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 | Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and $a_1=1$. If the sequence $\{S_n - n^2a_n\}$ is a constant sequence, then $S_n=$_______. | \frac{2n}{n+1} | 62 | 10 |
math | In the game of Guess the Card, two players each have a $\frac{1}{2}$ chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered $1,2,\dots,16$ clockwise. They participate in an $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly choos... | 164 | 177 | 3 |
math | Given the ellipse $\frac{x^2}{9}+\frac{y^2}{7}=1$, with foci $F_{1}$ and $F_{2}$ and a point $A$ on the ellipse such that $\angle AF_{1}F_{2}=45^{\circ}$, calculate the area of the triangle $AF_{1}F_{2}$. | \frac{7}{2} | 80 | 7 |
math | A four-digit number whose last digit is not 0, if the first two digits can divide 2014, and the product of the first two digits and the last two digits can be divided by 2014, then what is the largest four-digit number? | 5376 | 57 | 4 |
math | January 6, 2005, was "China's 1.3 Billion Population Day". To keep China's total population under 1.4 billion by 2015, the annual average natural population growth rate from January 6, 2005, for the next 10 years should be controlled within % (accurate to 0.01). | 0.74 | 81 | 4 |
math | Let \(\lfloor x\rfloor\) denote the greatest integer which is less than or equal to \(x\). For example, \(\lfloor\pi\rfloor = 3\). \(S\) is the integer equal to the sum of the 100 terms shown:
$$
S = \lfloor\pi\rfloor + \left\lfloor\pi + \frac{1}{100}\right\rfloor + \left\lfloor\pi + \frac{2}{100}\right\rfloor + \left\... | 314 | 178 | 3 |
math | The interval that contains a root of the function $f(x)=-\frac{1}{x}+\log_{2}x$ must be determined. | (1,2) | 32 | 5 |
math | At Pinehurst Academy, there is a five-square league with twelve players, including Amy and Bob. Each break, the twelve players split into two five-square games, each with six players. Throughout the year, each possible matchup of six players occurs exactly once. Additionally, two players, Chris and Dave, are always in ... | 28 | 81 | 2 |
math | If the sum of the coefficients of the expansion of $(2x^{2}-3x+a)^{5}$ is $1$, determine the coefficient of the term containing $x^{7}$. | -2040 | 40 | 5 |
math | Determine the number of ways to arrange the letters of the word "ROCKET" where one of the letters appears twice and all other letters are distinct. | 2520 | 31 | 4 |
math | Let $\theta$ be an acute angle, and let
\[\cos \frac{\theta}{2} = \sqrt{\frac{x - 2}{2x}}.\]
Express $\tan \theta$ in terms of $x.$ | -\frac{1}{2} \sqrt{x^2 - 4} | 49 | 16 |
math | Let $\{a_{n}\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted as $S_{n}$. Given that $S_{k}=5$, $a_{{k}^{2}+1}=-45$, and $a_{k+1}+a_{k+2}+\ldots +a_{2k}=-45$, where $k$ is a positive integer and $k\geqslant 2$, find the first term $a_{1}$ of this sequence. | 5 | 116 | 1 |
math | The distance from point P to the left focus of the ellipse, given that F1 and F2 are the left and right foci of the ellipse x^{2}/25 + y^{2}/16 = 1, point P is on the ellipse, M is the midpoint of F1P, and |OM| = 3. | 4 | 71 | 1 |
math | Given the function $f(x)=\frac{1}{3}x^{3}+x^{2}+ax$.
(1) Discuss the monotonicity of $f(x)$;
(2) Suppose $f(x)$ has two extreme points $x_{1}$, $x_{2}$, and the line $l$ passing through the two points $(x_{1},f(x_{1}))$, $(x_{2},f(x_{2}))$ intersects the $x$-axis at a point on the curve $y=f(x)$, find the value of $a... | a = \frac{2}{3} | 124 | 9 |
math | Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).
Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \). | -4 | 113 | 2 |
math | Determine the maximum potential salary for an individual player on a team, given that each team must have exactly 19 players, each player must receive a minimum salary of $20,000, and the total payroll for a team must not go beyond $800,000. | 440,000 | 62 | 7 |
math | A and B are running a 3000m race. When A is 500m away from the finish line, B is 600m away. If they continue at the same pace, how far will B be from the finish line when A reaches it? | 120 | 59 | 3 |
math | Given the following propositions:
\\(①\\) If the function \\(y=f(x)\\) satisfies \\(f(x-1)=f(x+1)\\), then the graph of the function \\(f(x)\\) is symmetric about the line \\(x=1\\);
\\(②\\) The symmetric point of point \\((2,1)\\) about the line \\(x-y+1=0\\) is \\((0,3)\\);
\\(③\\) The regression equation \\( ∧... | ②③ | 233 | 4 |
math | Given the proposition: "There exists an $x \in [1, 2]$ such that $x^2 + 2x + a \geq 0$" is a true statement, then the range of values for $a$ is. | [-8, +\infty) | 52 | 8 |
math | A rectangle can be divided into \( n \) equal squares. The same rectangle can also be divided into \( n + 76 \) equal squares. Find all possible values of \( n \). | 324 | 41 | 3 |
math | Observe the following equations: $2\times 4+1=9$, $4\times 6+1=25$, $6\times 8+1=49$, $\ldots$ Explore the pattern of the above equations and write the 5th equation as ______, the nth equation as ______. | 2n\times (2n+2)+1=(2n+1)^2 | 68 | 18 |
math | x + 2/3 = 7/15 + 1/5 - x/2 | 0 | 21 | 1 |
math | Let $g$ be a function defined for all nonnegative integers such that $g(1) = 3$, and the relationship
\[ g(m + n) + g(m - n) = 2g(m) + 3g(n) \]
holds for all nonnegative integers $m \geq n$. Find the sum of all possible values for $g(10)$. | 300 | 82 | 3 |
math | The minimum positive period of the function $f(x)=\tan \left(2x-\frac{\pi }{6}\right)$ is $\frac{\pi }{2}$. | \frac{\pi }{2} | 38 | 8 |
math | Given that circle O is tangent to circle C: x²+y²-6y+8=0 at point M(0,2), and passes through point N(2,0).
(1) Find the equation of circle O;
(2) If line L: y=kx-(k+1) intercepts two arc lengths on circle O with a ratio of 3:1, find the value of the real number k. | 1 | 90 | 1 |
math | A five-eighths sector of a circle of radius $6$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
A) $4\pi \sqrt{21}$
B) $5\pi \sqrt{22}$
C) $4.6875\pi \sqrt{21.9375}$... | 4.6875\pi \sqrt{21.9375} | 130 | 19 |
math | Given the function $f(x) = \sin\left(x - \frac{1}{2}\right)$, for $0 < x < 1$, the inequality $f(x) \cdot \log_{2}\left(x - 2^{m} + \frac{5}{4}\right) > 0$ is always satisfied. Find the range of the real number $m$. | m \in (-\infty, -2] | 82 | 11 |
math | Given $S_n$ is the sum of the first n terms of an arithmetic sequence $\{a_n\}$, if $a_4=1$ and $S_5=10$, determine the value of n when $S_n$ is maximized. | 5 | 55 | 1 |
math | Let \( y = \frac{\sum\limits_{n=1}^{50} \cos n^\circ}{\sum\limits_{n=1}^{50} \sin n^\circ} \). What is the greatest integer that does not exceed \( 100y \)? | 100 | 63 | 3 |
math | Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.
$(I)$ Find the standard equation of the ellipse;
$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx... | \dfrac{7}{6} | 158 | 7 |
math | Find the smallest digit \( d \) so that \( 85,21d,784 \) is divisible by 11. | 1 | 31 | 1 |
math | In a geometric sequence:
(1) If $a\_4=27$, $q=-3$, find $a\_7$ and $S\_4$;
(2) If $a\_5-a\_1=15$, $a\_4-a\_2=6$, find $a\_3$. | 4 | 66 | 1 |
math | Given a moving circle that passes through a fixed point $R(0,2)$ and intercepts a line segment $MN$ of length $4$ on the $x$-axis. A line $l$: $y=kx+t(t > 0)$ intersects the $y$-axis at point $Q$.
1. Find the equation of the trajectory $E$ of the moving circle's center.
2. The line $l$ intersects trajectory $E$ at poin... | (0,1) | 160 | 5 |
math | Given the function $f(x)=3\sin\left(2x-\frac{\pi}{3}\right)$.
$(1)$ Find the value of $f\left(\frac{3\pi}{4}\right)$;
$(2)$ Determine the intervals where $f(x)$ is monotonically increasing;
$(3)$ Find the range of $f(x)$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$. | \left[-3, \frac{3}{2}\right] | 99 | 14 |
math | Find the third derivative of the given function.
$$
y=\frac{\ln (2 x+5)}{2 x+5}, y^{\prime \prime \prime}=?
$$ | \frac{88 - 48 \ln(2 x+5)}{(2 x+5)^{4}} | 39 | 26 |
math | A company produces ceramics. According to the historical data, the fixed cost of producing ceramics per day is 14,000 yuan, and the cost increases by 210 yuan for each product produced. The daily sales volume $f(x)$ is related to the production quantity $x$ pieces by the following formula: $f\left(x\right)=\begin{cases... | 30000 | 276 | 5 |
math | In the polar coordinate system, the coordinates of point M are $$(3, \frac{\pi}{2})$$, and the equation of curve C is $$\rho = 2\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$$; taking the pole as the origin of the coordinate system and the positive half-axis of the x-axis as the polar axis, a line l with a slope of -... | \frac{3\sqrt{3}}{2} | 148 | 12 |
math | Given the equation in terms of $x$, $4+3ax=2a-7$, has a unique solution, and the equation in terms of $y$, $2+y=(b+1)y$, has no solution, determine the situation of the solution for the equation $az=b$ in terms of $z$. | z=0 | 66 | 3 |
math | One notebook, 3 notepads, and 2 pens cost 98 rubles. Additionally, 3 notebooks and 1 notepad cost 36 rubles less than 5 pens. How much does each item cost if a notebook costs an even number of rubles? (Each of these items costs a whole number of rubles.) | 4, 22, 14 | 72 | 9 |
math | Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is: | 22 | 29 | 2 |
math | Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ | 4944 | 61 | 4 |
math | Under what conditions is $\sqrt{a^2 + b^2 + c^2} = a + b + c$ true, where $a$, $b$, and $c$ are real numbers? | ab + bc + ca = 0 \quad \text{and} \quad a + b + c \geq 0 | 43 | 27 |
math | Given $f(x)= \begin{cases} \frac {x}{2},x\geqslant 0 \\ x^{2},x < 0\\ \end{cases}$, find the value of $f(f(-1))=$ \_\_\_\_\_\_, and the solution set for $f(f(x))\geqslant 1$ is \_\_\_\_\_\_. | (-\infty, -\sqrt{2}] \cup [4, \infty) | 85 | 20 |
math | Find the minimum value of $$ \big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big| $$ for real numbers $x$ not multiple of $\frac{\pi}{2}$ . | 2\sqrt{2} - 1 | 58 | 9 |
math | Given the base-nine representation of the number $N$ is $27,006,000,052_{\rm nine}$, find the remainder when $N$ is divided by 5. | 3 | 46 | 1 |
math | Given the set $A=\{x|(x-2)(x-3a-1) < 0\}$, the domain of the function $y=\log \left(\frac{2a-x}{x-(a^2+1)}\right)$ is set $B$.
(1) If $a=2$, find the set $B$;
(2) If $A=B$, find the value of the real number $a$. | -1 | 95 | 2 |
math | Given a triangular corner with side lengths $DB = EB = \sqrt{2}$ and $\angle DBE = 90^\circ$ is cut from equilateral triangle $ABC$ of side length $4$, calculate the perimeter of the remaining quadrilateral. | 10 | 53 | 2 |
math | Let $G_n$ be the Fibonacci sequence, defined by $G_0 = 0$, $G_1 = 1$, and $G_{n+2} = G_{n+1} + G_n$. Compute
\[
\sum_{n=0}^{\infty} \frac{G_n}{7^n}.
\] | \frac{49}{287} | 74 | 10 |
math | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c=4$, $C=\frac{\pi}{3}$.
1. If the area of $\triangle ABC$ is $4\sqrt{3}$, find $a$ and $b$.
2. If $\sin B=2\sin A$, find the area of $\triangle ABC$. | \frac{8\sqrt{3}}{3} | 101 | 12 |
math | Find the fraction with the smallest possible denominator that lies between \(\frac{7}{25}\) and \(\frac{8}{25}\). Provide all possible solutions. | \frac{2}{7} | 37 | 7 |
math | Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$ . Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}... | 11 | 194 | 2 |
math | What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$? | \frac{59\pi}{4} | 56 | 10 |
math | Given the function $f(x)=(2^{x}-2^{-x})\cdot x^{3}$, if the real number $a$ satisfies $f(\log _{2}a)+f(\log _{0.5}a)\leqslant 2f(1)$, determine the range of the real number $a$. | \left[\frac{1}{2}, 2\right] | 72 | 14 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $AD$ is the median to side $BC$.
$(1)$ If $a=4$, $b=2$, and $AD=1$, find the length of side $c$;
$(2)$ If $\overrightarrow{AB} \cdot \overrightarrow{AD} = c^{2}$, find the measure of angle $B$. | 90^{\circ} | 110 | 6 |
math | Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | 9999 | 69 | 4 |
math | Given the functions $f(x) = (x-a)^2e^x$ and $g(x) = x^3 - x^2 - 3$, where $a\in\mathbb{R}$,
1. If there exist $x_1, x_2 \in [0, 2]$ such that $g(x_1) - g(x_2) \geq M$ holds, find the maximum value of the real number $M$;
2. If for any $s, t \in [0, 2]$, $f(s) \geq g(t)$ always holds, find the range of values for the re... | (-\infty, -1] \cup [2 + \frac{1}{e}, +\infty) | 143 | 25 |
math | Given the sequence {a<sub>n</sub>}, where a<sub>1</sub> = 3 and n(n+1)(a<sub>n</sub> - a<sub>n+1</sub>) = 2 for all n ∈ N*.
1. Find the general term formula for the sequence {a<sub>n</sub>}.
2. Let b<sub>n</sub> = $$\frac{a_{1} \cdot a_{2} \ldots a_{n}}{(n+1) \cdot 2^{n}}$$, find the sum of the first n terms, S<sub>n<... | S_n = 2 - \frac{n+4}{2^{n+1}} | 150 | 18 |
math | The length of the line segment cut from the circle $(x-1)^2+y^2=1$ by the line $x+ \sqrt {3}y-2=0$ can be calculated. | \sqrt{3} | 43 | 5 |
math | A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid? | 70 | 31 | 2 |
math | The line $ax+by+c=0$ intersects the circle $x^2+y^2=4$ at two points $A$ and $B$. Given that $c^2=a^2+b^2$, and $O$ is the origin, then $\overrightarrow{OA} \cdot \overrightarrow{OB}=$ ______. | -2 | 72 | 2 |
math | For all real numbers $x$, let $[x]$ denote the greatest integer not greater than $x$. The function $f(x) = [x]$ is called the greatest integer function. If $a\_n = f(\dfrac{n}{10})$, $n \in \mathbb{N^*}$, and $S\_n$ is the sum of the first $n$ terms of the sequence ${a\_n}$, then $\dfrac{S_{2009}}{2010} = \_\_\_\_\_\_\... | \dfrac{S_{2009}}{2010} = 100 | 122 | 21 |
math | Given the function $f(x) = 2\sin \omega x\cos \omega x + 2 \sqrt {3}\sin^2 \omega x - \sqrt {3}$ ($\omega > 0$) has the smallest positive period of $\pi$.
(Ⅰ) Determine the intervals of increase for the function $f(x)$.
(Ⅱ) Translate the graph of the function $f(x)$ to the left by $\dfrac{\pi}{6}$ units and then upward... | \dfrac{59\pi}{12} | 160 | 11 |
math | In the Cartesian coordinate system $xoy$, the line $l$ passes through point $P(0,1)$ and has a slope of $1$. In the polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
(Ⅰ) Find the parametric equati... | \sqrt{6} | 147 | 5 |
math | In parallelogram ABCD, let the length of AB be $a$ ($a>0$), AD=1, $\angle BAD=60°$, and E is the midpoint of CD. If $\overrightarrow{AC} \cdot \overrightarrow{BE} = 1$, determine the value of $a$. | \frac{1}{2} | 68 | 7 |
math | Let $x$ be a value such that $9x^2 + 8x - 1 = 0$ and $27x^2 + 65x - 8 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. | \frac{1}{9} | 62 | 7 |
math | In the olympiad, each solved problem could earn 3, 8, or 10 points. Vasya scored 45 points. What is the minimum number of problems he could have solved? (It is necessary to explain why he could not have solved fewer problems.) | 6 | 59 | 1 |
math | Find the remainder when \( x^5 + 2x^3 + x + 3 \) is divided by \( x-4 \). | 1159 | 30 | 4 |
math | Let $q(x)$ be defined on $2 \le x \le 15$ such that $$q(x) = \begin{cases} x^2 + 1 &\quad \lfloor x \rfloor\text{ is even} \\ q(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the smallest prime factor of $\lfloor x \rfloor$. Determine the range of $q$ in interval notation. | [5, 197] | 115 | 8 |
math | Let $y$ be a positive real number. Find the minimum value of $3y^3 + 6y^{-2}$. | 9 | 28 | 1 |
math | Let \(c\) and \(d\) be the roots of \(x^2 - 6x + 10 = 0\). Compute
\[
c^3 + c^5 d^3 + c^3 d^5 + d^3.
\] | 16036 | 57 | 5 |
math | The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$. | -3 | 43 | 2 |
math | Given positive integers \( a, b, \) and \( c \) such that \( a < b < c \). If the product of any two numbers minus 1 is divisible by the third number, what is \( a^{2} + b^{2} + c^{2} \)? | 38 | 61 | 2 |
math | Given vectors $\overrightarrow{m}=(\cos x,1)$, $\overrightarrow{n}=(\sin x,\frac{\sqrt{3}}{2})$:
(1) When $\overrightarrow{m}$ is parallel to $\overrightarrow{n}$, find the value of $\frac{\sin x+\sqrt{3}\cos x}{\sqrt{3}\sin x-\cos x}$;
(2) In obtuse triangle $\Delta ABC$, angle $A$ is the obtuse angle, and $a,b,c$ a... | \frac{3}{4} | 175 | 7 |
math | Given that the left focus of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{m^{2}}=1 (m > 0)$ is $F\_1(-4,0)$, find the value of $m$ and the eccentricity. | e=\frac{4}{5} | 61 | 8 |
math | Given the functions $f(x)=|x-a|+a$ and $g(x)=4-x^{2}$, if there exists an $x \in \mathbb{R}$ such that $g(x) \geq f(x)$, find the range of possible values for $a$. | a \in \left(-\infty, \frac{17}{8}\right] | 62 | 20 |
math | Given line $l_{1}$: $y=2x+3$, and line $l_{2}$ is symmetric to $l_{1}$ with respect to the line $y=-x$, determine the slope of line $l_{2}$. | \dfrac{1}{2} | 52 | 7 |
math | Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.
Kristina: There were five people in front of me.
Marina: I was the first in line!
Liza: There was no one after me.
Nadya: I was standing ne... | 3 | 126 | 1 |
math | Given the complex numbers ${z_{1}}=x^{2}-1+(x^{2}-3x+2)i,$ and ${z_{2}}=x+(3-2x)i,$ where $x\in\mathbb{R}$.
(1) If ${z_{1}}$ is a purely imaginary number, find the value of the real number $x$;
(2) In the complex plane, if the point corresponding to ${z_{1}}$ lies in the fourth quadrant and the point corresponding to... | 1 < x < \frac{3}{2} | 133 | 11 |
math | Given the triangle $\triangle ABC$, the lengths of the sides opposite the three interior angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If in the triangle, $\cos A = \frac{1}{3}$, $S = 4\sqrt{2}$, and $\sin \left(A-B\right) = 2\sin B\left(1-2\cos A\right)$, calculate the length of side $c$. | 4 | 105 | 1 |
math | The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts. | \frac{1-\sqrt{2}}{2} | 25 | 12 |
math | Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$. | 40 | 45 | 2 |
math | If sin(π + α) = $- \frac {3}{5}$, find the value of cosα. | - \frac {4}{5} | 25 | 8 |
math | The interval that contains a root of the function $f(x) = e^x + x - 2$ is to be determined. | (0, 1) | 28 | 6 |
math | A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be from $A$ if $v^2 + w^2 = t^2$?
A) $\sqrt{8}$
... | \sqrt{10} | 139 | 6 |
math | Convert $235_{10}$ to base 2. Let $z$ be the number of zeros and $w$ be the number of ones in base 2. What is the value of $w-z?$ | 2 | 46 | 1 |
math | Given a positive term sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, $a_1=1$, and $(t+1)S_n = a_n^2 + 3a_n + 2$ for $t\in \mathbb{R}$.
$(1)$ Find the general term formula for the sequence $\{a_n\}$;
$(2)$ Suppose sequence $\{b_n\}$ satisfies $b_1=1$, $b_{n+1} - b_n = a_{n+1}$, find the sum of the first ... | \frac{3n^2+5n}{12(n+1)(n+2)} | 164 | 21 |
math | Given an integer \( n \geq 2 \), let \( f(n) \) be the second largest positive divisor of \( n \). For example, \( f(12)=6 \) and \( f(13)=1 \). Determine the largest positive integer \( n \) such that \( f(n)=35 \). | 175 | 71 | 3 |
math | 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 | 86 | 8 |
math | The entry fee for a concert is \$30 per adult and \$15 per child. On a particular day, the concert collected \$2250 in entry fees, with at least one adult and one child attending. Determine the ratio of adults to children on that day, such that the ratio is closest to 1. | 1 | 67 | 1 |
math | Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$, with left and right foci $F_{1}$, $F_{2}$, and the origin $O$, a perpendicular line is drawn from $F_{1}$ to a asymptote of $C$, with the foot of the perpendicular being $D$, and $|DF_{2}|=2\sqrt{2}|OD|$. Find the eccentricity of $C$. | \sqrt{5} | 119 | 5 |
math | Let $x$ be an angle such that $\tan x = \frac{a}{b}$ and $\tan 3x = \frac{b}{2a + b}.$ Then the least positive value of $x$ equals $\tan^{-1} k.$ Compute $k.$ | 1 | 61 | 1 |
math | Given X ∼ N(2, σ^2), and P(2 < X < 4) = 0.3, determine P(0 < X < 4). | 0.6 | 38 | 3 |
math | What is the largest quotient that can be formed using two numbers chosen from the set $\{ -30, -5, -3, 1, 3, 10, 15 \}$? | 15 | 44 | 2 |
math | Source: 1976 Euclid Part B Problem 2
-----
Given that $x$ , $y$ , and $2$ are in geometric progression, and that $x^{-1}$ , $y^{-1}$ , and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$ . | \frac{27}{2} | 82 | 8 |
math | The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \div k$ is an integer value? Express your answer as a common fraction. | \frac{1}{4} | 58 | 7 |
math | Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = 2\|\mathbf{b}\|\] and \[\|\mathbf{a} + \mathbf{b}\| = 2\|\mathbf{b}\|.\] Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | 104.48^\circ | 92 | 8 |
math | Given the function f(x) = |x - 2a| - |x - a|, where a ∈ R.
(I) Find the range of values for a if f(1) > 1.
(II) If a < 0, find the range of values for a such that the inequality f(x) ≤ |y + 2020| + |y - a| holds true for all x, y ∈ (-∞, a]. | [-1010, 0) | 96 | 9 |
math | Mr. Lee's students were asked to add two positive integers. Alice subtracted by mistake and got 4. Bob mistakenly multiplied and got 63. What was the correct answer? | 18 | 38 | 2 |
math | Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______. | 9+4 \sqrt {2} | 64 | 8 |
math | The polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(3+i) = 0$. What is $a+b+c+d$? | 49 | 53 | 2 |
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