task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Given that function $f(x)$ is a monotonically increasing function defined on $(0,+\infty)$, and for any positive numbers $x$, $y$, it satisfies $f(x\cdot y)=f(x)+f(y)$. If the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and $f(a_{n})=f(S_{n}+2)-f(4)(n∈N^{})$, then the general term formula of the se... | \frac {1}{2}\times( \frac {4}{3})^{n} | 127 | 19 |
math | Yihana walks for 10 minutes. A graph of her elevation in metres versus time in minutes is shown. Calculate the length of time for which she was walking uphill. | 5\ \text{minutes} | 36 | 7 |
math | How many bases between 2 and 10 inclusive make the number $729_{10}$ have a last digit of 5 in their respective base representations? | 2 | 35 | 1 |
math | The natural number \( a \) is divisible by 21 and has 105 distinct divisors, including 1 and \( a \). Find the smallest such \( a \). | 254016 | 39 | 6 |
math | A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$, parallel to the sides of $\triangle DEF$, divide it into three smaller triangles with areas $16$, $25$, and $36$ respectively. Find the area of $\triangle DEF$. | 77 | 60 | 2 |
math | An item's price is reduced by 15%. What percentage increase is required on this new price to bring it back to its original value? | 17.65\% | 29 | 7 |
math | If $\cos\left(\frac{7\pi}{2} + \alpha \right) = \frac{4}{7}$ and $\tan \alpha < 0$, find the value of $\cos(\pi - \alpha) + \sin\left(\frac{\pi}{2} - \alpha \right) \tan \alpha$. | \frac{4 + \sqrt{33}}{7} | 72 | 14 |
math | Solve the system of equations:
\[
\left\{
\begin{array}{l}
x^{2} y + x y^{2} + 3x + 3y + 24 = 0, \\
x^{3} y - x y^{3} + 3x^{2} - 3y^{2} - 48 = 0
\end{array}
\right.
\] | (-3, -1) | 91 | 6 |
math | A right rectangular prism has edge lengths \(\log_{3}x, \log_{5}x,\) and \(\log_{6}x,\) and its surface area and volume are numerically equal. Find the value of \(x\). | 8100 | 52 | 4 |
math | Given a hyperbola centered at the origin with its transverse axis on the $x$-axis, and its asymptote equations as $y= \pm \frac{\sqrt{6}}{3} x$, find the equation of the hyperbola, knowing that the line $y=2 x + \frac{\sqrt{210}}{3}$ intersects the hyperbola and the resulting chord has a length of 4. | \frac{x^2}{3} - \frac{y^2}{2} = 1 | 92 | 21 |
math | Leticia has a $9\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each ... | 486 | 195 | 3 |
math | Determine the values of $A$, $B$, $C$, and $D$ for the simplified form of the function $y=\frac{x^3 + 12x^2 + 47x + 60}{x+3}$, which simplifies to $y=Ax^2 + Bx + C$, where $x \neq D$. | 27 | 78 | 2 |
math | Five students stand in a circle and count in sequence, with the following rules: ① The first student starts counting from 1. The second student also starts from 1, and thereafter, each student's count is the sum of the previous two students' counts; ② If the count is a multiple of 3, the student must clap once. When th... | 7 | 96 | 1 |
math | A fair coin is tossed 4 times. What is the probability of getting exactly three consecutive heads? | \frac{1}{8} | 20 | 7 |
math | Let \( y = \frac{\sum\limits_{n=1}^{30} \sin n^\circ}{\sum\limits_{n=1}^{30} \cos n^\circ} \). Find the smallest integer that exceeds \( 50y \). | 14 | 59 | 2 |
math | Given the function $f(x) = \sin x \cos x - \sin^2 x + \frac{1}{2}$,
(Ⅰ) Determine the intervals where $f(x)$ is monotonically increasing;
(Ⅱ) In the triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, and let it be given that $b\cos 2A = b\cos A - a\sin... | \left[ - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right] | 138 | 26 |
math | A line $I$ intersects with the lines $y=1$ and $x-y-7=0$ at points $P$ and $Q$ respectively. The coordinates of the midpoint of the segment $PQ$ are $(1,-1)$. What is the slope of line $I$? | -\frac{2}{3} | 63 | 7 |
math | I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction. | \frac{3}{20} | 74 | 8 |
math | Given $M=\{1,2\}$ and $N=\{a,b\}$, where $a, b \in \mathbb{R}$, if $M=N$, then $2a+b=$ \_\_\_\_\_\_. | 5 | 52 | 1 |
math | Given the function $f(x) = \sin(2x + \varphi)$ $(0 < \varphi < \frac{\pi}{2})$, the graph of the function has a center of symmetry at $\left(\frac{3\pi}{8}, 0\right)$. Determine the interval in which the function $f(x)$ is monotonically decreasing. | \left[k\pi + \frac{\pi}{8}, k\pi + \frac{5\pi}{8}\right] | 78 | 28 |
math | Let $p$ , $q$ , $r$ , and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$ ? | 23 | 82 | 2 |
math | If the function $f(x) = x^2 - ae^x$ has three zeros, determine the range of values for $a$. | (0, \frac{4}{e^2}) | 29 | 12 |
math | A shop offers a sale of $25\%$ off on its products. Subsequently, the shop introduces an additional $15\%$ discount on the reduced prices and claims that the combined discount equals $40\%$ off the original price. Calculate, as a percentage of the original price, the difference between the actual combined discount and ... | 3.75\% | 79 | 6 |
math | Given the quadratic equation $x^{2}-2x+m=0$, determine the range of real numbers $m$ for which the equation has two distinct real roots. | m < 1 | 34 | 4 |
math | Using only the two operations "+1" (add 1) and "-i" (negative inverse), we can generate various sequences starting from an initial number. For example, starting with the number 3, we can form the sequence
$$
3 \xrightarrow{+1} 4 \xrightarrow{+1} 5 \xrightarrow{-i} -\frac{1}{5} \xrightarrow{+1} \frac{4}{5} \xrightarrow... | 0 \xrightarrow{+1} 1 \xrightarrow{-i} -1 \xrightarrow{+1} 0 | 184 | 27 |
math | A line $y=kx$ divides the area enclosed by the parabola $y=x-x^{2}$ and the $x$-axis into two equal parts. Find the value of $k$. | k=1- \frac {\sqrt[3]{4}}{2} | 42 | 16 |
math | Determine the values of \( a \) and \( b \) such that the five-digit decimal number \( \overline{a679b} \) is divisible by 72. | a = 3, \, b = 2 | 41 | 11 |
math | Given the function y = A sin(ωx + φ) (A > 0, ω > 0, |φ| < $\frac {π}{2}$), one of the highest points on the graph is at coordinates (2, $\sqrt {2}$). The graph from this highest point to its adjacent lowest point intersects the x-axis at point (6, 0). Determine the analytical expression for this function. | \sqrt {2} \sin(\frac {π}{8}x + \frac {π}{4}) | 88 | 23 |
math | A fair 8-sided die is rolled. If the roll is a multiple of 3, then you win that amount of dollars (for example, if you roll 6, then you win $\$6$). If the roll is not a multiple of 3, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value. | \$2.25 | 76 | 5 |
math | Calculate:
1. $(1)\left( \sqrt[3]{2} \times \sqrt{3}\right)^{6}+\left( \sqrt{2 \sqrt{2}}\right)^{ \frac{4}{3}}-4\left( \dfrac{16}{49}\right)^{- \frac{1}{2}}- \sqrt[4]{2}\times{8}^{0.25}-\left(-2009\right)^{0} $
2. $2\left(\lg \sqrt{2}\right)^{2}+\lg \sqrt{2}+\lg 5+ \sqrt{\left(\lg \sqrt{2}\right)^{2}-\lg 2+1} $ | 1 | 162 | 1 |
math | Given $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(\cos β,\sin β)(0 < α < \dfrac {π}{2}\,,\,- \dfrac {π}{2} < β < 0)$ with $|\overrightarrow{a}- \overrightarrow{b}|= \dfrac {2 \sqrt {5}}{5}$,
(I) find the value of $\cos (α-β)$;
(II) if $\cos β= \dfrac {12}{13}$, find the value of $\cos α$. | \dfrac {56}{65} | 129 | 9 |
math | The interval that contains the root of the function $f(x) = 2^x + x - 7$ must be determined. | (2, 3) | 28 | 6 |
math | Acute-angled triangle $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 150^\circ$ and $\stackrel \frown {BC} = 60^\circ$.
A point $E$ is chosen on the minor arc $AC$ of the circle such that $OE$ is perpendicular to $AC$. Determine the ratio of the magnitudes of $\angle OBE$ to $\ang... | 2 | 104 | 1 |
math | A company has designed a craft with a cost of $40$ yuan per unit. In order to set a reasonable price, they conducted a trial sale in the market. According to market research, when the selling price is $50$ yuan, the daily sales volume is 100 units. However, for every $1$ yuan increase in the selling price, the daily sa... | 55 \text{ yuan} | 156 | 7 |
math | Letters $P, Q, R,$ and $S$ represent four different digits selected from $0,1,2,\ldots,9.$ If $(P+Q)/(R+S)$ is an integer that is as small as possible, evaluate the value of $P+Q$. | 1 | 58 | 1 |
math | My three-digit code is 0A3. Reckha can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the positions of two digits (so A30 and 30A, for example, are forbidden, but 3A0 is fine). Reckha can otherwise choose any three-digit code where... | 4047 | 132 | 4 |
math | Given functions $f(x)=e^{x}-ax$ and $g(x)=\ln x-ax$, where $a\in \mathbb{R}$.
$(1)$ When $a=1$, find the monotonic interval of the function $y=f(g(x))$.
$(2)$ Let $h(x)=f(x)-g(x)$ and the minimum value of $h(x)$ be $m$. Find the minimum value of the function $F(x)=e^{x}-e^{m}\ln x$.
(Where $e\approx 2.71828$ is t... | 0 | 135 | 1 |
math | Given $f(α)= \dfrac {\sin (π+α)\cos (2π-α)\tan (-α+ \dfrac {3π}{2})}{\cos (-π -α )}$, find the value of $f(- \dfrac {31π}{3})$. | f\left(- \dfrac{31π}{3}\right) = \dfrac{1}{2} | 64 | 25 |
math | In a certain mechanism, three gears of different diameters are interconnected in such a way that the largest gear touches both of the smaller ones, and all three gears combined have 60 teeth. When the largest gear lacks 20 teeth to complete four full rotations, the second and third gears make 5 and 10 full rotations, r... | 30, 20, 10 \text{ teeth} | 80 | 15 |
math | The semi-focal distance of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ is $c$. If the line $y=2x$ intersects the ellipse at a point whose x-coordinate is exactly $c$, determine the eccentricity of the ellipse. | \sqrt{2} - 1 | 78 | 8 |
math | Among all the roots of the equation
\[ z^6 - z^4 + z^2 - 1 = 0, \]
find the maximum imaginary part of a root, which can be expressed as $\sin \phi$, where $-90^\circ \leq \phi \leq 90^\circ$. | 90^\circ | 69 | 4 |
math | Find the equation of line $l$ that passes through point $A(-1,1)$ and is parallel to the line $x+3y+4=0$. | x+3y-2=0 | 35 | 8 |
math | Given that 20% of the students scored 60 points, 25% scored 75 points, 40% scored 85 points, and the remainder scored 95 points, find the difference between the mean and median score of the students' scores in this competition. | 6 | 62 | 1 |
math | (1) Find the coefficient of $x^3$ in the expansion of $(\frac{1}{2} - x)^5$ and the sum of all coefficients in the expansion.
(2) From the numbers 0, 2, 3, 4, 5, 6, select any 4 to form a 4-digit number without repeating digits. Find the number of 4-digit numbers that meet the condition. | 300 | 91 | 3 |
math | In the arithmetic sequence $\{a_{n}\}$, it is known that $a_{1}+a_{3}=10$, and $a_{4}=9$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.
$(2)$ Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and the sum of the first $n$ terms of the sequence $\left\{{\frac{1}{{{S_n}}}}\right... | \lambda \in (-\infty, -1] \cup [\frac{1}{4}, +\infty) | 171 | 26 |
math | There are 30 cars in my building's parking lot. All of the cars are red or white, and a car can have either 2 doors or 4 doors. $\frac{1}{3}$ of them are red, $50\%$ of them are 4-door, and 8 of them are 2-door and white. How many of the cars are 4-door and red? | 3 | 87 | 1 |
math | Given that $f(x) = x^k$, where $k < 0$, what is the range of $f(x)$ on the interval $(0, 1]$? | (1, \infty) | 37 | 7 |
math | The cells of a \(5 \times 5\) grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for ... | 1680 | 112 | 4 |
math | Let these numbers be A and B. Let B be the smallest of these numbers. Then the greater number is $A=10B+n$, where $n$ is the crossed-out digit, and $0 \leq n \leq 9$. According to the condition, $A+B = 11B + n = 2022$. Hence, $n$ is the remainder when 2022 is divided by 11, giving $n = 9$. We get $B = 183$ and $A = 183... | \mathrm{B}=183, \mathrm{~A}=1839 | 123 | 19 |
math | Given that Ace travels at a constant speed, and when traveling east, Flash runs \( x \) times as fast as Ace, and when traveling west, Flash runs \( x+1 \) times as fast as Ace, with \( x>1 \), determine the total distance Flash must run to first catch Ace traveling east and then catch Ace again after turning around to... | \frac{2xy}{x-1} + \frac{(x+1)y}{x} | 109 | 21 |
math | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ with their respective sum of the first $n$ terms denoted as $S_n$ and $T_n$, it holds that for any positive integer $n$, the ratio $\frac {S_{n}}{T_{n}}$ is equal to $\frac {2n-3}{4n-3}$. Find the value of $\frac {a_{2}}{b_{3}+b_{13}}+ \frac {a_{14}}{b_{5}+b_{11}}$... | \frac {9}{19} | 125 | 8 |
math | Determine the value of $\cos 50^{\circ}\cos 20^{\circ}+\sin 50^{\circ}\sin 20^{\circ}$. | \frac{\sqrt{3}}{2} | 40 | 10 |
math | Given positive integers \(a, b, c\) that satisfy
\[ 1 < a < b < c, \quad a + b + c = 111, \quad b^2 = ac, \]
find \(b\). | 36 | 51 | 2 |
math | A circle is drawn through vertices $E$ and $H$ and tangent to side $FG$ of square $EFGH$ with side $12$ feet. Calculate the radius of the circle. | 7.5 | 42 | 3 |
math | For the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), the right vertex is $A$. A line $l$ passing through the origin intersects the ellipse $C$ at points $P$ and $Q$. If $|PQ| = a$ and $AP \perp PQ$, then the eccentricity of the ellipse $C$ is ______. | \frac{2\sqrt{5}}{5} | 102 | 12 |
math | Masha wrote a three-digit number on the board, and Vera wrote the same number next to it but swapped the last two digits. After that, Polina added the resulting numbers and got a four-digit sum, the first three digits of which are 195. What is the last digit of this sum? (The answer needs to be justified.) | 4 | 72 | 1 |
math | In the government of a country of knights and liars, there are 12 ministers. Some of them are liars, and the rest are knights. During a government meeting, the following opinions were expressed: the first minister said, "There is not a single honest person here," the second, "There is no more than one honest person her... | 6 | 121 | 1 |
math | Given that $\cos α= \frac{ \sqrt {5}}{5}$ and $α \in (0, \frac {π}{2})$:
(I) Find $\sin 2α$;
(II) Find $\tan (α+ \frac {π}{4})$. | -3 | 61 | 2 |
math | The Bank of Springfield's Super High Yield savings account compounds annually at a rate of three percent. If Bart invests $5000 in one of these accounts, how much interest will he earn after ten years? (Give your answer to the nearest dollar.) | 1720 | 53 | 4 |
math | Given the circles O$_{1}$: x$^{2}$+y$^{2}$+4x-4y+7=0 and O$_{2}$: x$^{2}$+y$^{2}$-4x-10y+13=0, determine the number of tangent lines shared by these circles. | 3 | 72 | 1 |
math | What is the smallest natural number that can be added to 25,751 to create a palindrome? | 1 | 23 | 1 |
math | Given that point P $(2-a, 3a+6)$ has equal distances to the two coordinate axes, then the coordinates of the point symmetric to P with respect to the origin O are. | (-3, -3) \text{ or } (-6, 6) | 40 | 17 |
math | Given the function $f(x)=\sin ^{6}x+\cos ^{6}x$, provide the following conclusions:
1. The range of $f(x)$ is $[0,2]$;
2. The smallest positive period of $f(x)$ is $\frac{\pi}{2}$;
3. The equation of the symmetry axis of the graph of $f(x)$ is $x=\frac{k\pi}{4}(k\in\mathbb{Z})$;
4. The center of symmetry of the... | 234 | 167 | 3 |
math | The ratio of the volume of the inscribed sphere to the circumscribed sphere of a regular tetrahedron. | 1:27 | 24 | 4 |
math | A store advertises a series of two discounts on a product. The first discount is 40% off the original price. Subsequently, a second discount of 30% off the new reduced price is applied. The store claims that the total discount offered is 70% off the original price. Calculate the actual total discount as a percentage of... | 12\% | 90 | 4 |
math | Among the divisors of the number \(1 \cdot 2 \cdot 3 \cdot \ldots \cdot 16 \cdot 17\), find the largest one that is: 1) a cube of a natural number; 2) a square of a natural number. | 120960^2 | 61 | 8 |
math | The function $g(x)$ satisfies
\[g(x) - 3g\left(\frac{1}{x}\right) = 3^x + x^2\]
for all $x \neq 0$. Find $g(3)$. | -\frac{3 \cdot 3^{1/3} + \frac{1}{3} + 36}{8} | 55 | 28 |
math | Simplify the following expressions:
(1) $a(a-b)-(a+b)(a-2b)$
(2) $(\frac{4x-9}{3-x}-x+3)÷\frac{{x}^{2}-4}{x-3}$. | -\frac{x}{x+2} | 59 | 8 |
math | The slope of the tangent line $l$ to the curve $y=a\cos x$ at $x=\frac{\pi}{6}$ is $\frac{1}{2}$. Find the equation of the tangent line $l$. | x - 2y - \sqrt{3} - \frac{\pi}{6} = 0 | 48 | 22 |
math | Solve the equation \(3x^3 + 2\sqrt{3}x^2 - 21x + 6\sqrt{3} = 0\), given that the product of two of its roots is equal to 1. | \sqrt{3}, \; \frac{\sqrt{3}}{3}, \; -2\sqrt{3} | 53 | 26 |
math | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle... | \frac{1}{3} | 229 | 7 |
math | A certain police officer patrols on a motorcycle on a north-south road. One day he set off from the guard post and stayed at point $A$ in the evening. The direction towards the north is defined as positive. The patrol record for that day is as follows: (unit: kilometers) $+10$, $-8$, $+6$, $-13$, $+7$, $-12$, $+3$, $-2... | 3.05\ \text{liters} | 166 | 11 |
math | Given the function $f(x) = xe^{x} - 2a(\ln x + x)$ has two zeros, determine the smallest integer value of $a$. | 2 | 35 | 1 |
math | Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[
\sqrt{1 + \sqrt{45 + 16 \sqrt{5}}} = \sqrt{a} + \sqrt{b}.
\] | (1,5) | 56 | 5 |
math | Given two lines $l_1: x-2y+4=0$ and $l_2: x+y-2=0$ intersect at point P
(1) Find the coordinates of point P;
(2) Let line $l_3: 3x-4y+5=0$, find the equations of the lines that pass through point P and are parallel and perpendicular to line $l_3$, respectively. | 4x+3y-6=0 | 92 | 9 |
math | Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = \arcsin x, \quad y = \arccos x, \quad y = 0
$$ | \frac{\pi}{2} | 51 | 7 |
math | In the diagram, $\overrightarrow{OA}\perp\overrightarrow{OC}$ and $\overrightarrow{OB}\perp\overrightarrow{OD}$. If $\angle{AOD}$ is 2.5 times $\angle{BOC}$, determine the measure of $\angle{AOD}$.
- Assume all labeled points originate from a common point O and spread outwards, defining angles between them as described... | 128.57\text{ degrees} | 88 | 11 |
math | In the right triangle $ABC$, the lengths of $AB$ and $BC$ are 15 and 20, respectively. The quadrilateral $BDEF$ is a square. If the height $EH$ of triangle $EMN$ is 2, then the area of square $BDEF$ is $\qquad$ . | 100 | 70 | 3 |
math | If $f(x)$ is defined for all integers $x \ge 0,$ $f(1) = 2,$ and
\[f(a + b) = f(a) + f(b) - 3f(ab)\] for all integers $a,$ $b \ge 0,$ compute $f(1986).$ | 2 | 72 | 1 |
math | Given that 48 students are divided into 4 groups of 12 people each, calculate the probability that Xiaoyu is assigned to group A. | \frac{1}{4} | 32 | 7 |
math | Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$. | 10 | 49 | 2 |
math | The median $AA_{0}$ of triangle $ABC$ is extended from point $A_{0}$ perpendicularly to side $BC$ outside of the triangle. Let the other end of the constructed segment be denoted as $A_{1}$. Similarly, points $B_{1}$ and $C_{1}$ are constructed. Find the angles of triangle $A_{1}B_{1}C_{1}$, given that the angles of tr... | 60^\circ | 123 | 4 |
math | In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km a... | 0.5 | 76 | 3 |
math | Insert parentheses in the expression 1-2-3-4-5-6-7=0 to make the equation true. | 1 - 2 - 3 - 4 - (5 - 6 - 7) = 0 | 27 | 23 |
math | Given that {a_n} is an arithmetic sequence, with the first term a_1 > 0, a_{2013} + a_{2014} > 0, and a_{2013} \cdot a_{2014} < 0, calculate the largest natural number n for which the sum of the first n terms S_n > 0. | 4026 | 83 | 4 |
math | From Zlatoust to Miass, "GAZ", "MAZ", and "KamAZ" started simultaneously. After reaching Miass, "KamAZ" immediately turned back and met "MAZ" at 18 km and "GAZ" at 25 km from Miass. After reaching Miass, "MAZ" also immediately turned back and met "GAZ" at 8 km from Miass. What is the distance from Zlatoust to Miass? | 60 | 102 | 2 |
math | Define the sequence \(\left\{x_{i}\right\}_{i \geq 0}\) by \(x_{0}=2009\) and \(x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}\) for all \(n \geq 1\). Compute the value of \(\sum_{n=0}^{2009} 2^{n} x_{n}\). | 2009 | 107 | 4 |
math | Given that two integers are selected from the integers $11$ to $15$, and the sum of the number of factors of these two numbers is $8$, calculate the probability of this event. | \frac{3}{10} | 41 | 8 |
math | In the sequence $\{a_n\}$, it is known that $a_1=1$, and $a_{n+1}+(-1)^{n}a_{n}=\cos (n+1)\pi$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Then, $S_{2015}=$ ______. | -1006 | 87 | 5 |
math | Given three points $P(5,2)$, $F_{1}(-6,0)$, $F_{2}(6,0)$:
(Ⅰ) Find the standard equation of the ellipse with foci $F_{1}$, $F_{2}$ that passes through point $P$;
(Ⅱ) Let the symmetric points of $P$, $F_{1}$, $F_{2}$ with respect to the line $y=x$ be $P'$, $F_{1}'$, $F_{2}'$ respectively. Find the standard equation ... | \dfrac{y^{2}}{20} - \dfrac{x^{2}}{16} = 1 | 150 | 26 |
math | Let $f(x)=\sin^{6}\frac{kx}{4}+\cos^{6}\frac{kx}{4}$, where $k$ is a positive integer. If for any real number $a$, we have $\{f(x)|a \lt x \lt a+1\}=\{f(x)|x\in \mathbb{R}\}$, then the minimum value of $k$ is ______. | 7 | 90 | 1 |
math | Given that $|\vec{a}| = 1$, $|\vec{b}| = \sqrt{2}$, $\vec{c} = \vec{a} + \vec{b}$, and $\vec{c} \perp \vec{a}$, find the angle between vectors $\vec{a}$ and $\vec{b}$. | \frac{3\pi}{4} | 75 | 9 |
math | Given that a real number $a$ is chosen arbitrarily within the interval $(-1,1)$ and a real number $b$ is chosen arbitrarily within the interval $(0,1)$, the probability that the line $ax-by=0$ intersects with the circle $(x-1)^{2}+(y-2)^{2}=1$ is $\_\_\_\_\_\_$. | \frac{5}{16} | 81 | 8 |
math | Find the number of positive integers $n \le 2000$ such that $21n$ is a perfect square. | 9 | 28 | 1 |
math | Given that \( a \) is a root of the equation \( x^2 - 5x + 1 = 0 \), calculate the last digit of \( a^4 + a^{-4} \). | 7 | 44 | 1 |
math | If $f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{1993}(3)=$ | \frac{1}{5} | 73 | 7 |
math | Given that $α, β ∈ (0, \frac{π}{2})$, and $\frac{\sin β}{\sin α} = \cos(α + β)$,
(1) If $α = \frac{π}{6}$, then $\tan β =$ _______;
(2) The maximum value of $\tan β$ is _______. | \frac{\sqrt{2}}{4} | 76 | 10 |
math | Given the moving straight line ${l_{0}}:ax+by+c-2=0 (a > 0, c > 0)$ that always passes through point $P(1,m)$, and the maximum distance from point $Q(4,0)$ to the moving straight line ${l_{0}}$ is $3$, find the minimum value of $\frac{1}{2a}+\frac{2}{c}$. | \frac{9}{4} | 91 | 7 |
math | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2\sin B=\sin A+\cos A\cdot \tan C$.
$(1)$ Find the value of $C$.
$(2)$ If the radius of the incircle of triangle $\triangle ABC$ is $\frac{{\sqrt{3}}}{2}$ and $b=4$, find $a-c$. | -1 | 105 | 2 |
math | Find the maximum value of the function $f\left(x\right)={\cos }^{2}x+\mathrm{sin}x$ on the interval $x\in [\frac{2\pi }{5},\frac{3\pi }{4}]$. | \frac{1+\sqrt{2}}{2} | 58 | 12 |
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