equals to what? | 0 | math | 46 |
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to . | 12 | math | 52 |
Consider a set of 16 integers from -8 to 7, inclusive, arranged to form a 4-by-4 square. Find the value of the common sum of the numbers in each row, column, and main diagonal. | -2 | math | 48 |
The probability it will rain on Friday is $40\%$, on Saturday is $50\%$, and on Sunday is $30\%$. Assuming the probability of rain on any given day is independent of the weather on any other day, what is the probability it will rain on all three days, expressed as a percent? | 6\% | math | 69 |
When simplified, what is the value of $(\sqrt{2} \times 2^{1/2} \times 2) + (18 \div 3 \times 2) - (8^{1/2} \times 4)$? | 16 - 8\sqrt{2} | math | 55 |
Given $x > 0, y > 0, \overrightarrow{a}=(x,1), \overrightarrow{b}=(1,y-1)$, and $\overrightarrow{a} \bot \overrightarrow{b}$, calculate the minimum value of $\frac{1}{x}+\frac{4}{y}$. | 9 | math | 72 |
A function $g(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 6x + 9\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 3000$. Compute the integer nearest to $T$. | 1332 | math | 94 |
Given $\sin x = \frac{3}{5}$, with $x \in \left( \frac{\pi}{2}, \pi \right)$, find the values of $\cos 2x$ and $\tan\left( x + \frac{\pi}{4} \right)$. | \frac{1}{7} | math | 63 |
Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | 1 | math | 99 |
Given six test scores have a mean of $85$, a median of $86$, and a mode of $88$. Determine the sum of the two lowest test scores. | 162 | math | 37 |
A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents "a female student who is number 43 in class 8 and enrolled in the year 2002," then the number for a male student who is number 23 in class 6 and enrolled i... | 086231 | math | 94 |
Given $(1+2x)(1-2x)^{7}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{8}x^{8}$, calculate the value of $a_{0}+a_{1}+a_{2}+\ldots+a_{7}$. | 253 | math | 72 |
A box contains ten cards. Four of the cards are black on both sides, three cards are black on one side and red on the other, and three of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the othe... | \frac{2}{3} | math | 78 |
Calculate the value of the definite integral $\int_{-\pi}^{0} (\cos x + e^x) \, dx$. | 1 - \frac{1}{e^{\pi}} | math | 28 |
Two subsets of the set S = {a, b, c, d, e, f} are to be chosen so that their union is S and their intersection contains exactly three elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? | 80 | math | 61 |
Alice is in Japan and she wants to buy a souvenir for $500$ yen. If one U.S. dollar is worth $110$ yen and the bank charges a $3\%$ commission fee for currency exchange, how much money, in USD to the nearest hundredth, does she have to spend for the souvenir? (You may use a calculator on this problem.) | 4.41\ \text{USD} | math | 80 |
Given the function $f(x)=-\sin ^{2}x+m\sin x+2$, when $x\in\left[ \frac {\pi}{6}, \frac {2\pi}{3}\right]$ the function has a maximum value of $\frac {3}{2}$. Find the value of $m$ at this time. | m=- \frac {1}{2} | math | 74 |
In triangle $DEF$, the side lengths are $DE = 15$, $EF = 20$, and $FD = 25$. A rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. Letting $WX = \lambda$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \la... | 16 | math | 153 |
A sequence of three real numbers forms an arithmetic progression with a first term of 5. If 3 is added to the second term and 15 is added to the third term, the three resulting numbers form a geometric progression. What is the largest possible value for the first term of the geometric progression? | 5 | math | 62 |
Compute the limit as \( x \) approaches 0 of the expression \( \frac{e^{x \cos x} - 1 - x}{\sin(x^2)} \). | \frac{1}{2} | math | 39 |
In \( \triangle ABC \), if \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), and \( |\overrightarrow{CA}| = 4 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \). | -\frac{29}{2} | math | 96 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sin B - \sin C = \sin \left(A-C\right)$.
$(1)$ Find angle $A$;
$(2)$ If $a=2$ and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the perimeter of $\triangle ABC$. | 2 + \sqrt{10} | math | 100 |
A function $f:\mathbb{Z} \to \mathbb{Z}$ satisfies
\begin{align*}
f(x+4)-f(x) &= 8x+20, \\
f(x^2-1) &= (f(x)-x)^2+x^2-2
\end{align*}for all integers $x.$ Enter the ordered pair $(f(0),f(1)).$ | (-1,1) | math | 91 |
A certain middle school plans to purchase a total of 200 sets of type $A$ and type $B$ desks and chairs. Through bidding, it is found that purchasing one set of type $A$ desks and chairs costs $40$ yuan less than purchasing one set of type $B$ desks and chairs, and that purchasing 3 sets of type $A$ and 5 sets of type ... | 40800 | math | 234 |
Given the general term formula of a sequence $\{a_n\}$ as $a_n=2^{n-1}+1$, find the value of the expression $a_1C_n^0+a_2C_n^1+a_3C_n^2+\cdots+a_{n+1}C_n^n=\_\_\_\_\_\_\_\_\_.$ | 3^n+2^n | math | 79 |
How many non-congruent triangles with only integer side lengths have a perimeter of 15 units? | 7 | math | 21 |
Marty wants to paint a box. He can choose to use either blue, green, yellow, or black paint. Also, he can style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose? | 12 | math | 60 |
Let \( a_{k} \) be the coefficient of \( x^{k} \) in the expansion of
\[ (x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99}. \]
Determine the value of \( \left\lfloor \frac{a_{4}}{a_{3}} \right\rfloor \). | 19 | math | 97 |
Find the range of the function
$$
f(x)=\sqrt{g^{2}(x)-245}, \text { where } g(x)=15-2 \cos 2x-4 \sin x
$$ | [0, 14] | math | 49 |
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible i... | 26 | math | 265 |
Observe the following inequalities:
\\(①1+\\dfrac{1}{{{2}^{^{2}}}}+\\dfrac{1}{{{3}^{^{2}}}}+\\dfrac{1}{{{4}^{^{2}}}}+\\dfrac{1}{{{5}^{^{2}}}}+\\dfrac{1}{{{6}^{^{2}}}} < \\dfrac{11}{6}\\)；
\\(②{{\\left( n+1 \\right)}^{2}}\\)；
\\(③\\dfrac{2n-1}{n}\\)；
Following this pattern, the fifth inequality is \_\_\_\_\_\_\_\_\_... | 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} < \frac{11}{6} | math | 159 |
With all angles measured in degrees, consider the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | 24 | math | 54 |
The graph of the function $f(x)=\sin \omega x$ ($\omega > 0$) is shifted to the right by $\dfrac{\pi}{12}$ units to obtain the graph of the function $y=g(x)$. Additionally, the function $g(x)$ is monotonically increasing in the interval $\left[\dfrac{\pi}{6}, \dfrac{\pi}{3}\right]$ and monotonically decreasing in the i... | 2 | math | 127 |
Find all functions $f:Q\rightarrow Q$ such that \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \] for all $x,y,z,t\in Q.$ | f(x) = c | math | 74 |
How many positive three-digit integers with each digit greater than 6 are divisible by 12? | 1 | math | 20 |
A hyperbola with asymptotes given by \(2x \pm 3y = 0\) and passing through the point \((1, 2)\) is... | 4 x^{2} - 9 y^{2} = -32 | math | 36 |
The function $f(x)$ is defined on the interval $(-1,1)$ and satisfies the equation $2f(x)-f(-x)=\log (x+1)$. Find $f(x)=$ _____. | f(x)= \frac {2}{3}\log (x+1)+ \frac {1}{3}\log (1-x) | math | 46 |
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