text stringlengths 222 11.8k |
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<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
From point \( M \) moving along a circle, perpendiculars \( MP \) and \( MQ \) are dropped on the diameters \( AB \) and \( CD \). Prove that the length of the segment \( PQ \) does not depend on the posit... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Express $\frac{165_7}{11_2}+\frac{121_6}{21_3}$ in base 10.<|im_end|>
<|im_start|>assistant
To solve the given problem, we first convert each number from its original base to base 10.
1. Converting $165_... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the inequality $|x+3|+|x+m| \geq 2m$ with respect to $x$ and its solution set is $\mathbb{R}$.
1. Find the range of values for the real number $m$.
2. Given $a > 0$, $b > 0$, $c > 0$ and $a + b + c ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that x and y = -2, the value of (x - 2y)^y is 0.001. What is the value of x?<|im_end|>
<|im_start|>assistant
Given that y = -2, we can substitute y into the equation (x - 2y)^y = 0.001.
So we have (... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Determine the value of $\log_{2}\left( \cos \frac{7\pi }{4} \right)$.
A: $-1$
B: $-\frac{1}{2}$
C: $\frac{1}{2}$
D: $\frac{\sqrt{2}}{2}$<|im_end|>
<|im_start|>assistant
1. **Analyze the problem**: This qu... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Dorothy earns $60000 a year from her work. She needs to pay 18% of this amount in taxes. She also has monthly bills totaling $800, an annual savings goal of $5000, and she invests 10% of her income in stoc... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the inequality $x^2 + x < \dfrac{a}{b} + \dfrac{b}{a}$ holds for any positive real numbers $a$ and $b$, determine the range of the real number $x$.
A: $(-2,0)$
B: $(-\infty,-2)\cup(1,+\infty)$
C... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
This line graph represents the prices of a rare book during the first 6 months of 2023. Determine the month during which the largest percentage drop in price occurred.
- January: The price change is $-10\... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The vertex coordinates of the parabola $y=x^2+2x-3$ are.<|im_end|>
<|im_start|>assistant
Since the parabola $y=x^2+2x-3$ can be rewritten as: $y=(x+1)^2-4$,
Therefore, the coordinates of its vertex are $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. How many uphill integers of four digits or fewer are there that are divisible by 9?
A) 2
B) 3
C) 4
D) 5... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In \(\triangle ABC\), the incenter is \(I\), and the incircle touches sides \(AB\) and \(AC\) at points \(M\) and \(N\) respectively. The extensions of \(BI\) and \(CI\) intersect \(MN\) at points \(K\) an... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find $97^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)<|im_end|>
<|im_start|>assistant
To solve for $97^{-1} \pmod{101}$, notice that $97 \equiv -4 \pmod{101}$ s... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $S_n$ is the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and $S_n = 2a_n - 1$, find the common ratio $q$ of the sequence $\{a_n\}$.<|im_end|>
<|im_start|>assistant
Since $S_n$ ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Stella wanted to buy a new dress for the upcoming dance. At the store, she found out that the dress she wanted was $50. The store was offering a certain discount on everything in the store, and the final c... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many different triangles can be formed having a perimeter of 7 units if each side must have integral length?<|im_end|>
<|im_start|>assistant
To determine the number of different triangles that can be f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The common point of the inscribed circle centered at \( O_1 \) and the circle tangent to side \( BC \) centered at \( O_2 \) with side \( BC \) are \( E_1 \) and \( E_2 \), respectively. The midpoint of si... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jenny is working on a school project. She spends 10 hours doing research for the project and some hours writing a proposal. If she has 20 hours total to work on the project and she has 8 hours left to writ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given vectors $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w'} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$, the projection of $\mathbf{v}$ on $\mathbf{w'}$ is found to be $\begin{pmatrix} ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Observe the following expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., then find the value of $a^{10}+b^{10}$.<|im_end|>
<|im_start|>assistant
We start by noting that the val... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $s_1, s_2, s_3$ be the sums of $n$, $3n$, and $5n$ terms, respectively, of the same arithmetic progression with first term $a$ and common difference $d$. Define $R = 2s_3 - 3s_2 + s_1$. Determine what ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The speed at which a man can row a boat in still water is 9.5 kmph. If he rows downstream, where the speed of current is 8.5 kmph, he takes a certain time to cover a distance. The time taken is 9.099272058... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Grandma has just finished baking a new large rectangular pan of brownies and plans to cut them into pieces as before, but now she wants the number of interior pieces to be twice the number of perimeter pie... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the solution set of the inequality $x^{2} - 2x + 3 \leq a^{2} - 2a - 1$ on $\mathbb{R}$ is empty, then the range of the real number $a$ is ______.<|im_end|>
<|im_start|>assistant
To find the range of $a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
This week, the fishmonger sold 50 kg of salmon. He sold three times more the following week. What was the total amount of fish sold in two weeks?<|im_end|>
<|im_start|>assistant
If the fishmonger sold 50 k... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If a = 105 and a ^ 3 = some number × 25 × 45 × 49, what is the value of that number?<|im_end|>
<|im_start|>assistant
Let's first simplify the expression a^3 = some number × 25 × 45 × 49.
We know that a = ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Evaluate: 30 - | - x^2 + 6x + 6 | + 5*cos(2*x) for x = 10<|im_end|>
<|im_start|>assistant
First, let's evaluate the expression inside the absolute value:
For x = 10:
-x^2 + 6x + 6 = -(10)^2 + 6*10 + 6
... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The absolute value of $-11$ is ( ).
A: $1$
B: $11$
C: $-\frac{1}{11}$
D: $\frac{1}{11}$<|im_end|>
<|im_start|>assistant
To solve the problem, we start by understanding what the absolute value function ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In the arithmetic sequence ${a_n}$, ${a_1=8,a_3=4}$.
(1) Let the sum of the first $n$ terms of the sequence ${a_n}$ be ${S_n}$. Find the maximum value of ${S_n}$ and the value of $n$ that makes ${S_n}$ ma... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The number of roots of the function $f(x) = x^2 - 2x + 3$ is ( )
A: 0
B: 1
C: 2
D: 3<|im_end|>
<|im_start|>assistant
Since $f(x) = x^2 - 2x + 3$, the discriminant $\Delta = b^2 - 4ac = (-2)^2 - 4 \times 3 ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=(1-k)x+ \frac{1}{e^{x}}$.
(I) Find the monotonic intervals of the function $f(x)$;
(II) When $k=0$, there exists a tangent line to the function curve $f(x)$ passing through point $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A plane that is 1 unit away from the center of a sphere cuts the sphere, resulting in a cross-sectional area of $\pi$. The volume of the sphere is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
Solution: Cu... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Show that among the students in the class, there are 2 who know the same number of people. The acquaintance is a reciprocal relation.<|im_end|>
<|im_start|>assistant
1. **Introduction and Restatement of t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Evaluate $\frac{8}{29} - \frac{5}{87}$ and reduce the result to its simplest form.<|im_end|>
<|im_start|>assistant
To solve the problem, we first need a common denominator. Noticing that 87 is a multiple o... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Yoongi collected 4, and Jungkook collected 6 divided by 3, and Yuna collected 5. Who collected the biggest number?<|im_end|>
<|im_start|>assistant
Jungkook collected 6 divided by 3, which equals 2. So, Yo... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In an acute triangle \\(ABC\\), prove that: \\(\sin A + \sin B + \sin C > \cos A + \cos B + \cos C\\).<|im_end|>
<|im_start|>assistant
Proof: Since \\(\triangle ABC\\) is an acute triangle,
it follows tha... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In the non-decreasing sequence of odd integers $\{a_1, a_2, a_3, \ldots\} = \{1, 3, 3, 3, 5, 5, 5, 5, 5, \ldots\}$ where each odd positive integer $k$ appears $k+2$ times, determine integers $b, c$, and $d... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given $x+y=1$, where $x$ and $y$ are positive numbers, find the minimum value of $\frac{1}{x}+\frac{4}{y}$.<|im_end|>
<|im_start|>assistant
To find the minimum value of $\frac{1}{x}+\frac{4}{y}$ given $x+y... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A storage crate 4 centimeters high, 3 centimeters wide, and 6 centimeters long can hold 72 grams of coffee. A second crate, with 1.5 times the height and width, but the same length as the first crate, is u... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \(x_1, x_2, x_3\) be positive real numbers such that \(x_1 + 3x_2 + 4x_3 = 72\). Additionally, \(x_1 = 3x_2\). Find the smallest possible value of
\[
x_1^2 + x_2^2 + x_3^2.
\]<|im_end|>
<|im_start|>ass... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If an integer can be expressed in the form of $a^{2}+b^{2}$ (where $a$ and $b$ are positive integers), then this number is called a "Fengli number." For example, $2$ is a "Fengli number" because $2=1^{2}+1... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Sarah interviewed some students at her school and asked them which fruit they liked best - oranges, apples, pears or strawberries. 70 students said they liked oranges, 120 students said they liked pears, a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A Senate committee has 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee consisting of 4 Republicans and 3 Democrats?<|im_end|>
<|im_start|>assistant
1. **Choose Republicans**: Th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A point $(x,y)$ in the plane is termed a lattice point if both $x$ and $y$ are integers. Find the area of the largest square that contains exactly five lattice points in its interior.
A) 15.0
B) 17.0
C... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\]<|im_end|>
<|im_start|>assistant
To find the maximum value of the expression $\sqrt{abc} + \sqrt{(1 -... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider the quadratic equation $x^2 + px + q = 0$, where $p$ and $q$ are positive numbers. If the quadratic has complex roots such that the real part of each root is $\frac{1}{2}$, find the value of $p$.
... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The line with equation \( y = x + 4 \) is translated down 6 units. The \( y \)-intercept of the resulting line is
(A) 6
(B) 4
(C) 10
(D) -6
(E) -2<|im_end|>
<|im_start|>assistant
1. **Identify the given l... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If set $A=\{x|x^{2}+x-6=0\}$, $B=\{x|x\cdot m+1=0\}$, and $B\subseteq A$, then the set of possible values for $m$ is ____.<|im_end|>
<|im_start|>assistant
To solve for the set $A$, we start with the equati... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The line $l$ passes through point $P(1,4)$ and intersects the positive half of the $x$-axis and the positive half of the $y$-axis at points $A$ and $B$, respectively, with $O$ being the origin.
$(1)$ When ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A new cell phone plan costs $25$ per month, plus $8$ cents per text message sent, plus $15$ cents for each minute used over $25$ hours. In February, Sam sent $150$ text messages and talked for $26$ hours. ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The straight line passes through the origin and the point $(-1, -1)$. Calculate the inclination angle of the line.<|im_end|>
<|im_start|>assistant
The inclination angle of a line is the angle that the line... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A certain high school has a total of 1200 students in the senior year. A stratified sampling method is used to select a sample of 200 students for a health survey. If the sample contains 10 more boys than ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Is there a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that \( \lim _{x \rightarrow \infty} f(x)=+\infty \) and for every \( n \),
\[
\lim _{x \rightarrow \infty} \frac{f(x)}{\lg (\lg \ldots... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $a=3$, $b=2\sqrt{6}$, $\angle B = 2\angle A$.
(I) Find the value of $\cos A$;
(II) Find the value of $c$.<|im_end|>
<|im_start|>assistant
(I) Given that $a=3$, $b=2\sqrt{6}$, and $\angl... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A certain shopping mall purchased a batch of daily necessities at a unit price of $20$ yuan. If they are sold at a unit price of $30$ yuan, then 400 items can be sold in half a month. According to sales ex... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the sum of the interior angles of an \(n\)-sided convex polygon is \(x^{\circ}\), find \(x\).<|im_end|>
<|im_start|>assistant
1. **Recall the formula for the sum of the interior angles of an \( n \)-si... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let circle $\Theta$ be both the incircle of $\triangle DEF$ and the circumcircle of $\triangle PQR$. The point $P$ lies on $\overline{EF}$, point $Q$ on $\overline{DE}$, and point $R$ on $\overline{DF}$. G... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Triangle \(PQR\) is located entirely in the second quadrant. Let \(P=(a,b)\), \(Q=(c,d)\), and \(R=(e,f)\) with \(a, c, e<0\) and \(b, d, f > 0\). Points \(P\), \(Q\), and \(R\) are reflected across the li... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The sum of three numbers is 660. The first number is twice the second, and the third number is one-third of the first. What is the second number?<|im_end|>
<|im_start|>assistant
Let's denote the second num... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the one-variable quadratic equation $x^{2}-2x+m=0$ has two distinct real roots with respect to $x$, then the range of $m$ is ______.<|im_end|>
<|im_start|>assistant
To determine the range of $m$ for whi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A circle whose equation is \(x^2 + y^2 = 4\) and a parabola with the equation \(y = x^2\) are placed in a coordinate plane. Given that a line intersects both the circle and the parabola and is not tangent ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all numbers \( d \in (0, 1] \) that have the following property: if \( f(x) \) is an arbitrary continuous function defined for \( x \in [0, 1] \), where \( f(0) = f(1) \), then there exists a number \... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The arithmetic square root of $\sqrt{(-4)^{2}}$ is:
A: $\pm 2$
B: $\pm 4$
C: $2$
D: $4$<|im_end|>
<|im_start|>assistant
To solve for the arithmetic square root of $\sqrt{(-4)^{2}}$, we proceed as follo... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given a function $f(x)$ defined on $(0, +\infty)$, its derivative $f′(x)$ satisfies $$x^{2}f′(x) + 2xf(x) = \frac{\ln x}{x},$$ and $f(e) = \frac{1}{4e}$, where $e$ is the base of the natural logarithm. Fin... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $a$ and $b$ are real numbers, and $b^{2}+(4+i)b+4+ai=0$ (where $i$ is the imaginary unit), find the value of $|a+bi|$.<|im_end|>
<|im_start|>assistant
[Analysis]
This problem involves the operat... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given a sector with a radius of $10cm$ and a perimeter of $45cm$, find the central angle of the sector in radians.<|im_end|>
<|im_start|>assistant
**Analysis**
From the given information, we can calculate... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then
\[... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \( A(n) \) denote the largest odd divisor of the number \( n \). For example, \( A(21) = 21 \), \( A(72) = 9 \), and \( A(64) = 1 \). Find the sum \( A(111) + A(112) + \ldots + A(218) + A(219) \).<|im_... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
For each natural number \( n \in \mathbf{N} \), find the maximum value of the product of natural numbers with a fixed sum \( n \).<|im_end|>
<|im_start|>assistant
Given a fixed natural number \( n \in \ma... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The sum of the first 2005 terms of the sequence 1, 2, 3, 4, 1, 2, 3, 4, ... is:
(A) 5011
(B) 5110
(C) 5020
(D) 5010
(E) 501 <|im_end|>
<|im_start|>assistant
1. **Identify the repeating pattern**:... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the greatest number of consecutive integers whose sum is $45?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$<|im_end|>
<|im_start|>a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many two-digit prime numbers can be formed by choosing two different digits from the set $\{3, 7, 8, 9\}$ to be used as the tens digit and units digit?<|im_end|>
<|im_start|>assistant
To find the solut... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In recent years, running has become a lifestyle for more and more people. According to official data, the number of participants in the half marathon in Shanghai reached 78,922 in the year 2023. Express th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A sum fetched a total simple interest of 4016.25 at a certain rate per annum in 5 years. The sum is 5737.5. What is the rate of interest per annum?<|im_end|>
<|im_start|>assistant
To find the rate of inter... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Which of the following statements about a parallelepiped is correct?
A: A parallelepiped has 12 vertices.
B: A cube is not a parallelepiped.
C: A parallelepiped has 12 edges.
D: Each face of a parallel... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider a geometric figure consisting of three triangles as follows: Triangle ABC, Triangle DEF, and Triangle GHI, where point D is on line segment AC, point F is on line segment BI, and point H is on lin... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Calculate the positive difference between the solutions of the equation $|x - 3| = 15$.<|im_end|>
<|im_start|>assistant
Let's denote the two solutions to the equation $|x - 3| = 15$ as $x_1$ and $x_2$. The... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A six-digit number is formed by repeating a three-digit number, such as 256256, or 678678, etc. This type of number cannot be divided by ( )
A: 11
B: 101
C: 13
D: 1001<|im_end|>
<|im_start|>assistant
**A... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the \(45^{\text{th}}\) even positive integer?<|im_end|>
<|im_start|>assistant
Every even positive integer can be written in the form \(2x\), where \(x\) is a positive integer. The sequence of even ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Ma Xiaohu's home is $1800$ meters away from school. One day, Ma Xiaohu went to school from home. After leaving for $10$ minutes, his father found out that he had forgotten his math textbook and immediately... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the domain of the function $g(x) = \frac{x-3}{\sqrt{x^2 - 5x + 6}}$?<|im_end|>
<|im_start|>assistant
For the function $g(x) = \frac{x-3}{\sqrt{x^2 - 5x + 6}}$ to be defined:
1. The expression under... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle $ABC$, $BC = 6$, $AC = 3$, and $\angle C = 30^\circ$. Altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. Find $AH:HD$.<|im_end|>
<|im_start|>assistant
First, we calculate $AB$ and... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Define a function $f(x)$ on $\mathbb{R}$ that satisfies: $f'(x) > 1 - f(x)$, $f(0) = 6$, where $f'(x)$ is the derivative of $f(x)$. The solution set for the inequality $e^x f(x) > e^x + 5$ (where $e$ is th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Lucille made an agreement with her mom that she would earn six cents for every weed she pulled in her mom's garden. There are eleven weeds in the flower bed, fourteen in the vegetable patch, and thirty-two... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Brad has a cylindrical barrel with radius 8 inches and height 24 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 8 inches. How many trips must Brad mak... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\{a\_n\}$ be a geometric sequence with a common ratio $q > 1$. If $a_{2016}$ and $a_{2017}$ are the roots of the equation $4x^2-8x+3=0$, then $a_{2018}+a_{2019}=$ \_\_\_\_\_\_.<|im_end|>
<|im_start|>a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A piece of plywood has a total area of 35.7 square feet, with a width of 3.8 feet. If you double the width of this plywood, what is the new length required to maintain the same total area?<|im_end|>
<|im_s... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Define the operation $a \otimes b =
\begin{cases}
a & \text{if } a \geq b, \\
b & \text{if } a < b
\end{cases}$. Let the function $f(x) = x \otimes (2-x)$.
(1) Use algebraic methods to prove: The graph... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the function $f\left(x\right)$ defined on $R$ is monotonically increasing on $\left[0,+\infty \right)$, and the function $f\left(x\right)-1$ is an odd function, then the solution set of $f\left(... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What value of $x$ will give the maximum value for the function $-2x^2 + 8x + 16$?<|im_end|>
<|im_start|>assistant
We begin by finding the vertex of the parabola defined by $f(x) = -2x^2 + 8x + 16$.
First,... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
An equilateral triangle has sides 10 units long. An equilateral triangle with sides 6 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the equations of the two altitudes in triangle $ABC$ are $2x-3y+1=0$ and $x+y=0$, with vertex $A$ at $(1,2)$, find:
(1) The equation of the line on which side $BC$ lies;
(2) The area of $\triangl... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, with $a_1=1$, and $S_{n+1}=4a_n+2$ ($n\in\mathbb{N}^+$).
(1) If $b_n=a_{n+1}-2a_n$, find $b_n$;
(2) If $$c_{n}= \frac {1}{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If $p$, $q$, and $N$ are positive numbers and $q < 50$, determine under what conditions the number obtained by increasing $N$ by $3p\%$ and then decreasing the result by $2q\%$ is greater than $N$.
- **(A)... |
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