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
Consider the terms of an arithmetic sequence: $-\frac{1}{3}, y+2, 4y, \ldots$. Solve for $y$.<|im_end|>
<|im_start|>assistant
For an arithmetic sequence, the difference between consecutive terms must be eq... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Suppose that $g(x) = 5x - 3$. What is $g^{-1}(g^{-1}(14))$?<|im_end|>
<|im_start|>assistant
First, we need to find the inverse function $g^{-1}(x)$. Given $g(x) = 5x - 3$, solve for $x$:
\[ y = 5x - 3 \]
\... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A farmer has a rectangular field with dimensions $3m+8$ and $m-3$ where $m$ is a positive integer. If the field has an area of 76 square meters, find the value of $m$.<|im_end|>
<|im_start|>assistant
Using... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the functions $f(x) = \log_a(1+x)$ and $g(x) = \log_a(1-x)$, where $a>0$ and $a \neq 1$.
1. Find the domain of the function $f(x) - g(x)$.
2. Determine the parity of the function $f(x) - g(x)$.
3. Fi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all solutions to the equation $\displaystyle\sqrt[3]{3 - \frac{x}{3}} = -2$.<|im_end|>
<|im_start|>assistant
Start by isolating the cube root:
$$ \sqrt[3]{3 - \frac{x}{3}} = -2 $$
Cube both sides to ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\cos \frac{C}{2} = \frac{\sqrt{5}}{3}$ and $a \cos B + b \cos A = 2$, find ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Julian is writing a comic book. On average, his story has 280 frames per page. In his 25-page book, 10 pages have 305 frames, 7 pages have 250 frames, and the remaining pages have the average number of fra... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If an arc of $60^{\circ}$ on circle $C$ has the same length as an arc of $40^{\circ}$ on circle $D$, what is the ratio of the area of circle $C$ to the area of circle $D$? Express your answer as a common f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $P$ is any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \overrightarrow{OP}+ \overrightarrow{OQ}|$.<|im_end|>
<|i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute $\cos 225^\circ$.<|im_end|>
<|im_start|>assistant
Let $Q$ be the point on the unit circle that corresponds to $225^\circ$ measured counterclockwise from the positive $x$-axis.
The angle $225^\cir... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that $b$ is a multiple of $2373$, find the greatest common divisor of $b^2 + 13b + 40$ and $b + 5$.<|im_end|>
<|im_start|>assistant
We use the Euclidean Algorithm:
\[
\text{gcd}(b^2 + 13b + 40, b + 5... |
<|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 real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \).<|im_end|... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Tom decides to open a theater. He knows it will cost $5 per square foot of space that he needs and he needs 12 square feet for every seat in his theater. He wants a 500 seat theater. He also realizes const... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Factorize: $x^{3}-6x^{2}+9x=\_\_\_\_\_\_.$<|im_end|>
<|im_start|>assistant
To factorize the given expression $x^{3}-6x^{2}+9x$, we start by identifying a common factor in all terms. Observing the expressio... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \cdot DF = DA^2$.<|im_end|>
<|i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider the product of $x^5$, $x^2 + \frac{1}{x^2}$, and $1 + \frac{2}{x} + \frac{3}{x^2}$. Determine the degree of the resulting polynomial.
$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\q... |
<|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 $a$, $b$, and $c$ is 120. If we increase $a$ by 8, decrease $b$ by 3 and multiply $c$ by 3, the three resulting numbers are equal. What is the value of $b$?<|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 vessel of capacity 2 liters has a certain percentage of alcohol and another vessel of capacity 6 liters has 40% alcohol. The total liquid of 8 liters was poured out into a vessel of capacity 10 liters, a... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the sum of the first $n$ terms of a sequence $\{a\_n\}$ is $S\_n$, and $S\_n=2^{n}a\_n-1$, find the sum of the first $n$ terms of the sequence $\{\frac{a\_n}{a_{n+1}}\}$, denoted as $T\_n=$ `___... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the value of $k$ such that all roots of the polynomial $x^4 - 4x^3 + 4x^2 + kx - 4$ are real.<|im_end|>
<|im_start|>assistant
Let $p, q, r, s$ be the real roots of the polynomial. According to Vieta's... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
At its Grand Opening, the Guthrie Market distributed two different types of souvenirs. One type cost 20 cents each, and the other type cost a certain amount. One thousand souvenirs were distributed in all,... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \)<... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Three vertices of a cube are $A = (5, 9, 6)$, $B = (5, 14, 6)$, and $C = (5, 14, 11)$. What is the surface area of the cube?<|im_end|>
<|im_start|>assistant
Calculate the distances between the points $A, B... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$. If $QR = 2001$, $SU = 2002$, and $TV = 2003$, find the distance b... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=x\ln x$, if $0 \lt x_{1} \lt x_{2}$, then which of the following options is correct?
A: $x_{1}+f(x_{1}) \lt x_{2}+f(x_{2})$
B: $x_{2}f(x_{1}) \lt x_{1}f(x_{2})$
C: When ${x_2}＞{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given \\(f(x)=2a-\sin x\\), then \\(f''(x)=\\)
A: \\(\cos x\\)
B: \\(-\cos x\\)
C: \\(2+\cos x\\)
D: \\(2-\cos x\\)<|im_end|>
<|im_start|>assistant
**Analysis**
This question mainly tests the oper... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jimmy decides to make sandwiches for a picnic. He makes 8 sandwiches in total, using two slices of bread each. How many packs of bread does he need to buy to make these sandwiches, assuming he starts wit... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many two-digit positive integers are congruent to 1 (mod 5)?<|im_end|>
<|im_start|>assistant
If an integer is congruent to 1 mod 5, then it can be expressed as $5k+1$. We need it to fulfill the conditi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the following four propositions:
① The negation of the proposition "For all $x \in \mathbb{R}, \cos(x) > 0$" is "There exists an $x \in \mathbb{R}$ such that $\cos(x) \leq 0$".
② If $0 < a < 1$, ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The molecular weight of a certain substance is 74. What is the molecular weight of Bleach?<|im_end|>
<|im_start|>assistant
Bleach is a common name for a solution of sodium hypochlorite (NaOCl) in water. Th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
We flip a fair coin 12 times. What is the probability that we get heads in exactly 9 of the 12 flips?<|im_end|>
<|im_start|>assistant
- The total number of possible outcomes when flipping a coin 12 times i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many distinct, positive factors does $1320$ have?<|im_end|>
<|im_start|>assistant
First, perform the prime factorization of $1320$.
\[ 1320 = 2^3 \times 3 \times 5 \times 11 \]
Each factor of $1320$ c... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Observe the pattern of the following equations and answer the following questions:
The first equation: $1^{2}+2^{2}-3^{2}=1\times a-b$,
The second equation: $2^{2}+3^{2}-4^{2}=2\times 0-b$,
The third... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the product of the solutions of the equation: $|z| = 3(|z| - 2)$.<|im_end|>
<|im_start|>assistant
First, express the equation in the basic form:
$$ |z| = 3(|z| - 2). $$
Expanding and rearranging ter... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10?<|im_end|>
<|im_start|>assistant
Given the constraints \(1 \leq x, y, z... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A square of side $4$ is dissected into four pieces such that $E$ and $F$ are the midpoints of adjacent sides and $AG$ is perpendicular to $BF$. These four pieces are then reassembled into a rectangle. Calc... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Dilan, Martha, Phillip, and Veronica went to the park together to have some fun. They all had a different number of marbles. At the end of the day, they redistributed the marbles so they each had 15 marble... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The following line is parameterized, so that its direction vector is of the form \(\begin{pmatrix} b \\ 1 \end{pmatrix}\). Find $b$.
Consider a line passing through the points \(\begin{pmatrix} -4 \\ 6 \e... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let \( a_0, a_1, \ldots \) be a sequence of positive integers, and let \( (b_n) \) be the sequence defined by \( b_n = \operatorname{gcd}(a_n, a_{n+1}) \) for all \( n \geq 0 \). Show that it is possible t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Casey is going to stay at a motel while she works her summer job as a ski instructor. The hotel charges a certain amount per week or $1000/month. Casey will be staying 3 months, each with exactly 4 weeks. ... |
<|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 3000 terms of a geometric sequence is 500. The sum of the first 6000 terms is 950. Find the sum of the first 9000 terms.<|im_end|>
<|im_start|>assistant
Let the first term be $a$, and ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
David obtained 96 marks in English, 98 in Mathematics, 99 in Physics, some marks in Chemistry, and 98 in Biology. His average marks are 98.2. What are his marks in Chemistry?<|im_end|>
<|im_start|>assistan... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the least positive multiple of 25 that is greater than 500?<|im_end|>
<|im_start|>assistant
To find the least positive multiple of 25 that is greater than 500, we start by dividing 500 by 25:
\[ \f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that an interior angle of a regular polygon is $144^{\circ}$, then the number of sides of this regular polygon is ( )
A: $12$
B: $10$
C: $8$
D: $6$<|im_end|>
<|im_start|>assistant
Given that an i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
if p is the product of the integers from 1 to 35 , inclusive , what is the greatest integer k for which 3 ^ k is a factor of p ?<|im_end|>
<|im_start|>assistant
To find the greatest integer \( k \) for whi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=(ax^2+x+a)e^{-x}$ $(a\in \mathbb{R})$.
$(1)$ If $a\geqslant 0$, the maximum value of the function $f(x)$ is $\frac{3}{e}$, find the value of the real number $a$;
$(2)$ If for any... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $g_0(x) = x + |x-200|-|x+200|$, and for $n \geq 1$, let $g_n(x) = |g_{n-1}(x)|-1$. For how many values of $x$ is $g_{150}(x)=0$?<|im_end|>
<|im_start|>assistant
First, simplify $g_0(x)$:
\[ g_0(x) = \l... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, where $\overrightarrow{a}=(1,2)$
(Ⅰ) If $|\overrightarrow{c}|=3\sqrt{5}$ and $\overrightarrow{a} \parallel \overri... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle \( DEF \), \( DE = 6 \), \( EF = 8 \), and \( DF = 10 \).
Point \( Q \) is randomly selected inside triangle \( DEF \). What is the probability that \( Q \) is closer to \( F \) than it is to ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \((a - b)^{2}\) divides \( ab \) for every distinct \( a, b \in S \).<|im_end|>
<|im_start|>assistant
Given the pro... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$.<|im_end|>
<|im_start|>assistant
Sinc... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute
\[
\sum_{n=1}^\infty \frac{3n + 2}{n(n + 1)(n + 3)}.
\]<|im_end|>
<|im_start|>assistant
First, decompose $\frac{3n + 2}{n(n + 1)(n + 3)}$ into partial fractions:
\[
\frac{3n + 2}{n(n + 1)(n + 3)} =... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Solve the inequality
\[ -8x^2 + 6x - 1 < 0. \]<|im_end|>
<|im_start|>assistant
Firstly, find the discriminant of the quadratic equation \(-8x^2 + 6x - 1 = 0\):
\[
\Delta = 6^2 - 4(-8)(-1) = 36 - 32 = 4.
\]... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Each edge of a cube is colored either red or black. Every face of the cube has at least two black edges. The smallest number possible of black edges is:
$\text{(A) } 5\quad \text{(B) } 6\quad \text{(C) } 7... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If $3x - y = 9$ and $x + 4y = 11$, what is the value of $x$?<|im_end|>
<|im_start|>assistant
To solve for $x$, we can use elimination to remove $y$. First, multiply the first equation by 4 and the second e... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider a quadratic polynomial \[x^2 - tx + q,\] where the roots \(r_1\) and \(r_2\) satisfy \[r_1 + r_2 = r_1^2 + r_2^2 = r_1^4 + r_2^4.\] Determine the minimum possible value of \[\dfrac{1}{r_1^5} + \df... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jane has sixteen slips of paper with numbers $2, 2, 2.5, 2.5, 3, 3, 3.5, 3.5, 4, 4, 4.5, 4.5, 5, 5, 5.5,$ and $6$ which she wants to distribute into six cups labeled $A$, $B$, $C$, $D$, $E$, $F$. She decid... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Jason drives past 3 convenience stores on his way to work. The distance between the first store and the second store is 6 miles. The distance between the second store and third store is 2/3rds longer than ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the function $f(x)=-x^{5}-3x^{3}-5x+3$, if $f(a)+f(a-2) > 6$, then the range of the real number $a$ is $(\quad)$
A: $(-\infty,3)$
B: $(3,+\infty)$
C: $(1,+\infty)$
D: $(-\infty,1)$<|im_end|>
... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all prime numbers \( p \) such that for any prime number \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), then there does not exist an integer \( a > 1 \) such that \( a^2 \) divides \( r \).... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Sally had 27 Pokemon cards. Dan has some new Pokemon cards. Sally bought 20 Pokemon cards. Now, Sally has 6 more Pokemon cards than Dan has. How many Pokemon cards does Dan have?<|im_end|>
<|im_start|>assi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The correct statement is \_\_\_\_\_\_ (only fill in the number of the correct statement)
$①$ If set $A={y|y=x-1}$, $B={y|y=x^{2}-1}$, then $A∩B={(0,-1)}$, $((1,0))$;
$②$ $y= \sqrt {x-3}+ \sqrt {2-x}$ is a ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that ( ( ( 13 ! ) ^ some_power ) - ( ( 13 ! ) ^ 8 ) ) / ( ( ( 13 ! ) ^ 8 ) + ( ( 13 ! ) ^ 4 ) ) = a, the units digit for a / ( ( 13 ! ) ^ 4 ) is 9. What is the value of some_power?<|im_end|>
<|im_sta... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
For the nonzero numbers $a$, $b$, and $c$, define $$
\text{{J}}(a,b,c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
$$ Find $\text{{J}}(3,18, 27)$.<|im_end|>
<|im_start|>assistant
We start by substituting th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the sequence ${a_n}$, the sum of the first n terms, S<sub>n</sub>, is equal to $n^2$. Find the value of $a_3^2 - a_2^2$.
A: 9
B: 16
C: 21
D: 11<|im_end|>
<|im_start|>assistant
Since $S_n = n^2$ by de... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$.<|im_en... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given a complex number $z$ satisfying $\frac {2i}{z}=- \sqrt {3}-i$ (where $i$ is the imaginary unit), then $z$ equals
A: $1$
B: $-1$
C: $- \frac {1}{2}- \frac { \sqrt {3}}{2}i$
D: $- \frac {1}{2}+ \fr... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Alice's white water rafting class is composed of some students and 10 instructors, including herself. She has 20 life vests on hand. 20% of her students are bringing life vests. Alice will need to get 22 m... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=3$, $c=2\sqrt{3}$, and $A=30^{\circ}$, find the values of angles $B$, $C$, and side $a$.... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Define a regular \(n\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \(m\) steps where \(m\) is coprime with \(n\) and \(m\) is not a multi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The greatest prime number that is a divisor of $16,385$ is $3$ because $16,385 = 3 \times 5 \times 1093$. What is the sum of the digits of the greatest prime number that is a divisor of $16,385$?
$\textbf{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the circle $(x-a)^2+(x+4)^2=9$ is tangent to the y-axis, then $a=$ ___.<|im_end|>
<|im_start|>assistant
From the equation of the circle, we can find that the center of the circle is $A(a, -4)$, and the ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If the tangent line to the curve $y=e^{x}$ at $x=1$ is perpendicular to the line $2x+my+1=0$, then $m=$ .
A: $-2e$
B: $2e$
C: $-\dfrac{2}{e}$
D: $\dfrac{2}{e}$<|im_end|>
<|im_start|>assistant
The derivati... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The value range of the function $f(x)=\sin x-\cos \left(x+ \frac {\pi }{6}\right)$ is ( )
A: $[-2,2]$
B: $[- \sqrt {3}, \sqrt {3}]$
C: $[-1,1]$
D: $[- \frac { \sqrt {3}}{2}, \frac { \sqrt {3}}{2}]$<|im_en... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[
f(y-f(x)) = f(x) - 2x + f(f(y))
\]<|im_end|>
<|im_start|>assistant
Nous voulons trouver ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given ellipse $C_{1}: \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ and hyperbola $C_{2}: x^{2}-y^{2}=4$ have the same right focus $F_{2}$. Point $P$ is a common point of $C_{1}$ and $C_{2}$. ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
To earn money for her new computer, Tina sells handmade postcards. In a day, she can make some postcards. For each postcard sold, Tina gets $5. Tina earned $900 if she managed to sell all the postcards she... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Eric has a chicken farm with some chickens. His chickens lay 3 eggs each day. After 3 days, Eric collected 36 eggs. How many chickens does Eric have on his farm?<|im_end|>
<|im_start|>assistant
If Eric col... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given the sequence $a_n$: $\frac{1}{1}$, $\frac{2}{1}$, $\frac{1}{2}$, $\frac{3}{1}$, $\frac{2}{2}$, $\frac{1}{3}$, $\frac{4}{1}$, $\frac{3}{2}$, $\frac{2}{3}$, $\frac{1}{4}$, ..., according to the pattern... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Nine families have an average of 3 children per family. If exactly three of these families are childless, what is the average number of children in the families with children?<|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
Let $n$ and $k$ be integers, $1\le k\le n$ . Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:
(i) No product of $k-1$ distinct elements of $A$ is divisi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A rectangle is divided into three equal parts along its length and two equal parts along its width. Two points are drawn on each division point, and semicircle arcs are drawn connecting these points on adj... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Kaleb revises his definition of a $\emph{clever integer}$ as an even integer between 50 and 150 whose digits sum to 10. What fraction of all clever integers within this new range is divisible by 11? Expres... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Suppose $\sin y = 0$ and $\cos(y+w) = -\frac{1}{2}$. What is the smallest possible positive value of $w,$ in radians?<|im_end|>
<|im_start|>assistant
Given $\sin y = 0$, y can be $y = n\pi$ for any integer... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The reciprocal of the real number $-2$ is ( )
A: $2$
B: $-2$
C: $\frac{1}{2}$
D: $-\frac{1}{2}$<|im_end|>
<|im_start|>assistant
To find the reciprocal of a real number, we use the formula $\frac{1}{x}$... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
How many four-digit numbers satisfy the property that the second digit is the average of the first and the third digits?<|im_end|>
<|im_start|>assistant
For the second digit to be the average of the first ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary. Enter all the solutions, separated by commas.<|im_end|>
<|im_start|>assistant
To find all real numbers ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A triangle with vertices \(A = (2, 6)\), \(B = (5, -2)\), and \(C = (7, 3)\) is reflected about the line \(y = 2\) to create a second triangle. What is the area of the union of the two triangles?
\(\textbf... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Consider the following figure:
We seek to ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $11$. What is the maximum number of elements in $T$?<|im_end|>
<|im_start|>assistant
1. **Cal... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In a class of 50 students, numbered from 1 to 50, the class is equally divided into 5 groups according to their numbers in ascending order. Using systematic sampling, 5 students are selected for a survey. ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle \(ABC\), the lengths are \(AB = 6\), \(AC = 10\). The point \(M\) is the midpoint of \(BC\), and \(AM = 5\). Determine the length of \(BC\).
\(\textbf{(A)}\ 2\sqrt{45}\)
\(\textbf{(B)}\ 2\sqrt{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given that the sum of the first $n$ terms of the sequence ${a\_n}$ is $S\_n$, and $S\_n = 2a\_n - n$, find the maximum value of $n$ that satisfies $a\_n \leq 10n$.<|im_end|>
<|im_start|>assistant
We know t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
A four-inch wooden cube has its top and four side faces painted blue, leaving the bottom face unpainted. The cube is then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1, |\overrightarrow{b}| = 2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\frac{1}{2}$, find:
1. The val... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the sum of all positive real solutions \(x\) to the equation \[3\cos2x \left(\cos2x - \cos{\left( \frac{2000\pi^2}{x} \right) } \right) = \cos4x - 1,\] where \(x\) is measured in radians.<|im_end|>
<|... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
If \( f(x) = 5 - 4x \) and \( g(x) = x^2 + 2 \), find \( f(g(2)) \).<|im_end|>
<|im_start|>assistant
First, we calculate \( g(2) \):
$$ g(2) = (2)^2 + 2 = 4 + 2 = 6. $$
Then, substitute \( g(2) \) into \(... |
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