source stringclasses 1
value | problem stringlengths 92 2.6k | solution stringlengths 945 4.35k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | What amazing conclusion can be drawn from the answer: "There is something instead of nothing"? | 1. Firstly, we need to understand the setup of the problem. We are given that there are two priests regarding the Question of Questions.
2. The first priest knows the correct answer to the Question of Questions and always lies.
3. The second priest, if he knows the correct answer, he also lies.
4. The philosopher poses... | [
{
"content": "What amazing conclusion can be drawn from the answer: \"There is something instead of nothing\"?",
"role": "user"
},
{
"content": "1. Firstly, we need to understand the setup of the problem. We are given that there are two priests regarding the Question of Questions.\n2. The first prie... |
olympiads | In a circle, chords \( AB \) and \( AC \) are drawn such that \( AB = 2 \), \( AC = 1 \), and \( \angle CAB = 120^\circ \). Find the length of the chord of the circle that bisects the angle \( CAB \).
Answer: 3 |
Given:
- Chords \( AB \) and \( AC \) are drawn in a circle.
- \( AB = 2 \)
- \( AC = 1 \)
- \( \angle CAB = 120^\circ \)
We need to find the length of the chord that bisects \(\angle CAB\).
1. Let's denote the midpoint of \( \angle CAB \) as \( E \). Therefore, \( \angle EAC = \angle EAB = 60^\circ \).
2. Since \(... | [
{
"content": "In a circle, chords \\( AB \\) and \\( AC \\) are drawn such that \\( AB = 2 \\), \\( AC = 1 \\), and \\( \\angle CAB = 120^\\circ \\). Find the length of the chord of the circle that bisects the angle \\( CAB \\).\n\nAnswer: 3",
"role": "user"
},
{
"content": "\nGiven:\n- Chords \\( A... |
olympiads | In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646. |
To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets.
1. **Case 1: Buying the set "keyboard and mouse" and additional headphones.**
- There are 4 such sets available.
- There ar... | [
{
"content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.",
"role": "user"
},
... |
olympiads | The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land equally among his sons. If a man had no sons, the land was transferred to the state. No other family members gained or lost land in any other way. There were a total of 180 people in the family. What is the s... |
Let's consider a problem where each male in a noble family divides his land equally among his sons upon death. If a person has no sons, his land reverts to the state. Given there are 180 members in the family, we need to find the smallest share of the original land that any member could inherit.
1. **Understanding th... | [
{
"content": "The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land equally among his sons. If a man had no sons, the land was transferred to the state. No other family members gained or lost land in any other way. There were a total of 180 people in the... |
olympiads | On the side \( DE \) of a regular hexagon \( ABCDEF \), point \( K \) is chosen so that the line \( AK \) divides the area of the hexagon in the ratio \( 3:1 \). In what ratio does point \( K \) divide the side \( DE \) ? Answer: \( 3:1 \). |
1. Since the area $S$ of the regular hexagon $ABCDEF$ can be assumed to be 6, let's establish the areas of some of its triangles:
- The area of each equilateral triangle within the hexagon is $S(\triangle ABC) = S(\triangle AEF) = 1$.
- The remaining areas can be calculated from these known triangles. So, we hav... | [
{
"content": "On the side \\( DE \\) of a regular hexagon \\( ABCDEF \\), point \\( K \\) is chosen so that the line \\( AK \\) divides the area of the hexagon in the ratio \\( 3:1 \\). In what ratio does point \\( K \\) divide the side \\( DE \\) ? Answer: \\( 3:1 \\).",
"role": "user"
},
{
"conten... |
olympiads | A (convex) trapezoid \(ABCD\) is called "good" if it has a circumcircle, the sides \(AB\) and \(CD\) are the parallel sides, and \(CD\) is shorter than \(AB\). For a good trapezoid \(ABCD\), the following designations are made:
- The line parallel to \(AD\) passing through \(B\) intersects the extension of side \(CD\)... |
1. **Condition Analysis:**
- We need to find an equivalent condition for the angles $ \Varangle BSE $ and $ \Varangle FSC $ to be equal in an inscribed trapezoid $ ABCD $ with parallel sides $ AB $ and $ CD $, where $ CD < AB $.
- $S$ is defined such that the parallel to $ AD $ through $ B $ intersects the ext... | [
{
"content": "A (convex) trapezoid \\(ABCD\\) is called \"good\" if it has a circumcircle, the sides \\(AB\\) and \\(CD\\) are the parallel sides, and \\(CD\\) is shorter than \\(AB\\). For a good trapezoid \\(ABCD\\), the following designations are made:\n\n- The line parallel to \\(AD\\) passing through \\(B\... |
olympiads | There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty boxes. How many boxes wer... |
Given that:
- Initially, there were 2013 empty boxes.
- In each operation, 13 new boxes are added (not nested within each other).
- After several operations, the number of non-empty boxes became 2013.
- We need to determine the total number of boxes.
Here is the detailed step-by-step solution:
1. **Understanding the... | [
{
"content": "There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there were 2013 non-empty box... |
olympiads | Using an electric stove with a power of \( P = 1000 \, \text{W} \), a certain amount of water is heated. When the stove is turned on for \( t_1 = 2 \, \text{minutes} \), the temperature of the water increases by \( \Delta T = 2^{\circ} \text{C} \). After turning off the heater, the temperature decreases back to the ini... |
1. **Understanding the problem and notation:**
- Power of the electric stove: \( P = 1000 \, \mathrm{W} \)
- Time for heating: \( t_1 = 2 \, \mathrm{min} = 120 \, \mathrm{s} \)
- Temperature increase: \( \Delta T = 2^{\circ} \mathrm{C} \)
- Time for cooling back to initial temperature: \( t_2 = 1 \, \mathr... | [
{
"content": "Using an electric stove with a power of \\( P = 1000 \\, \\text{W} \\), a certain amount of water is heated. When the stove is turned on for \\( t_1 = 2 \\, \\text{minutes} \\), the temperature of the water increases by \\( \\Delta T = 2^{\\circ} \\text{C} \\). After turning off the heater, the te... |
olympiads | How can a $3 \times 3$ square be cut into unit squares with the least number of straight-line cuts:
a) if the parts cannot be overlapped (i.e., each time, only one part can be cut)?
b) if the parts can be overlapped?
c) if the square can be folded before cutting? (Answer: one cut is sufficient) |
To solve the problem of determining the minimum number of straight cuts required to divide a $3 \times 3$ square into unit squares under different conditions, let's address each part separately.
#### Part b: If the parts can overlap.
1. **Initial Configuration**:
- We start with a $3 \times 3$ square grid.
- T... | [
{
"content": "How can a $3 \\times 3$ square be cut into unit squares with the least number of straight-line cuts:\n\na) if the parts cannot be overlapped (i.e., each time, only one part can be cut)?\n\nb) if the parts can be overlapped?\n\nc) if the square can be folded before cutting? (Answer: one cut is suff... |
olympiads | A student began an examination task between 9 and 10 o’clock and finished between 1 and 2 o’clock. Find the exact time the student spent on the task, given that at the beginning and at the end of the work, the hour and minute hands of the clock, when swapped, occupied the same positions on the clock face.
Answer: $60 ... |
1. Analyze the initial and final positions of the hour hand and minute hand on the clock. It is given that the exam starts between 9:00 and 10:00 and ends between 13:00 and 14:00. The hour and minute hands switch positions at the start and end of the exam.
2. Let's denote the exact amount of time taken to complete... | [
{
"content": "A student began an examination task between 9 and 10 o’clock and finished between 1 and 2 o’clock. Find the exact time the student spent on the task, given that at the beginning and at the end of the work, the hour and minute hands of the clock, when swapped, occupied the same positions on the clo... |
olympiads | In a triangle with side lengths \(a, b, c\) and an area equal to \(\frac{1}{4}\) and a circumradius of 1, if
$$
s=\sqrt{a}+\sqrt{b}+\sqrt{c}, \quad t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},
$$
then the relationship between \(s\) and \(t\) is:
(A) \(s>t\);
(B) \(s=t\);
(C) \(s<t\);
(D) Uncertain.
Answer: ( ) |
To determine the relationship between \( s \) and \( t \) where \( a, b, c \) are the side lengths of a triangle, we first translate the given conditions and use appropriate theorems.
1. **Given Conditions and Initial Equations:**
\[
\text{Area of the triangle} = \frac{1}{4}, \quad \text{and the circumradius} ... | [
{
"content": "In a triangle with side lengths \\(a, b, c\\) and an area equal to \\(\\frac{1}{4}\\) and a circumradius of 1, if\n$$\ns=\\sqrt{a}+\\sqrt{b}+\\sqrt{c}, \\quad t=\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c},\n$$\nthen the relationship between \\(s\\) and \\(t\\) is:\n(A) \\(s>t\\);\n(B) \\(s=t\\);\n(C) \... |
olympiads | Vovochka approached a game machine, on the screen of which the number 0 was displayed. The rules of the game stated: "The screen shows the number of points. If you insert a 1-ruble coin, the number of points increases by 1. If you insert a 2-ruble coin, the number of points doubles. If you score 50 points, the machine ... |
1. Let's consider the scenario from the target, which is to achieve a score of 50 points, and we need to understand how to reach there starting from 0. We should minimize the total amount of money spent in terms of $1$-rubles and $2$-rubles.
2. The game rules are:
- Inserting a $1$ ruble coin increases the score ... | [
{
"content": "Vovochka approached a game machine, on the screen of which the number 0 was displayed. The rules of the game stated: \"The screen shows the number of points. If you insert a 1-ruble coin, the number of points increases by 1. If you insert a 2-ruble coin, the number of points doubles. If you score ... |
olympiads | On the island of knights and liars live knights who always tell the truth and liars who always lie. One day, five inhabitants of this island were asked in turn how many of them are knights.
- "One," said the first.
- "Two," said the second.
- "Three," said the third.
- "Don't believe them, they are all liars," said th... |
1. **Identifying Constraints:**
- We have five inhabitants on the island, each either a knight (always tells the truth) or a liar (always lies).
- The statements provided by each inhabitant are as follows:
1. The first claims that there is one knight.
2. The second claims that there are two kni... | [
{
"content": "On the island of knights and liars live knights who always tell the truth and liars who always lie. One day, five inhabitants of this island were asked in turn how many of them are knights.\n\n- \"One,\" said the first.\n- \"Two,\" said the second.\n- \"Three,\" said the third.\n- \"Don't believe ... |
olympiads | For the natural numbers \( x \) and \( y \) and an odd integer \( z \), it is known that \( x! + y! = 8z + 2017 \). Find all possible triples of numbers \( (x, y, z) \). (Recall that \( 1! = 1, 2! = 1 \cdot 2, n! = 1 \cdot 2 \cdot \ldots \cdot n \)).
Answer: \( (1, 4, -249), (4, 1, -249), (1, 5, -237), (5, 1, -237) \... |
1. **Understanding the Equation and Initial Constraints**:
The given problem is:
\[
x! + y! = 8z + 2017
\]
where \(x\) and \(y\) are natural numbers, and \(z\) is an odd integer.
By the properties of factorials, we can infer that both \(x!\) and \(y!\) are natural numbers. The sum \(x! + y!\) must ... | [
{
"content": "For the natural numbers \\( x \\) and \\( y \\) and an odd integer \\( z \\), it is known that \\( x! + y! = 8z + 2017 \\). Find all possible triples of numbers \\( (x, y, z) \\). (Recall that \\( 1! = 1, 2! = 1 \\cdot 2, n! = 1 \\cdot 2 \\cdot \\ldots \\cdot n \\)). \n\nAnswer: \\( (1, 4, -249), ... |
olympiads | The airplane departed from Perm on September 28 at noon and arrived in Kirov at 11:00 AM (all times mentioned are local times). At 7:00 PM on the same day, the airplane departed from Kirov to Yakutsk and arrived there at 7:00 AM. After three hours, it departed from Yakutsk to Perm and returned there at 11:00 AM on Sept... |
To determine the amount of time the airplane was in the air, follow these steps:
1. **Calculate Total Time Away from Perm**: The airplane departed from Perm at 12:00 PM on September 28 and returned at 11:00 AM on September 29.
\[
\text{Total time away} = \text{Time of return in Perm (September 29, 11:00 AM)} ... | [
{
"content": "The airplane departed from Perm on September 28 at noon and arrived in Kirov at 11:00 AM (all times mentioned are local times). At 7:00 PM on the same day, the airplane departed from Kirov to Yakutsk and arrived there at 7:00 AM. After three hours, it departed from Yakutsk to Perm and returned the... |
olympiads | In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646. |
To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets.
1. **Case 1: Buying the set "keyboard and mouse" and additional headphones.**
- There are 4 such sets available.
- There ar... | [
{
"content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.",
"role": "user"
},
... |
olympiads | Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives... |
Given the game rules that involve starting with a score of \(0\) and using 1-ruble and 2-ruble coins to adjust the points, we need to find the minimum rubles necessary to achieve exactly \(50\) points without exceeding that amount. Below, we will carefully analyze the steps:
1. **Understanding Coin Operations:**
-... | [
{
"content": "Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: \"The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 po... |
olympiads | Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 caskets, and it is known that only one of them contains the treasure, while the others will cause a stone slab to fall on the person trying to open them. The caskets are numbered from left to right. The first, fourth, and fif... |
1. **Identify and Translate Clues:**
- Box 1: "The treasure is in me or in Box 4."
- Box 2: "The treasure is in the box to the left of me."
- Box 3: "The treasure is in me or in the rightmost box."
- Box 4: "There is no treasure in the boxes to the left of me."
- Box 5: "All other boxes contain lie... | [
{
"content": "Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 caskets, and it is known that only one of them contains the treasure, while the others will cause a stone slab to fall on the person trying to open them. The caskets are numbered from left to right. The ... |
olympiads |
Given that replacing \( x \) with \( -x \) does not change the inequality due to the properties of absolute value (\( |x| = |-x| \)), the region described by the inequality will be symmetric relative to the \( OY \)-axis. Therefore, it is sufficient to find the area of the figure for \( x \geq 0 \). In the region \( x... | Let's analyze the first problem step-by-step.
1. **Problem Statement and Understanding Symmetry**:
Given the inequality involving the absolute value, we utilize the property of the modulus: \( |x| = |-x| \). Thus, replacing \( x \) with \( -x \) does not change the inequality. This implies the region defined by t... | [
{
"content": "\nGiven that replacing \\( x \\) with \\( -x \\) does not change the inequality due to the properties of absolute value (\\( |x| = |-x| \\)), the region described by the inequality will be symmetric relative to the \\( OY \\)-axis. Therefore, it is sufficient to find the area of the figure for \\(... |
olympiads | Evaluate the solution and answer provided below. Point out all the errors and shortcomings.
Find the range of the function \( f(x) = x + \frac{2}{x^2} \) for \( x > 0 \).
Solution: According to the inequality between the arithmetic mean and the geometric mean, the following inequality holds:
\[ x + \frac{2}{x^2} \ge... |
Given the function \( f(x) = x + \frac{2}{x^2} \) for \( x > 0 \), we need to find the range of the function \( f(x) \).
1. **Finding the minimum value of the function:**
The first step to identify the range is to find the critical points of the function \( f(x) \). We start by computing the first derivative of ... | [
{
"content": "Evaluate the solution and answer provided below. Point out all the errors and shortcomings.\n\nFind the range of the function \\( f(x) = x + \\frac{2}{x^2} \\) for \\( x > 0 \\).\n\nSolution: According to the inequality between the arithmetic mean and the geometric mean, the following inequality h... |
olympiads | There are 7 empty boxes on a circle, numbered in a clockwise direction as $1, 2, 3, 4, 5, 6, 7$. Xiao Ming first puts a white piece in box number 1, then puts the second white piece in box number 3, and then puts the third white piece in box number 6, and so on. After placing the $(k-1)$-th white piece, Xiao Ming count... |
1. **Label the Boxes:**
- There are 7 empty boxes in a circular arrangement, labeled sequentially in a clockwise direction as 1, 2, 3, 4, 5, 6, and 7.
2. **First Placement Rule:**
- Xiaoming places the first white checker in box 1.
- The second checker goes to box 3 (count 2 positions from box 1).
- T... | [
{
"content": "There are 7 empty boxes on a circle, numbered in a clockwise direction as $1, 2, 3, 4, 5, 6, 7$. Xiao Ming first puts a white piece in box number 1, then puts the second white piece in box number 3, and then puts the third white piece in box number 6, and so on. After placing the $(k-1)$-th white ... |
olympiads | A $7 \times 7$ table is filled with zeroes. In one operation, the minimum number in the table is found (if there are several such numbers, any one of them is chosen), and one is added to it as well as to all the numbers in the cells adjacent to it by side or corner. What is the largest number that can appear in one of ... |
Given a $7 \times 7$ matrix filled entirely with zeros initially, we need to determine the maximum number that can appear in any cell after 90 operations. Each operation involves selecting a cell containing the minimum value (if multiple such cells exist, any can be chosen) and incrementing that cell and its adjacent ... | [
{
"content": "A $7 \\times 7$ table is filled with zeroes. In one operation, the minimum number in the table is found (if there are several such numbers, any one of them is chosen), and one is added to it as well as to all the numbers in the cells adjacent to it by side or corner. What is the largest number tha... |
olympiads | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively ($b \neq 1$). If both $\frac{C}{A}$ and $\frac{\sin B}{\sin A}$ are the roots of the equation $\log_{\sqrt{6}} x = \log_{b}(4x-4)$, then the triangle $\triangle ABC$ is:
(A) An isosceles triangle, but not a ... | 1. Given:
\[
\log_{\sqrt{6}} x = \log_b(4x - 4)
\]
and the roots of this equation are \(\frac{C}{A}\) and \(\frac{\sin B}{\sin A}\).
2. Begin by rewriting the logarithmic equation in a common base. We know:
\[
\log_{\sqrt{6}} x = \log_b(4x - 4)
\]
3. Using the change of base formula, we expres... | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively ($b \\neq 1$). If both $\\frac{C}{A}$ and $\\frac{\\sin B}{\\sin A}$ are the roots of the equation $\\log_{\\sqrt{6}} x = \\log_{b}(4x-4)$, then the triangle $\\triangle ABC$ is:\n(A) ... |
olympiads | Alice saw a mysterious caterpillar while wandering in Wonderland. It was very long, with a visible head but an infinitely extending body. She noticed that it resembled the repeating decimal 3.141592, with the head being the digit 3 and its body repeating the sequence $14159215921592 \ldots \ldots$.
The caterpillar ask... | ### Problem:
爱丽丝梦游仙境时看到一只奇怪的毛毛虫，它很长，见首不见尾。爱丽丝仔细观察发现它就像循环小数 \( 3.141592 \), 头是数字 \( 3 \), 各节身体就像小数点后的数 \( 141592141592\ldots \ldots \)。毛毛虫问爱丽丝: “从我的头开始往后数 2020 个数字, 和是多少?”
1. **Identify the pattern in the repeating sequence**:
The decimal expansion of \( \pi \) is given as \( 3.141592... \). The repeating part aft... | [
{
"content": "Alice saw a mysterious caterpillar while wandering in Wonderland. It was very long, with a visible head but an infinitely extending body. She noticed that it resembled the repeating decimal 3.141592, with the head being the digit 3 and its body repeating the sequence $14159215921592 \\ldots \\ldot... |
olympiads |
The altitudes \( CF \) and \( AE \) of the acute-angled triangle \( ABC \) intersect at point \( H \). Points \( M \) and \( N \) are the midpoints of segments \( AH \) and \( CH \), respectively. It is known that \( FM = 1 \), \( EN = 4 \), and \( FM \parallel EN \). Find the measure of \( \angle ABC \), the area of ... |
1. We are considering an acute-angled triangle $ABC$ with altitudes $CF$ and $AE$ intersecting at $H$. Points $M$ and $N$ are the midpoints of segments $AH$ and $CH$ respectively, and we are given that $FM = 1$, $EN = 4$, and $FM \parallel EN$.
2. First, notice that since $FM$ is the median in triangle $AFH$ and $EN$... | [
{
"content": "\nThe altitudes \\( CF \\) and \\( AE \\) of the acute-angled triangle \\( ABC \\) intersect at point \\( H \\). Points \\( M \\) and \\( N \\) are the midpoints of segments \\( AH \\) and \\( CH \\), respectively. It is known that \\( FM = 1 \\), \\( EN = 4 \\), and \\( FM \\parallel EN \\). Find... |
olympiads | Given the equation \(\arccos \frac{4}{5} - \arccos \left(-\frac{4}{5}\right) = \arcsin x\), determine the value of \(x\):
(A) \(x = \frac{24}{25}\);
(B) \(x = -\frac{24}{25}\);
(C) \(x = 0\);
(D) Such an \(x\) does not exist.
Answer: \(\ \) |
1. Start with the given equation:
\[
\arccos \frac{4}{5} - \arccos \left( -\frac{4}{5} \right) = \arcsin x
\]
2. Use the property of the arccos function:
\[
\arccos \left( -\frac{4}{5} \right) = \pi - \arccos \frac{4}{5}
\]
3. Substitute the property into the original equation:
\[
\arccos ... | [
{
"content": "Given the equation \\(\\arccos \\frac{4}{5} - \\arccos \\left(-\\frac{4}{5}\\right) = \\arcsin x\\), determine the value of \\(x\\):\n(A) \\(x = \\frac{24}{25}\\);\n(B) \\(x = -\\frac{24}{25}\\);\n(C) \\(x = 0\\);\n(D) Such an \\(x\\) does not exist.\nAnswer: \\(\\ \\)",
"role": "user"
},
... |
olympiads | In the figure, the circle $K$ touches the line $L_{3}\left(d_{1}<d_{2}\right)$. The radii of the circles $K_{1}$, $K_{2}$, and $K$ are denoted by $r_{1}=\frac{d_{1}}{2}$, $r_{2}=\frac{d_{2}}{2}$, and $r$. Their centers are $O_{1}$, $O_{2}$, and $O$. Points $B$, $C$, and $D$ are points of tangency of the circles with th... |
1. From the diagram, the given relationships between the radii are:
\[ r_1 = \frac{d_1}{2}, \quad r_2 = \frac{d_2}{2} \]
2. We have two sets of relations from the external tangency and triangles involving the centers of the circles:
\[
\begin{cases}
\left(r + r_1 \right)^2 = \left(r - 2r_2 - r_1\right)^2 + y^2 \\
\... | [
{
"content": "In the figure, the circle $K$ touches the line $L_{3}\\left(d_{1}<d_{2}\\right)$. The radii of the circles $K_{1}$, $K_{2}$, and $K$ are denoted by $r_{1}=\\frac{d_{1}}{2}$, $r_{2}=\\frac{d_{2}}{2}$, and $r$. Their centers are $O_{1}$, $O_{2}$, and $O$. Points $B$, $C$, and $D$ are points of tange... |
olympiads | Danil drew several lines through a single point. Among all the formed angles, he only considered angles with integer degree measures. Danil claims that among these angles, there are exactly 15 more angles with odd measures than those with even measures. Can this be true? Answer: No, it cannot be true. |
1. **Identify the Problem Context:**
Danil has drawn several straight lines through a single point, creating multiple angles. He then investigates only those angles that have integer degree measures. We need to determine if Danil's claim that there are exactly 15 more angles with odd degree measures than with even ... | [
{
"content": "Danil drew several lines through a single point. Among all the formed angles, he only considered angles with integer degree measures. Danil claims that among these angles, there are exactly 15 more angles with odd measures than those with even measures. Can this be true? Answer: No, it cannot be t... |
olympiads | Is the equation \( y^{2} + y = x^{3} - x \) solvable in coprime natural numbers? Answer: No. |
To determine if the equation \( y^2 + y = x^3 - x \) can be solved in mutually prime natural numbers, we proceed as follows:
1. Rewrite the equation in the following form:
\[
y(y + 1) = x(x^2 - 1)
\]
2. Since \(\gcd(x, y) = 1\) (i.e., \(x\) and \(y\) are mutually prime), it follows that \(y + 1\) must be a ... | [
{
"content": "Is the equation \\( y^{2} + y = x^{3} - x \\) solvable in coprime natural numbers? Answer: No.",
"role": "user"
},
{
"content": "\nTo determine if the equation \\( y^2 + y = x^3 - x \\) can be solved in mutually prime natural numbers, we proceed as follows:\n\n1. Rewrite the equation i... |
olympiads | Let \( m \geq n \) be positive integers. Frieder is given \( mn \) posters of Linus with different integer dimensions \( k \times l \) with \( 1 \leq k \leq m \) and \( 1 \leq l \leq n \). He must put them all up on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty... |
1. **Define Key Terms:**
- A *chain* is a sequence of posters, each poster covering the previous one completely.
- An *antichain* is a collection of posters where no poster can completely cover any other poster in the group.
- The *degree* of a poster is defined as the sum of its length and width.
2. **An... | [
{
"content": "Let \\( m \\geq n \\) be positive integers. Frieder is given \\( mn \\) posters of Linus with different integer dimensions \\( k \\times l \\) with \\( 1 \\leq k \\leq m \\) and \\( 1 \\leq l \\leq n \\). He must put them all up on his bedroom wall without rotating them. Every time he puts up a po... |
olympiads | As shown in the figure, in an isosceles triangle \( \triangle ABC \), \( CD \) is the altitude to the base \( AB \), \( E \) is the midpoint of \( BC \), and \( AE \) intersects \( CD \) at \( F \). Given the three routes:
(a) \( A \rightarrow F \rightarrow C \rightarrow E \rightarrow B \rightarrow D \rightarrow A \);
... |
Given an isosceles triangle \( \triangle ABC \), where \( CD \) is the altitude from vertex \( C \) to the base \( AB \), and \( E \) is the midpoint of side \( BC \). Line \( AE \) intersects \( CD \) at point \( F \). We need to analyze the three given paths:
1. \( A \rightarrow F \rightarrow C \rightarrow E \right... | [
{
"content": "As shown in the figure, in an isosceles triangle \\( \\triangle ABC \\), \\( CD \\) is the altitude to the base \\( AB \\), \\( E \\) is the midpoint of \\( BC \\), and \\( AE \\) intersects \\( CD \\) at \\( F \\). Given the three routes:\n(a) \\( A \\rightarrow F \\rightarrow C \\rightarrow E \\... |
olympiads | As shown in the figure, in triangle $\mathrm{ABC}$, point $\mathrm{D}$ is on $\mathrm{BC}$, and $\angle \mathrm{ABC} = \angle \mathrm{ACB}$, $\angle \mathrm{ADC} = \angle \mathrm{DAC}$, and $\angle \mathrm{DAB} = 21^{\circ}$. Find the measure of $\angle \mathrm{ABC}$; also, answer: which triangles in the figure are acu... |
1. Given:
- In $\triangle \mathrm{ABC}$, $\angle \mathrm{ABC} = \angle \mathrm{ACB}$.
- Point $\mathrm{D}$ is on $\mathrm{BC}$.
- $\angle \mathrm{ADC} = \angle \mathrm{DAC}$.
- $\angle \mathrm{DAB} = 21^\circ$.
2. We are asked to find $\angle \mathrm{ABC}$.
3. Let $\angle \mathrm{ABC} = \angle \mathrm{AC... | [
{
"content": "As shown in the figure, in triangle $\\mathrm{ABC}$, point $\\mathrm{D}$ is on $\\mathrm{BC}$, and $\\angle \\mathrm{ABC} = \\angle \\mathrm{ACB}$, $\\angle \\mathrm{ADC} = \\angle \\mathrm{DAC}$, and $\\angle \\mathrm{DAB} = 21^{\\circ}$. Find the measure of $\\angle \\mathrm{ABC}$; also, answer:... |
olympiads | Find the values of \(a\) and \(b\).
**Answer:**
\[a = \pm \frac{4}{3 \sqrt{3}}, \quad b = \pm \frac{2}{3 \sqrt{3}}\] |
1. Consider the scenario where the numbers \(a\) and \(b\) have the same sign. In this case, we have \( |a| + |b| = |a+b| \).
2. Let \( x = \frac{\pi}{3} \). Then,
\[
2x = \frac{2\pi}{3}, \quad \sin x = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin 2x = \sin \frac{2\pi}{3} = \frac{\sqrt{3... | [
{
"content": "Find the values of \\(a\\) and \\(b\\). \n\n**Answer:**\n\\[a = \\pm \\frac{4}{3 \\sqrt{3}}, \\quad b = \\pm \\frac{2}{3 \\sqrt{3}}\\]",
"role": "user"
},
{
"content": "\n1. Consider the scenario where the numbers \\(a\\) and \\(b\\) have the same sign. In this case, we have \\( |a| + ... |
olympiads | The accrued salary of a citizen from January to June inclusive was 23,000 rubles per month, and from July to December, it was 25,000 rubles. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs to be paid to the bu... |
#### Step 1: Calculate income tax (НДФЛ) from salary
1. **Calculate January to June earnings**:
\[
23000 \text{ руб. per month} \times 6 \text{ months} = 138000 \text{ руб.}
\]
2. **Calculate July to December earnings**:
\[
25000 \text{ руб. per month} \times 6 \text{ months} = 150000 \text{ руб.}
... | [
{
"content": "The accrued salary of a citizen from January to June inclusive was 23,000 rubles per month, and from July to December, it was 25,000 rubles. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax need... |
olympiads |
Calculate
$$
\frac{2 a b\left(a^{3}-b^{3}\right)}{a^{2}+a b+b^{2}}-\frac{(a-b)\left(a^{4}-b^{4}\right)}{a^{2}-b^{2}} \quad \text{ when } \quad a=-1, 555 \ldots 566 \quad (\text{2010 fives}), \quad b=5, 444 \ldots 4444 \quad (\text{2011 fours})
$$
Answer: 343. |
1. **Expression Breakdown**: Write down the expression we need to simplify:
\[
\frac{2ab(a^3 - b^3)}{a^2 + ab + b^2} - \frac{(a - b)(a^4 - b^4)}{a^2 - b^2}
\]
2. **Substitute Values for \( a \) and \( b \)**:
Given:
\[
a = -1\underbrace{5 \ldots 5}_{2010}6
\]
\[
b = 5\underbrace{4 \ldots 44... | [
{
"content": "\nCalculate\n\n$$\n\\frac{2 a b\\left(a^{3}-b^{3}\\right)}{a^{2}+a b+b^{2}}-\\frac{(a-b)\\left(a^{4}-b^{4}\\right)}{a^{2}-b^{2}} \\quad \\text{ when } \\quad a=-1, 555 \\ldots 566 \\quad (\\text{2010 fives}), \\quad b=5, 444 \\ldots 4444 \\quad (\\text{2011 fours})\n$$\n\nAnswer: 343.",
"role"... |
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