text stringlengths 593 16.2k |
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<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 17 Given that $a, b, c, d$ are non-zero real numbers, $f(x)=\frac{a x+b}{c x+d}, x \in \mathbf{R}$, and $f(19)=$ 19, $f(97)=9... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5. Given $\triangle A B C$ with three sides $A B=\sqrt{34}, B C$ $=5 \sqrt{10}, C A=2 \sqrt{26}$. Then the area of $\triangle A B C$ ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
3. The polynomial $P(x)$ of the third degree, as well as its first, second, and third derivatives, take the value 1 at $x=-3$. Find $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
\left.\begin{array}{l}{[\text { Theorem of the sum of squares of diagonals }]} \\ {[\text { Pythagorean Theorem (direct and inverse) ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
【Question 4】
A father wants to divide his property among his children: first, he gives 1000 yuan and $\frac{1}{10}$ of the remaining ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
10. Let $f$ be a function from non-negative real numbers to non-negative real numbers, satisfying
$$
f\left(a^{3}\right)+f\left(b^{3}... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. (3 points) Find the total length of intervals on the number line where the inequalities $x<1$ and $\sin \log _{2} x<0$ are satisfi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example $1$, $, $ Given line segments $b, c$. Find the line segment $a$ such that
$$
\frac{1}{a}=\frac{1}{b}+\frac{1}{c} .
$$<|im_end... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Problem 5.3. A cuckoo clock is hanging on the wall. When a new hour begins, the cuckoo says "cuckoo" a number of times equal to the n... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
3. The solution set of the inequality $\sqrt{x+\frac{1}{x^{2}}}-\sqrt{x-\frac{1}{x^{2}}}<\frac{1}{x}$ is<|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
Please reason step by step, and put your final answer within \boxed{}.
2. Let $K$ and $L$ be points on sides $BC$ and $AC$ of triangle $ABC$, respectively, such that $|BK|=\frac{1}{3}|BC|,|AL|=\frac{1}{3}... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. On the side $AC$ of triangle $ABC$, a point $K$ is taken which divides $AC$ in the ratio $1:3$, and on the side $BC$ a point $L$ i... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
3. The pond has a square shape. On the first frosty day, the part of the pond that is no more than 10 meters away from the nearest po... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Which is the three-digit number that is 1 less than twice the number obtained by swapping its extreme digits?<|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
Please reason step by step, and put your final answer within \boxed{}.
9. (16 points) Given that $x$ is a real number. Find the range of the function
$$
f(x)=4 \sin ^{3} x+\sin ^{2} x-4 \sin x+8
$$<|im_en... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Task A-1.2.
All natural numbers from 1 to some number were written on a board. After one of the numbers was erased, the arithmeti... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Task 4
Represent the number 14 as a sum of odd numbers, find at least 4 examples!<|im_end|>
<|im_start|>assistant
Okay, let's se... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Task 2 - 010612
In the Soviet rocket tests conducted in October 1961, at a target distance of about $12500 \mathrm{~km}$, all hit... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
7. Given that box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Now, 3 balls are randomly taken from... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
9. Given $-1<a<0$. Simplify
$$
\sqrt{\left(a+\frac{1}{a}\right)^{2}-4}+\sqrt{\left(a-\frac{1}{a}\right)^{2}+4}
$$
to get $\qquad$ .<... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. In the aquarium, there are 1000 neon tetras, clownfish, and guppies. Among them, the neon tetras are the most numerous, and the gu... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$
\[x_{n+1}=\ln(e^{x_n}-x_n)\]
(as usual, the function $\ln... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. If the three sides of a right-angled triangle are positive integers, and the numerical value of its perimeter is equal to the nume... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
1. Let $x$ and $y$ be real numbers, and satisfy
$$
\left\{\begin{array}{l}
(x-1)^{5}+1999(x-1)=2519, \\
(y-1)^{5}+1999(y-1)=-2519 .
\... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
16. Rearrange the digits of any three-digit number to form the largest and smallest numbers possible, and their difference will form ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. In a chess tournament, $n$ players participate in a round-robin competition. After players A and B each played two games, they wit... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
8. Line $l$ :
$$
m(2 x-y-5)+(3 x-8 y-14)=0
$$
is intercepted by circle $\odot A$ with center $A(1,0)$ and radius 2, the length of th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, for $n \geqslant 1$, we have
$$
\begin{array}{l}
4\left(x_{1} x_{n}... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
[ Systems of nonlinear algebraic equations ] Higher degree equations (miscellaneous) $\quad]$
Solve the system of equations:
$x^{3}... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
The [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ is the smallest [positive](https://artofproblemsolving.com/... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. Calculate with an accuracy of one-tenth the value of the expression $\sqrt{86+41 \sqrt{86+41 \sqrt{86+\ldots}}}$
#<|im_end|>
<|im... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
$\begin{array}{l}\quad \text { 3. } \arcsin \frac{1}{\sqrt{10}}+\arccos \frac{7}{\sqrt{50}}+\operatorname{arctg} \frac{7}{31} \\ +\op... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Problem 10.8. Real numbers $x$ and $y$ are such that $x^{3}+21 x y+y^{3}=343$. What can $x+y$ be? List all possible options.<|im_end|... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 2 Given the function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying $f(0) \neq 0$, and for any $x, y \in \mathbf{R}$ we h... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Problem Statement
Calculate the definite integral:
$$
\int_{-\frac{2 \pi}{3}}^{0} \frac{\cos x \, dx}{1+\cos x-\sin x}
$$<|im_en... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
2. In the parliament of a certain state, there are 2016 deputies, who are divided into 3 factions: "blues," "reds," and "greens." Eac... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
1. Calculate the expression with mixed numbers
$895231755 \frac{234}{357118} \cdot 895231754 \frac{234}{357118}-895231756 \frac{234}... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. For any $x, y \in \mathbf{R}$, the function $f(x, y)$ satisfies
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f(x, f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
3. Solve the equation $2021 x=2022 \cdot \sqrt[2022]{x^{2021}}-1 .(10$ points $)$<|im_end|>
<|im_start|>assistant
Okay, so I need to ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 3 Find the maximum value of the function $y=6 \sqrt{x-5}+8 \sqrt{6-x}$.<|im_end|>
<|im_start|>assistant
Okay, so I need to fi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 10 Find all functions defined on the set of positive real numbers, which take positive real values, and satisfy the condition... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Sets $A, B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A\cap B| = 16$, $|A\cap C| = 51$, $|B\cap C| = 19$. Compute the nu... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5 Let $a, b$ be two positive numbers, and $a>b$. Points $P, Q$ are on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. If the... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. Write an arithmetic sequence in the following manner:
1st row: 2
2nd row: 5, 8
3rd row: $11, 14, 17$
4th row: $20, 23, 26, 29$
$\... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Problem 4. A circle is inscribed in a circular sector (the circle touches both radii and the arc of the sector). The radius of the se... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Fernando is a cautious person, and the gate of his house has 10 distinct padlocks, each of which can only be opened by its respective... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
By how many zeros does the number 2012! end?<|im_end|>
<|im_start|>assistant
Okay, so I need to figure out how many zeros the number ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 1: 20 teams participate in the national football championship. To ensure that in any three teams that have already played, th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
8,9}
Side $A B$ of triangle $A B C$ is a chord of a certain circle. Sides $A C$ and $B C$ lie inside the circle, the extension of si... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
(solved by Victor Quach). Find all real solutions of the equation:
$$
4 x^{2}-40[x]+51=0
$$
where $[x]$ denotes the integer part of... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
The diameter of one base circle of a frustum is $100 \mathrm{~mm}$. If the diameter is increased by $21 \%$, while the height and the... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. The smallest natural number $a$ that makes the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}$ $<a-1995 \frac{1}{3... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Task 3 - 050913
Compare the two numbers!
$$
A=\frac{5678901234}{6789012345} \quad \text { and } \quad B=\frac{5678901235}{678901... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5. [5] Five marbles of various sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, eac... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Five cities are to be connected by a railway network consisting of four straight segments so that it is possible to travel from any c... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
What is the maximum number of edges we can select from a regular dodecahedron so that no two of them have a common point?<|im_end|>
<... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
10. In the race between the tortoise and the hare, the hare's and tortoise's speed ratio at the start was $3: 2$. When the hare reach... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. If the equations $x^{2}+2 a x+b^{2}=0$ and $x^{2}+2 c x-$ $b^{2}=0$ have a common root, and $a, b, c$ are exactly the three sides ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
10. There are 8 red, 8 white, and 8 yellow chopsticks. Without looking, how many chopsticks must be taken out to ensure that at least... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
18. A collection of 2011 circles divide the plane into $N$ regions in such a way that any pair of circles intersects at two points an... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5. What is the greatest number of acute angles that can occur in a convex polygon?<|im_end|>
<|im_start|>assistant
Okay, so the probl... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
For all integers $n \geqslant 0$, we denote $s(n)$ as the sum of the digits of $n$. Determine all integers $n \geqslant 0$ such that ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
26・10 In the Cartesian coordinate system, if the equation $m\left(x^{2}+y^{2}+2 y+1\right)=$ $(x-2 y+3)^{2}$ represents an ellipse, t... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
[ Pythagorean Theorem (direct and inverse). ] [ Properties and characteristics of an isosceles triangle. ]
In an isosceles triangle... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example + Given that $f(x)$ is an even function defined on $\mathbf{R}$, if $g(x)$ is an odd function, and $g(x)=f(x-1)$, $g(1)=2003$... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
15. Three tired cowboys entered a saloon and hung their hats on a buffalo horn at the entrance. When the cowboys left deep into the n... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
To celebrate her birthday, Ana is going to prepare pear pies and apple pies. At the market, an apple weighs $300 \mathrm{~g}$ and a p... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
22. A year is called a leap year if it is either divisible by 4 but not divisible by 100 , or divisible by 400 . Hence, the years 200... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
9.25 On a bookshelf, 30 volumes can fit. In how many ways can they be arranged so that the first and second volumes do not stand next... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
6. The greatest common divisor (GCD) between each pair of three numbers is $3$, $4$, and $5$. What is the minimum sum of these three ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.
[img]https://cdn.artofprobl... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$.... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 3.35. Find the derivative of the function $u=x^{3} y^{3} z^{3}$ at the point $M(1,1,1)$ in the direction that forms angles of... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
$P$ is a polynomial of degree 4. $P(0)=P(1)=1, P(2)=4, P(3)=9, P(4)=16$. Calculate $P(-1)$<|im_end|>
<|im_start|>assistant
Okay, so I... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
1. Calculate:
$$
\frac{2013^{2}+2011}{2011^{2}-2013} \times \frac{4020^{2}-8040}{2011 \times 2014-4}=
$$
$\qquad$<|im_end|>
<|im_star... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
\section*{Problem 4 - 031044}
How many trailing zeros does the product
\[
p_{1}^{1} \cdot\left(p_{1}^{2} \cdot p_{2}^{1}\right) \cd... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
14. Find
$$
2 \sum_{k=1}^{n} k^{3} \mathrm{C}_{n}^{k}-3 n \sum_{k=1}^{n} k^{2} \mathrm{C}_{n}^{k}+n^{2} \sum_{k=1}^{n} k \mathrm{C}_{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5 Let $f(x)=x^{3}+\log _{2}\left(x+\sqrt{x^{2}+1}\right)$, then for any real numbers $\dot{a}, b$, $a+b \geqslant 0$ is a ( ) for $f(... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 5 As shown in Figure 4, place 3 identical coins sequentially into a $4 \times 4$ square grid (each square can only hold 1 coi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
The total number of edges in two regular polygons is $2014$, and the total number of diagonals is $1,014,053$. How many edges does th... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
11.3. The perimeter of triangle $ABC$ is 24 cm, and the segment connecting the point of intersection of its medians with the point of... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. Let $a, b, c$ and $d$ be real numbers greater than 1. Calculate the value of the expression
$$
a^{\left(\log _{b} c\right)-1} b^{... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5. Given $f(x)=\sqrt{\frac{1-x}{1+x}}$. If $\alpha \in\left(\frac{\pi}{2}, \pi\right)$, then $f(\cos \alpha)+f(-\cos \alpha)$ can be ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\ln 2 x-\ln \pi}{\sin \left(\f... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
(15) Real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, then the maximum value of $\sqrt{2} x y+y z$ is
$\qquad$ .<|im_end|>
<|im_... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 22 Skew lines $a, b, a \perp b, c$ forms a $30^{\circ}$ angle with $a$, find the range of the angle $\theta$ formed between $... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
2. For the ellipse $\frac{x^{2}}{12}+\frac{y^{2}}{3}=1$, the foci are $F_{1}$ and $F_{2}$, and point $P$ is on the ellipse. If the mi... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
4. Given real numbers $a, b, c$ satisfy
$$
a+b+c=a^{2}+b^{2}+c^{2} \text {. }
$$
Then the maximum value of $a+b+c$ is $\qquad$<|im_e... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Example 1 If a positive integer has 21 positive divisors, what is the smallest positive integer with this property?<|im_end|>
<|im_st... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
1. Find the smallest natural number $k$, for which the following holds: If we select any $k$ different numbers from the set $\{1,4,7,... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Find all real solutions to: $\mathrm{x}+\mathrm{y}-\mathrm{z}=-1 ; \mathrm{x}^{2}-\mathrm{y}^{2}+\mathrm{z}^{2}=1,-\mathrm{x}^{3}+\ma... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
5. Every school from a certain region sent 3 students to the competition. Andrej, Blaž, and Žan represented the same school. When all... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
Knowing that $a$ is a natural number, and that $4 a^{2}$ and $\frac{4}{3} \times a^{3}$ are natural numbers with 4 digits, determine ... |
<|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Please reason step by step, and put your final answer within \boxed{}.
A $400 \mathrm{~m}$ track is constructed so that the points $A, B, C$, and $D$ divide the track into four segments of equal length. T... |
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