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
If the equation $(5x+ \frac {5}{x})-|4x- \frac {4}{x}|=m$ has exactly four distinct real roots in $(0,+\infty)$, then the range of the real number $m$ is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
When ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x) = e^x(x-b) \ (b \in \mathbb{R})$. If there exists $x \in \left[ \frac{1}{2},2 \right]$ such that $f(x) + xf''(x) > 0$, then the range of the real number $b$ is $(\ )$.
A: $(-\infty... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the initial side of angle $\alpha$ coincides with the positive half of the x-axis, and its terminal side lies on the ray defined by $3x-4y=0$ (where $x<0$), find the value of $\sin\alpha - \cos\... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Calculate:
\\((1) \sqrt{48}+ \sqrt{8}- \sqrt{18}- \sqrt{12} \\) \\((2) 2( \sqrt{2}+ \sqrt{3})-( \sqrt{3}- \sqrt{2})^{2} \\).<|im_end|>
<|im_start|>assistant
This problem examines the mixed opera... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $f(x) = x^3 - 1$ and $g(x) = 3x^2 + 3x + 1$. What is $g(f(-3))$?<|im_end|>
<|im_start|>assistant
First, compute $f(-3)$:
$f(-3) = (-3)^3 - 1 = -27 - 1 = -28$.
Next, substitute $-28$ into $g(x)$:
$g(-28... |
<|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 algorithms and the three types of logical structures (sequential structure, conditional structure, loop structure) is correct?
A: An algorithm can contain at most tw... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Selene has 120 cards numbered from 1 to 120, inclusive, and she places them in a box. Selene then chooses a card from the box at random. What is the probability that the number on the card she chooses is a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the area enclosed by the region defined by the equation $x^2 + y^2 - 8x + 6y = -9$?<|im_end|>
<|im_start|>assistant
1. **Complete the square for $x$:**
\( x^2 - 8x \rightarrow (x^2 - 8x + 16) ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=\ln x+ \frac{1-x}{ax}$, where $a$ is a positive constant.
(1) Find the range of values for $a$ such that the function $f(x)$ is monotonically increasing in the interval $[1,+\infty... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $U={1,2,...,100}$. For the sequence ${a_n}$ ($n∈N^*$) and the subset $T$ of $U$, if $T=∅$, define $S_T=0$; if $T={t_1,t_2,...,t_k}$, define $S_T=a_{t_1}+a_{t_2}+...+a_{t_k}$. For example, when $T={1,3,... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \%$ faster, he would have arrived at th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given five distinct real numbers, consider all their pairwise sums. The smallest two sums are 20 and 200, while the largest two sums are 2014 and 2000. Determine the range of the smallest number, \( a \), ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \( C \) and \( D \) be two distinct points on a semicircle of diameter \([A B]\). The lines \((A C)\) and \((B D)\) intersect at \( F \), and the lines \((A D)\) and \((B C)\) intersect at \( E \). Sho... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The expression $2y^2 - 5y - 12$ can be written as $(2y + a)(y + b),$ where $a$ and $b$ are integers. What is $a - b$?<|im_end|>
<|im_start|>assistant
We start by expressing the quadratic $2y^2 - 5y - 12$ i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let set $U = \{ x \mid x \text{ is a positive integer less than } 9\}$, set $A = \{1, 2, 3\}$, and set $B = \{3, 4, 5, 6\}$. Then the intersection of the complements of $A$ and $B$ with respect to $U$, den... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Simplify first, then evaluate: $3(a^{2}-ab+7)-2(3ab-a^{2}+1)+3$, where $a=2$ and $b=\frac{1}{3}$.<|im_end|>
<|im_start|>assistant
To simplify and then evaluate the given expression $3(a^{2}-ab+7)-2(3ab-a^{... |
<|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 is correct? ( )
A: The square of number $a$ is positive.
B: The absolute value of number $a$ is positive.
C: The square root of 16 equals 4.
D: 3 is the square root of 9.... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the graph of a linear function passes through the points (3, 5) and (-4, -9),
(1) Find the equation of this linear function;
(2) Determine whether the point (-1, -3) is on the graph of this line... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider the function \( g(x) = \log \left(\frac{2+x}{2-x}\right) \) for \( -2 < x < 2 \). Evaluate \( g\left(\frac{4x+x^2}{2+2x}\right) \) in terms of \( g(x) \).
A) \( g(x) \)
B) \( 2g(x) \)
C) \( 3g(x) ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
On a board, nine quadratic trinomials are written: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A list of five positive integers has mean $15$ and range $20$. The mode and median are both $10$. Determine how many different values are possible for the second largest element of the list.
$\mathrm{(A) \... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Ray writes a two-digit number. He sees that the number exceeds 4 times the sum of its digits by a certain value. If the number is increased by 18, the result is the same as the number formed by reversing t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the length of the longest pole that can be placed in a room 12 m long, 8 m broad, and 9 m high.<|im_end|>
<|im_start|>assistant
The longest pole that can be placed in a room would be the diagonal of t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
$a=0$ is the condition for the complex number $z=a+bi$ ($a, b \in \mathbb{R}$) to be a purely imaginary number ( ).
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a$ and $b$ be the roots of the equation $x^2 - 6x + 8 = 0$. Compute:
\[a^2 + a^5 b^3 + a^3 b^5 + b^2.\]<|im_end|>
<|im_start|>assistant
First, by Vieta's formulas:
- $a + b = 6$
- $ab = 8$
Compute $a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:
$$
\begin{array}{ccccccc}
X & 0 & 1 & 2 & \cdots & k & \cdots \\
P & \mathrm{e}^{-\lambda} ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two stations p and q are 200 km apart on a straight track. One train starts from p at 7 a.m. and travels towards q at 20 kmph. Another train starts from q at a certain time and travels towards p at a speed... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $m$ and $n$ are opposites, $a$ and $b$ are negative reciprocals, and the absolute value of $x$ equals 3, then the value of the expression $x^3 - (1+m+n+ab)x^2 + (m+n)x^{2004} + (ab)^{2005}$ equa... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Tom and Tim both brought 4, six-sided dice to school. How many total sides are there?<|im_end|>
<|im_start|>assistant
To solve the problem, we start by calculating the total number of dice Tom and Tim brou... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
For all positive integer $n$ , define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$ .
Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$ -digit integers $n$ (in de... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Evaluate $\cos 420^{\circ} + \sin 330^{\circ}$.
A: $1$
B: $-1$
C: $\dfrac{1}{2}$
D: $0$<|im_end|>
<|im_start|>assistant
We have $\cos 420^{\circ} + \sin 330^{\circ} = \cos (360^{\circ} + 60^{\circ}) + ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Prove that
$$
\int_{0}^{\pi}|\ln \sin x| \, dx \leq \frac{\pi^{3}}{12}
$$
where $\ln$ denotes the natural logarithm with base $e = 2.71\ldots$.<|im_end|>
<|im_start|>assistant
1. **Understanding the Int... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least $(\sqrt{2}-1)$ times the radius of the circumcircle.<|im_end|>
<|im_start|>assistant
1. **Identifi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the polynomial expression \[2a(b - c)^3 + 3b(c - a)^3 + 2c(a - b)^3,\] it is known to be factorable into the form $(a - b)(b - c)(c - a) p(a,b,c)$. Find the polynomial $p(a,b,c)$.<|im_end|>
<|im_sta... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two pipes A and B can fill a tank in 10 hours and some hours respectively. If both the pipes are opened simultaneously, the tank will be filled in approximately 6 hours. How much time will pipe B take to f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The line $y=kx-2$ intersects the parabola $y^2=8x$ at points A and B, and the x-coordinate of the midpoint of AB is 2. Find the value of $k$.<|im_end|>
<|im_start|>assistant
Since the line $y=kx-2$ and the... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In the book "Nine Chapters on the Mathematical Art," there is a mathematical problem stated as follows: During the Qingming Festival outing to the park, a total of eight boats were taken. Each large boat h... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the maximum value of $3 \cos x + 4 \sin x$ over all angles $x$.<|im_end|>
<|im_start|>assistant
We start by identifying that the expression $3 \cos x + 4 \sin x$ can be rewritten in the form of $R \co... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Calculate the volume of the body bounded above by the surface \(z = xy^2\) and below by the rectangle \(0 \leq x \leq 1\), \(0 \leq y \leq 2\).<|im_end|>
<|im_start|>assistant
1. **Identify the problem:**... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Vasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equa... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Harry, Tom, and Dick start a 150-mile journey together. Tom and Dick initially travel by car at 30 mph, while Harry walks at 4 mph. After 50 miles, Harry continues walking, and Tom goes back to pick up Dic... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jackson is making dinner. He makes a salad out of lettuce (50 calories), carrots (twice the calories of the lettuce) and dressing (210 calories). He also makes a pizza with 600 calories for the crust, 1/3 ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A unit has 200 employees. Now, 40 employees need to be selected as a sample using the systematic sampling method. All employees are randomly numbered from 1 to 200 and evenly divided into 40 groups accordi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given sets $A=\{x|(x-2)(x-(a+1)) < 0\}$ and $B=\left\{x\middle| \frac{x-2a}{x-(a^{2}+1)} < 0\right\}$. Find the range of real values of $a$ for which $B \subseteq A$.<|im_end|>
<|im_start|>assistant
When $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
At North High School, 2200 students have choices among basketball, football, and baseball. At South Academy, there are 2600 students. In the two schools combined, the percentage of students who prefer bask... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider two equiangular polygons $P_1$ and $P_2$. Each angle of $P_1$ is $x$ degrees, and each angle of $P_2$ is $\frac{x}{2}$ degrees. Determine the number of possibilities for the integer number of side... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $α\in\mathbb{R}$ and $\sin α + 2\cos α = \frac{\sqrt{10}}{2}$, find the value of $\tan α$.<|im_end|>
<|im_start|>assistant
Since we are given that $\sin α + 2\cos α = \frac{\sqrt{10}}{2}$, we ca... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A team of 60 workers constructed a bridge in 5 days working at the same efficiency. If the subsequent team worked at 80% efficiency and consisted of 40 workers, how many days would it have taken this team ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The slope angle of the tangent line to the graph of the function $f(x) = \ln(x^2+1)$ at the point $(1, f(1))$ is
A: 0
B: $\frac{\pi}{2}$
C: $\frac{\pi}{3}$
D: $\frac{\pi}{4}$<|im_end|>
<|im_start|>assista... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In the given figure, semicircles with centers at points \( A \) and \( B \) and radii 2 and 1, respectively, are inscribed in a larger semicircle with diameter \( JK \). The two smaller semicircles are als... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Expand $(3t^3 + 2t^2 - 4t + 3)(-2t^2 + 3t - 4)$.<|im_end|>
<|im_start|>assistant
Using the distributive property, the expansion is given by:
\[
(3t^3 + 2t^2 - 4t + 3)(-2t^2 + 3t - 4) = 3t^3(-2t^2 + 3t - 4)... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the remainder when $$123456 + 123457 + 123458 + 123459 + 123460 + 123461$$ is divided by 11.<|im_end|>
<|im_start|>assistant
First, we reduce each number modulo 11:
\[
123456 \equiv 1, \quad 123457 \e... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Among all the roots of
\[ z^6 - z^4 + z^2 - 1 = 0, \]
the maximum imaginary part of a root can be expressed as $\cos \phi$, where $-90^\circ \le \phi \le 90^\circ$. Find $\phi$.<|im_end|>
<|im_start|>assis... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The tower in King Arthur's castle is topped with a roof that represents a triangular pyramid in which all plane angles at the apex are right angles. The three roof slopes are painted in different colors. T... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
m = { - 9 , - 7 , - 5 , - 3 , 0, 5 , 7 }
t = { - 8 , - 6 , - 2 , 0, 3 , 4 , 6 , 7 }
If an integer is to be randomly selected from set m above and an integer is to be randomly selected from set t above, w... |
<|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\\), and it is known that \\(S_n = 2a_n - 2^{n+1} (n \in \mathbb{N}^*)\).
\\((1)\\) Find the general formula for the seq... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given sets $M=\{2, 4\}$, $N=\{1, 2\}$, and $P=\{x|x= \frac{a}{b}, a\in M, b\in N\}$, then the number of subsets of set $P$ is ( )
A: 3
B: 4
C: 8
D: 16<|im_end|>
<|im_start|>assistant
Since $M=\{2, ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \( S_{n} \) be the number of sequences \((a_{1}, a_{2}, \ldots, a_{n})\), where \( a_{i} \in \{0,1\} \), in which no six consecutive blocks are equal. Prove that \( S_{m} \rightarrow \infty \) as \( n ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits).<|im_end|>
<|im_start|>assista... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Every day, Lou works out by running three miles on a circular track that is one-quarter of a mile long and has a constant incline of 5%. His wife, Rosie, also runs on the same track at the same time as her... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jim spends 8 hours scuba diving. In that time he finds a treasure chest with 100 gold coins in it. He also finds some smaller bags that have half as much gold each. He finds 25 gold coins per hour. How man... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The zero of the function $f(x) = 1 - \log_{2}x$ is ( )
A: $(1, 1)$
B: $1$
C: $(2, 0)$
D: $2$<|im_end|>
<|im_start|>assistant
Solution: Let $f(x) = 1 - \log_{2}x = 0$, we can get $x = 2$.
Therefo... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A student was asked to find 4 / 5 of a number. But the student divided the number by 4 / 5, thus the student got a certain amount more than the correct answer. The number is 60.00000000000002. How much mor... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The price of rice falls by 20%. With the money that was sufficient to buy a certain amount of rice previously, now 25 kg of rice can be bought. How much rice could be bought previously with the same amount... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The sides of an isosceles triangle are $\sin x,$ $\sin x,$ and $\sin 5x,$ and its vertex angle is $2x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas.<|im... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $z$ be a complex number such that
\[|z - 8| + |z + 6i| = 17.\] Find the smallest possible value of $|z|$.<|im_end|>
<|im_start|>assistant
First, by the Triangle Inequality,
\[ |z - 8| + |z + 6i| = |z -... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Among the following numbers, the positive integers are:
A: $3$
B: $2.1$
C: $0$
D: $-2$<|im_end|>
<|im_start|>assistant
To determine which of the given options are positive integers, we evaluate each op... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In the plane rectangular coordinate system $xOy$, it is known that the initial side of angle $\alpha$ is the non-negative half-axis of the $x$-axis, and the terminal side passes through the point $P(-1,2)$... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many positive integers $n$ satisfy \[\dfrac{n+800}{80} = \lfloor \sqrt{n} \rfloor?\]
(Recall that $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)<|im_end|>
<|im_start|>assistant
1. **Se... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the sets $A=\{-2,-1,0,1,2\}$ and $B=\{x|x^{2}\leqslant 1\}$, find the intersection $A\cap B=$ .
A: $\{-2,-1,0,1\}$
B: $\{-1,1\}$
C: $\{-1,0\}$
D: $\{-1,0,1\}$<|im_end|>
<|im_start|>assistant
Sin... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$ . Define an **operation** to be 22 consecutive application... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
One fourth of one third of two fifth of a number is 16. What is the value of two fifth of that number?<|im_end|>
<|im_start|>assistant
Let's call the number we're looking for "N".
According to the proble... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
It is known that the length of the segment representing the product of two segments depends on the choice of the unit segment. How does the length of the product segment change if the unit segment is halve... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A lecture was given for 90 minutes to an audience of 200 people. Thirty percent of the audience listened to the full lecture, while fifteen percent were not attentive throughout the lecture. Of the remaini... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Edward spent $250 on clothes, then spent 35% of his remaining money on food. After buying food, he used 50% of the remaining money to purchase electronics, and at that point had $200 left. How much did Edw... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A firecracker was thrown vertically upward with a speed of \(20 \text{ m/s}\). One second after the flight began, it exploded into two fragments of equal mass. The first fragment flew horizontally immediat... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the sum \( S_{n} = \sin \alpha \cdot \sin ^{2} \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin ^{2} \frac{\alpha}{4} + \cdots + 2^{n-1} \cdot \sin \frac{\alpha}{2^{n-1}} \cdot \sin ^{2} \frac{\a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=ax^{2}+bx+c$, where $a\in\mathbb{N}^{*}$, $b\in\mathbb{N}$, $c\in\mathbb{Z}$.
$(1)$ If $b > 2a$, and $f(\sin x)$ ($x\in\mathbb{R}$) has a maximum value of $2$ and a minimum value ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the function $y=f(x)$ has a domain of $(0,+\infty)$, and when $x > 1$, $f(x) > 0$. For any $x$, $y\in(0,+\infty)$, the equation $f(x)+f(y)=f(xy)$ holds. If the sequence $\{a_n\}$ satisfies $a_1=... |
<|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 and third of three consecutive integers is 118. What is the value of the second integer?<|im_end|>
<|im_start|>assistant
Let's denote the second integer as $n$. Given that these are co... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ $c$ be three terms in an arithmetic series where all terms are positive and the common difference is $d$. If $abc = 125$, find the smallest possible value of $b$.<|im_end|>
<|im_start|>assist... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The graph of the function $y = a^{x+1} - 1$ ($a > 0$ and $a \neq 1$) must pass through the point $\_\_\_\_\_\_$.<|im_end|>
<|im_start|>assistant
For the function $y = a^{x+1} - 1$ ($a > 0$ and $a \neq 1$),... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In $\triangle ABC$, point $D$ lies on segment $BC$ and does not coincide with $B$ or $C$. The circumcircle of $\triangle ABD$ intersects segment $AC$ at another point $E$, and the circumcircle of $\triangl... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Dana has 15 more pencils than Jayden, who has twice as much as Marcus. How many more pencils does Dana have than Marcus, if Jayden has 20 pencils?<|im_end|>
<|im_start|>assistant
If Jayden has 20 pencils, ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
After spending Rs. 5000 on rent, Rs. 1500 on milk, Rs. 4500 on groceries, Rs. 2500 on children's education, some amount on petrol, and Rs. 5650 on miscellaneous expenses, Mr. Kishore saved 10% of his month... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given a circle O with equation $x^2+y^2=13$. A perpendicular line to the y-axis is drawn through any point P on circle O, with Q as the foot of the perpendicular. Find the equation of the locus of the midp... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If $7$ lunks can be traded for $4$ kunks, and $3$ kunks will buy $5$ apples, how many lunks are needed to purchase two dozen apples?<|im_end|>
<|im_start|>assistant
1. First, determine the number of kunks ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the inequality $x^2 - ax + 2a > 0$ is always true for all $x$ in $\mathbb{R}$, find the range of the real number $a$.<|im_end|>
<|im_start|>assistant
Since the inequality $x^2 - ax + 2a > 0$ is ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
On the sides of a convex centrally symmetric hexagon \( A B C D E F \), equilateral triangles are constructed externally. Prove that the midpoints of the segments connecting the vertices of adjacent triang... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
There are 50 boys and 50 girls lined up in some order. In this row, there exists exactly one group of 30 consecutive children where the number of boys and girls is equal. Prove that there exists a group of... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=3^n-2$, find the general formula for the term $a_n=$ ______.<|im_end|>
<|im_start|>assistant
To solve, we know the sum of the first $n$... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \in \{70,85,88,90,98,100\}$ and it satisfies $f(1)<f(2) \leq f(3)<f(4)$, then the tot... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A person writes \( n \) letters for \( n \) individuals, each receiving one letter, and prepares \( n \) envelopes labeled with the recipients' names and addresses. How many different ways are there to pla... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the ellipse $\dfrac {x^{2}}{a^{2}}+y^{2}=1$ passes through the point $P(1, \dfrac {\sqrt {6}}{3})$, then the eccentricity $e$ of the ellipse is
A: $\dfrac {\sqrt {3}}{2}$
B: $\sqrt {3}-1$
C: $\dfrac {\s... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A car drives 60 miles on local roads at a certain speed, and 120 miles on the highway at 60 mph. The average speed of the entire trip is 36 mph. What is the speed of the car on local roads?<|im_end|>
<|im_... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Evaluate the infinite geometric series: $$\frac{1}{4} + \frac{1}{12} + \frac{1}{36} + \frac{1}{108} + \dots$$<|im_end|>
<|im_start|>assistant
1. **Identify the first term and common ratio**: The first term... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
During the baseball season, Judy had $50$ hits. Among her hits were $2$ home runs, $3$ triples, and $8$ doubles. The rest of her hits were singles. What percent of her hits were singles?
A) 68%
B) 70%
C) 7... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A jacket is originally priced at \$50 and is on sale for 15% off. John has a special member card that grants an additional 30% discount on the already reduced price. What single percent discount would resu... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given $x$ and $y$ as positive real numbers such that $x \leq y$, show that $\frac{1}{y^{2}} \leq \frac{1}{x^{2}}<|im_end|>
<|im_start|>assistant
1. **Given**: We have two positive real numbers \(x\) and \... |
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