problem stringlengths 15 7.47k | answer stringlengths 1 261 | input stringlengths 157 7.61k |
|---|---|---|
## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | <|im_start|>user
Problem:## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm... |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | 180 | <|im_start|>user
Problem:3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a ... |
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$? | 8 | <|im_start|>user
Problem:Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
Solve this problem step by step. Put the fina... |
4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).
(A) $1<x<7$.
(B) $5 \ll x<7$.
(C) $1<x<\sqrt{7}$.
(D) $5<x<7$ or $1<x<\sqrt{7}$. | D | <|im_start|>user
Problem:4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).
(A) $1<x<7$.
(B) $5 \ll x<7$.
(C) $1<x<\sqrt{7}$.
(D) $5<x<7$ or $1<x<\sqrt{7}$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assis... |
1. Solve the equation: $\frac{8 x+13}{3}=\frac{19-12 x}{2}-\left(16 x-\frac{7-4 x}{6}\right)$. | \frac{1}{4} | <|im_start|>user
Problem:1. Solve the equation: $\frac{8 x+13}{3}=\frac{19-12 x}{2}-\left(16 x-\frac{7-4 x}{6}\right)$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5? | 0 | <|im_start|>user
Problem:A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$ | -\ln 2 | <|im_start|>user
Problem:Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$. | \sinA=\frac{1}{2}or\sinA=1 | <|im_start|>user
Problem:11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$.
Solve this problem step by step. Put the final answer in a boxed for... |
Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\mathbb{C}$. | x_{1,2}=\(10^{2014}+1),\quadx_{3,4}=\(10^{2014}+1)\cdoti | <|im_start|>user
Problem:Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\mathbb{C}$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>ass... |
6. As shown in Figure 2, let $P$ be a point inside the equilateral $\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is | 11 \sqrt{3} | <|im_start|>user
Problem:6. As shown in Figure 2, let $P$ be a point inside the equilateral $\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of ... |
8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Mathematics Examination, 1979... | 1 | <|im_start|>user
Problem:8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Math... |
9.6. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer. | -27 | <|im_start|>user
Problem:9.6. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer.
Solve this problem step by step. Put the final answer in a boxed format (\bo... |
Given $0 \leqslant x, y, z \leqslant 1$, solve the equation:
$$\frac{x}{1+y+z x}+\frac{y}{1+z+x y}+\frac{z}{1+x+y z}=\frac{3}{x+y+z} .$$ | x=y=z=1 | <|im_start|>user
Problem:Given $0 \leqslant x, y, z \leqslant 1$, solve the equation:
$$\frac{x}{1+y+z x}+\frac{y}{1+z+x y}+\frac{z}{1+x+y z}=\frac{3}{x+y+z} .$$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$ | \frac{3\pi}{4}-\frac{9\sqrt{3}}{8} | <|im_start|>user
Problem:## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Example 6 The rules of a "level-up game" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:
(1) What is the maximum number of levels a person can pass in this game?
(2) What is the pro... | \frac{100}{243} | <|im_start|>user
Problem:Example 6 The rules of a "level-up game" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:
(1) What is the maximum number of levels a person can pass in this ... |
2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | 18 | <|im_start|>user
Problem:2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_... |
Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum n... | 9 | <|im_start|>user
Problem:Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coinci... |
7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if soluti... | 10 | <|im_start|>user
Problem:7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a tripl... |
Example 1 (Question from the 13th "Hope Cup" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\sqrt[5]{x}+1=0$ are $\alpha, \beta$ respectively, then $\alpha+\beta$ equals ( ).
A. -1
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. 1 | -1 | <|im_start|>user
Problem:Example 1 (Question from the 13th "Hope Cup" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\sqrt[5]{x}+1=0$ are $\alpha, \beta$ respectively, then $\alpha+\beta$ equals ( ).
A. -1
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. 1
Solve this problem step by step. Put the f... |
Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$. | 402005 | <|im_start|>user
Problem:Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im... |
Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.
---
The translation preserves the original text's formatting and ... | 2020 | <|im_start|>user
Problem:Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.
---
The translation preserves the origin... |
1. If the set
$$
A=\{1,2, \cdots, 10\}, B=\{1,2,3,4\},
$$
$C$ is a subset of $A$, and $C \cap B \neq \varnothing$, then the number of such subsets $C$ is $(\quad)$.
(A) 256
(B) 959
(C) 960
(D) 961 | C | <|im_start|>user
Problem:1. If the set
$$
A=\{1,2, \cdots, 10\}, B=\{1,2,3,4\},
$$
$C$ is a subset of $A$, and $C \cap B \neq \varnothing$, then the number of such subsets $C$ is $(\quad)$.
(A) 256
(B) 959
(C) 960
(D) 961
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im... |
Augusto has a wire that is $10 \mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:
. MATHEMATICE YoUTH CLUE A. 25
B. 31
C. 37
D. 43 | 31 | <|im_start|>user
Problem:29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25
B. 31
C. 37
D. 43
Solve this problem step by step. Put the final answer in a boxed format (\boxe... |
Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea? | 6 | <|im_start|>user
Problem:Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_... |
12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4. | \frac{5}{13} | <|im_start|>user
Problem:12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|... |
3. Let $AB$ be a chord of the unit circle $\odot O$. If the area of $\odot O$ is exactly equal to the area of the square with side $AB$, then $\angle AOB=$ $\qquad$ (to 0.001 degree). | 124.806^{\circ} | <|im_start|>user
Problem:3. Let $AB$ be a chord of the unit circle $\odot O$. If the area of $\odot O$ is exactly equal to the area of the square with side $AB$, then $\angle AOB=$ $\qquad$ (to 0.001 degree).
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assis... |
83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation) | 1536 | <|im_start|>user
Problem:83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation)
Solve this problem step by step. Put the fi... |
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 8 | <|im_start|>user
Problem:On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the ... |
4. The number of real solutions to the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$ is ( ).
(A) 0
(B) 1
(C) 2
(D) 4 | C | <|im_start|>user
Problem:4. The number of real solutions to the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$ is ( ).
(A) 0
(B) 1
(C) 2
(D) 4
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$ | 4 | <|im_start|>user
Problem:## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Condition of the problem
Find the derivative.
$$
y=\frac{1}{24}\left(x^{2}+8\right) \sqrt{x^{2}-4}+\frac{x^{2}}{16} \arcsin \frac{2}{x}, x>0
$$ | \frac{x^{3}-x}{8\sqrt{x^{2}-4}}+\frac{x}{8}\cdot\arcsin\frac{2}{x} | <|im_start|>user
Problem:Condition of the problem
Find the derivative.
$$
y=\frac{1}{24}\left(x^{2}+8\right) \sqrt{x^{2}-4}+\frac{x^{2}}{16} \arcsin \frac{2}{x}, x>0
$$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
\section*{Problem 5 - 071225}
All ordered pairs of real numbers \((x, y)\) are to be determined for which the system of equations
\[
\begin{aligned}
x \cdot\left(a x^{2}+b y^{2}-a\right) & =0 \\
y \cdot\left(a x^{2}+b y^{2}-b\right) & =0
\end{aligned}
\]
is satisfied. Here, \(a\) and \(b\) are real numbers with \(a ... | (0,-1),(0,0),(0,1),(-1,0),(1,0) | <|im_start|>user
Problem:\section*{Problem 5 - 071225}
All ordered pairs of real numbers \((x, y)\) are to be determined for which the system of equations
\[
\begin{aligned}
x \cdot\left(a x^{2}+b y^{2}-a\right) & =0 \\
y \cdot\left(a x^{2}+b y^{2}-b\right) & =0
\end{aligned}
\]
is satisfied. Here, \(a\) and \(b\) a... |
6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{3}{3}... | 1078 | <|im_start|>user
Problem:6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}... |
40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers. | 25,27,292185,2187,2189 | <|im_start|>user
Problem:40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$ | 10 | <|im_start|>user
Problem:Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
19. Given $m \in\{11,13,15,17,19\}$, $n \in\{1999,2000, \cdots, 2018\}$.
Then the probability that the unit digit of $m^{n}$ is 1 is ( ).
(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{3}{10}$
(D) $\frac{7}{20}$
(E) $\frac{2}{5}$ | E | <|im_start|>user
Problem:19. Given $m \in\{11,13,15,17,19\}$, $n \in\{1999,2000, \cdots, 2018\}$.
Then the probability that the unit digit of $m^{n}$ is 1 is ( ).
(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{3}{10}$
(D) $\frac{7}{20}$
(E) $\frac{2}{5}$
Solve this problem step by step. Put the final answer in a boxed ... |
1. (6 points) Today is January 31, 2015, welcome to the 2015 "Spring Cup" Final. The calculation result of the expression $\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}$ is | 6500 | <|im_start|>user
Problem:1. (6 points) Today is January 31, 2015, welcome to the 2015 "Spring Cup" Final. The calculation result of the expression $\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}$ is
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
| \frac{\pi \sqrt{3}}{6} | <|im_start|>user
Problem:Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$. | \frac{a^2 + b^2 + c^2}{3} | <|im_start|>user
Problem:Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
Solve this problem step by step. Put the ... |
1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is | \frac{\sqrt{2}}{2} | <|im_start|>user
Problem:1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
The function $f$ maps the set of positive integers into itself, and satisfies the equation
$$
f(f(n))+f(n)=2 n+6
$$
What could this function be? | f(n)=n+2 | <|im_start|>user
Problem:The function $f$ maps the set of positive integers into itself, and satisfies the equation
$$
f(f(n))+f(n)=2 n+6
$$
What could this function be?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
30. Find the remainder when the 2018-digit number $\underbrace{\overline{55 \cdots}}_{2018 \text { 555 }}$ is divided by 13. | 3 | <|im_start|>user
Problem:30. Find the remainder when the 2018-digit number $\underbrace{\overline{55 \cdots}}_{2018 \text { 555 }}$ is divided by 13.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4 | B | <|im_start|>user
Problem:1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
1. Let the universal set be the set of real numbers. If $A=\{x \mid \sqrt{x-2} \leqslant 0\}, B=\left\{x \mid 10^{x^{2}-2}=10^{x}\right\}$, then $A \cap \bar{B}$ is
A. $\{2\}$
B. $\{-1\}$
C. $\{x \mid x \leqslant 2\}$
D. $\varnothing$ | D | <|im_start|>user
Problem:1. Let the universal set be the set of real numbers. If $A=\{x \mid \sqrt{x-2} \leqslant 0\}, B=\left\{x \mid 10^{x^{2}-2}=10^{x}\right\}$, then $A \cap \bar{B}$ is
A. $\{2\}$
B. $\{-1\}$
C. $\{x \mid x \leqslant 2\}$
D. $\varnothing$
Solve this problem step by step. Put the final answer in a b... |
4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ).
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{... | D | <|im_start|>user
Problem:4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ).
(A) $30^{\circ}$
(... |
7.242. $\left(16 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}-0.048\right) \lg \left(x^{3}+2 x+1\right)=0$. | 0 | <|im_start|>user
Problem:7.242. $\left(16 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}-0.048\right) \lg \left(x^{3}+2 x+1\right)=0$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Example 1 In $\triangle ABC$, it is known that $x \sin A + y \sin B + z \sin C = 0$. Find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$. | 0 | <|im_start|>user
Problem:Example 1 In $\triangle ABC$, it is known that $x \sin A + y \sin B + z \sin C = 0$. Find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_st... |
In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$ | 400 | <|im_start|>user
Problem:In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text... |
11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. | \sqrt{3} | <|im_start|>user
Problem:11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
$\textbf{... | \frac{7}{27} | <|im_start|>user
Problem:The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knigh... |
A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out ... | 9 | <|im_start|>user
Problem:A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child,... |
Example 11 Let $x>0, y>0, \sqrt{x}(\sqrt{x}+2 \sqrt{y})$ $=\sqrt{y}(6 \sqrt{x}+5 \sqrt{y})$. Find the value of $\frac{x+\sqrt{x y}-y}{2 x+\sqrt{x y}+3 y}$. | \frac{1}{2} | <|im_start|>user
Problem:Example 11 Let $x>0, y>0, \sqrt{x}(\sqrt{x}+2 \sqrt{y})$ $=\sqrt{y}(6 \sqrt{x}+5 \sqrt{y})$. Find the value of $\frac{x+\sqrt{x y}-y}{2 x+\sqrt{x y}+3 y}$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? | \frac{1}{2} | <|im_start|>user
Problem:9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Let's determine all the triples of numbers $(x, y, m)$ for which
$$
-2 x + 3 y = 2 m, \quad x - 5 y = -11
$$
and $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number. | (x,y,)=(-1,2,4)or(-6,1,7.5) | <|im_start|>user
Problem:Let's determine all the triples of numbers $(x, y, m)$ for which
$$
-2 x + 3 y = 2 m, \quad x - 5 y = -11
$$
and $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_s... |
## Zadatak B-1.2.
Na slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.
| 27 | <|im_start|>user
Problem:## Zadatak B-1.2.
Na slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.
Solve this problem step by step. Put the... |
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 6 | <|im_start|>user
Problem:The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point ... |
Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$. | <-\frac{1}{3} | <|im_start|>user
Problem:Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}... |
2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$. | 9599 | <|im_start|>user
Problem:2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \leqslant x \leqslant 0)$ is $(\quad)$.
(A) $(-\infty, 4]$
(B) $[3,12]$
(C) $[-12,4]$
(D) $[4,12]$ | C | <|im_start|>user
Problem:Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \leqslant x \leqslant 0)$ is $(\quad)$.
(A) $(-\infty, 4]$
(B) $[3,12]$
(C) $[-12,4]$
(D) $[4,12]$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,
$$\begin{array}{l}
\vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right) \\
\vec{f}_{2}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right]
\end{... | 2.1 | <|im_start|>user
Problem:4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,
$$\begin{array}{l}
\vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right) \\
\vec{f}_{2}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\... |
6. Given that $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. If $t \in[0,1]$, then
$$
|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|
$$
the minimum value is ( ).
(A) $2 \sqrt{193}$
(B) 2... | 26 | <|im_start|>user
Problem:6. Given that $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. If $t \in[0,1]$, then
$$
|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|
$$
the minimum value is ( ).... |
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$ | 8 | <|im_start|>user
Problem:All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 1... |
1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$. | 1 | <|im_start|>user
Problem:1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistan... |
2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).
(A) $\frac{\sqrt{5}+1}{2}$
(B) $\frac{\sqrt{5}-1}{2}$
(C) 1
(D) 2 | A | <|im_start|>user
Problem:2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).
(A) $\frac{\sqrt{5}+1}{2}$
(B) $\frac{\sqrt{5}-1}{2}$
(C) 1
(D) 2
Solve this problem step by step. Put the final answer in a boxed... |
V-2 If one side of the rectangle is reduced by $3 \mathrm{~cm}$, and the other side is reduced by $2 \mathrm{~cm}$, we get a square whose area is $21 \mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.
.<|im_end|>
<|im_start|>assistant
|
4. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$. | 88327 | <|im_start|>user
Problem:4. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.
Solve this pro... |
In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer? | \frac{1}{2} | <|im_start|>user
Problem:In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?
Solve this problem step by step. Put the final answer in a bo... |
18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original ridin... | 72 | <|im_start|>user
Problem:18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5... |
53. How many four-digit numbers contain at least one even digit? | 8375 | <|im_start|>user
Problem:53. How many four-digit numbers contain at least one even digit?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ . | [-\sin 1+\cos 1-\tan 1+\frac{\pi}{4}, \sin 1+\cos 1+\tan 1+\frac{3 \pi}{4}] | <|im_start|>user
Problem:1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ .
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i] | \left\lfloor \frac{n^2}{2} \right\rfloor - 1 | <|im_start|>user
Problem:Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i]
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).... |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 4 | <|im_start|>user
Problem:There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k ... |
10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid. | a | <|im_start|>user
Problem:10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 2343 | <|im_start|>user
Problem:8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|... |
1. Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$. (Here, a sequence with ... | [\frac{n^{2}}{4}] | <|im_start|>user
Problem:1. Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$... |
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ ca... | 503 | <|im_start|>user
Problem:Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible v... |
9.171. $0.6^{\lg ^{2}(-x)+3} \leq\left(\frac{5}{3}\right)^{2 \lg x^{2}}$. | x\in(-\infty;-0.1]\cup[-0.001;0) | <|im_start|>user
Problem:9.171. $0.6^{\lg ^{2}(-x)+3} \leq\left(\frac{5}{3}\right)^{2 \lg x^{2}}$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
2. Find all integer solutions of the inequality
$$
x^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \leq 0
$$ | (6,2,7),(20,0,19) | <|im_start|>user
Problem:2. Find all integer solutions of the inequality
$$
x^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \leq 0
$$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.
| =24,b=30,=36,\alpha=4124^{\}34^{\\},\beta=5546^{\}16^{\\},\gamma=8249^{\}10^ | <|im_start|>user
Problem:Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \] | (1, 11, 3) | <|im_start|>user
Problem:Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \]
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
15. As shown in the figure, the area of square $\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefor... | 72 | <|im_start|>user
Problem:15. As shown in the figure, the area of square $\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 sq... |
25. Anna, Bridgit and Carol run in a $100 \mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?
A $37 \mathrm{~m}$
B $41 \mathrm{~m}... | 37\mathrm{~} | <|im_start|>user
Problem:25. Anna, Bridgit and Carol run in a $100 \mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?
A $37 \math... |
Given the complex number $z$ has a modulus of 1. Find
$$
u=\frac{(z+4)^{2}-(\bar{z}+4)^{2}}{4 i} \text {. }
$$
the maximum value. | \frac{\sqrt{9+24 \sqrt{6}}}{2} | <|im_start|>user
Problem:Given the complex number $z$ has a modulus of 1. Find
$$
u=\frac{(z+4)^{2}-(\bar{z}+4)^{2}}{4 i} \text {. }
$$
the maximum value.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p) | 700 | <|im_start|>user
Problem:## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p)
Solve this problem step by step... |
1. The curve $(x+2 y+a)\left(x^{2}-y^{2}\right)=0$ represents three straight lines intersecting at one point on the plane if and only if
A. $a=0$
B. $a=1$
C. $a=-1$
D. $a \in \mathbf{R}$ | 0 | <|im_start|>user
Problem:1. The curve $(x+2 y+a)\left(x^{2}-y^{2}\right)=0$ represents three straight lines intersecting at one point on the plane if and only if
A. $a=0$
B. $a=1$
C. $a=-1$
D. $a \in \mathbf{R}$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>as... |
9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train? | 16^{100} | <|im_start|>user
Problem:9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).
(A) the I quadrant
(B) the II quadrant
(C) the III quadrant
(D) the IV quadrant | C | <|im_start|>user
Problem:2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).
(A) the I quadrant
(B) the II quadrant
(C) the III quadrant
(D) the IV quadrant
Solve this proble... |
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(... | 50 | <|im_start|>user
Problem:Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\t... |
Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.
How many games did father play with uncle?
(M. Volfová) | 16 | <|im_start|>user
Problem:Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.
How many games did father play with uncle?
(M. Volfová)
Solve this problem... |
## Problem 4
Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.
Selected problems by Prof. Cicortas Marius
Note: a) The actual working t... | 499 | <|im_start|>user
Problem:## Problem 4
Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.
Selected problems by Prof. Cicortas Marius
Note... |
Problem 5.3. In five of the nine circles in the picture, the numbers 1, 2, 3, 4, 5 are written. Replace the digits $6, 7, 8, 9$ in the remaining circles $A, B, C, D$ so that the sums of the four numbers along each of the three sides of the triangle are the same.
=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$.
| \frac{7}{3} | <|im_start|>user
Problem:The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$.
Solve th... |
18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$. | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | <|im_start|>user
Problem:18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$.
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$. | 14\sqrt[4]{3} | <|im_start|>user
Problem:5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$.
Sol... |
4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$. | 11 | <|im_start|>user
Problem:4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.
Solve this problem... |
7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$ | -\frac{1}{4}\\frac{\sqrt{15}}{4}i | <|im_start|>user
Problem:7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between point... | 4 \sqrt{13-6 \sqrt{2}} | <|im_start|>user
Problem:Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shorte... |
5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\sqrt{1+a^{2}}+\sqrt{1+2 b}$ the integer part is | 2 | <|im_start|>user
Problem:5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\sqrt{1+a^{2}}+\sqrt{1+2 b}$ the integer part is
Solve this problem step by step. Put the final answer in a boxed format (\boxed{}).<|im_end|>
<|im_start|>assistant
|
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 8