conversations listlengths 2 2 | question stringlengths 11 28.8k |
|---|---|
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\nsize(150... | A cone has a volume of $12288\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]
import markers;
size(150);
import geometry;
draw(scale(1,.2)*arc((0,0),1,0,180),dashed);
draw(scale(1,.2)*arc((0,... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the projection of the vector $\\bar{a}=3 \\bar{i}+4 \\bar{j}-\\bar{k}$ in the direction of the vector $\\bar{b}=\\bar{i}+\\bar{j}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the projection o... | Find the projection of the vector $\bar{a}=3 \bar{i}+4 \bar{j}-\bar{k}$ in the direction of the vector $\bar{b}=\bar{i}+\bar{j}$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( P \\in \\mathbb{R}[X] \\) be a monic polynomial of degree 2020 such that \\( P(n) = n \\) for all \\( n \\in \\{0, 1, \\ldots, 2019\\} \\). Calculate \\( P(2020) \\)."
},
{
"from": "assistant",
"value": "<think>\nOk... | Let \( P \in \mathbb{R}[X] \) be a monic polynomial of degree 2020 such that \( P(n) = n \) for all \( n \in \{0, 1, \ldots, 2019\} \). Calculate \( P(2020) \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all subsequences \\( \\{a_{1}, a_{2}, \\cdots, a_{n}\\} \\) of the sequence \\(\\{1, 2, \\cdots, n\\}\\) that satisfy the property \\( i + 1 \\mid 2(a_{1} + a_{2} + \\cdots + a_{j}) \\) for \\(1 \\leq i \\leq n\\)."
},
{
"... | Find all subsequences \( \{a_{1}, a_{2}, \cdots, a_{n}\} \) of the sequence \(\{1, 2, \cdots, n\}\) that satisfy the property \( i + 1 \mid 2(a_{1} + a_{2} + \cdots + a_{j}) \) for \(1 \leq i \leq n\). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. During a trip, person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the... | During a trip, person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the coastal road, how much time did the entire trip take in minutes?
(A) 50
(B) 70
(C) 8... |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.You are given two images img1 and img2 both of size n x n, represented as binary, square matrices of the same size. (A binary matrix has only... | Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.You are given two images img1 and img2 both of size n x n, represented as binary, square matrices of the same size. (A binary matrix has only 0s and 1s as values.)
We translate one ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given an acute triangle \\(ABC\\). The altitude \\(AA_1\\) is extended through vertex \\(A\\) by a segment \\(AA_2 = BC\\). The altitude \\(CC_1\\) is extended through vertex \\(C\\) by a segment \\(CC_2 = AB\\). Find the angles of tri... | Given an acute triangle \(ABC\). The altitude \(AA_1\) is extended through vertex \(A\) by a segment \(AA_2 = BC\). The altitude \(CC_1\) is extended through vertex \(C\) by a segment \(CC_2 = AB\). Find the angles of triangle \(A_2BC_2\). |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in [Russian] and [Bengali].\n\nThere's a zebra crossing appearing in the mi... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in [Russian] and [Bengali].
There's a zebra crossing appearing in the middle of nowhere with N blocks in it. The c... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given the function\n$$\ny(x) = (\\sqrt{1+x} + \\sqrt{1-x} + 2) \\left(\\sqrt{1-x^2} + 1 \\right),\n$$\n\nwhere \\(x \\in [0,1]\\). Determine the minimum value of \\(y(x)\\)."
},
{
"from": "assistant",
"value": "<think>\nOka... | Given the function
$$
y(x) = (\sqrt{1+x} + \sqrt{1-x} + 2) \left(\sqrt{1-x^2} + 1 \right),
$$
where \(x \in [0,1]\). Determine the minimum value of \(y(x)\). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at... | Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways ca... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a sequence of \\( n \\) distinct numbers, it is required to simultaneously find the largest and the second largest numbers.\n\na) Prove that this can be done by comparing \\( n+m-2 \\) pairs of numbers, where the integer \\( m \\) i... | In a sequence of \( n \) distinct numbers, it is required to simultaneously find the largest and the second largest numbers.
a) Prove that this can be done by comparing \( n+m-2 \) pairs of numbers, where the integer \( m \) is determined by the inequalities \( m-1 < \log_{2} n \leq m \).
b) Prove that it is generall... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. A hexahedron \\(ABCD E\\) is made of two regular congruent tetrahedra \\(ABCD\\) and \\(ABCE\\). Prove that there exists only one isometry \\(\\mathcal{Z}\\) that maps points \\(A, B, C, D, E\\) onto \\(B, C, A, E, D\\), respectively. ... | A hexahedron \(ABCD E\) is made of two regular congruent tetrahedra \(ABCD\) and \(ABCE\). Prove that there exists only one isometry \(\mathcal{Z}\) that maps points \(A, B, C, D, E\) onto \(B, C, A, E, D\), respectively. Find all points \(X\) on the surface of the hexahedron whose distance from \(\mathcal{Z}(X)\) is m... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the smallest positive integer \\( p \\) for which \\(\\sqrt{2^{3} \\times 5 \\times p}\\) is an integer?\n(A) 2\n(B) 5\n(C) 10\n(D) 1\n(E) 20"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to ... | What is the smallest positive integer \( p \) for which \(\sqrt{2^{3} \times 5 \times p}\) is an integer?
(A) 2
(B) 5
(C) 10
(D) 1
(E) 20 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \\angle A \\equal{} 50^{\\circ}$ . Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \\angle KAC \\equal{} \\angle LAB \\equal{} 10^{\\circ}$ . Det... | (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$ . Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$ . Determine the ratio $ CK/LB$ . |
[
{
"from": "user",
"value": "\"What is the expected temperature range for dark atoms or dark molecules to remain stable in a dark matter halo based on current theoretical models?\""
},
{
"from": "assistant",
"value": "<think>\nOkay, so the user is asking about the expected temperature range for d... | "What is the expected temperature range for dark atoms or dark molecules to remain stable in a dark matter halo based on current theoretical models?" |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a line \\( l \\) and a point \\( A \\). An arbitrary line passing through \\( A \\) and skew to \\( l \\) is constructed. Let \\( M \\) and \\( N \\) be the endpoints of the common perpendicular to this arbitrary line and \\( l \... | Given a line \( l \) and a point \( A \). An arbitrary line passing through \( A \) and skew to \( l \) is constructed. Let \( M \) and \( N \) be the endpoints of the common perpendicular to this arbitrary line and \( l \) (where \( M \) is on the line passing through \( A \)). Find the locus of points \( M \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. There are 2013 cards, each with a unique number written on them, all face down. In one move, you can point to ten cards, and you will be told one of the numbers written on those ten cards (but not which card the number is on). What is ... | There are 2013 cards, each with a unique number written on them, all face down. In one move, you can point to ten cards, and you will be told one of the numbers written on those ten cards (but not which card the number is on). What is the largest number \( t \) for which you can guarantee to determine the exact number ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$ $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0... | Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$ $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$ *Proposed by Kyle Lee* |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. If we connect the point of intersection $A$ of two tangents to an ellipse with one of its foci $G$, the line $AG$ bisects the angle $KGH$, where $K$ and $H$ are the points where the tangents touch the ellipse. Prove this theorem if the... | If we connect the point of intersection $A$ of two tangents to an ellipse with one of its foci $G$, the line $AG$ bisects the angle $KGH$, where $K$ and $H$ are the points where the tangents touch the ellipse. Prove this theorem if the equation of the ellipse is $9x^{2} + 25y^{2} = 225$, and the coordinates of the poin... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nKing Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \\( p \\), or an incorrect answer with a probability \\( q = 1-p \\).\n\nIf both advisors give ... |
King Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \( p \), or an incorrect answer with a probability \( q = 1-p \).
If both advisors give the same answer, the king listens to them. If they give opposite answers, the king makes a d... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all natural numbers \\( n \\) such that the sum of the digits of the number \\( 2^n \\) in decimal notation equals 5."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all natural numbers n such tha... | Find all natural numbers \( n \) such that the sum of the digits of the number \( 2^n \) in decimal notation equals 5. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given \\( f(x) = \\frac{1+x}{2-x} \\), for \\( n \\in \\mathbf{N} \\), define \\( f_{1}(x)=f(x) \\) and \\( f_{n+1}(x)=f\\left[f_{n}(x)\\right] \\). If \\( f_{13}(x)=f_{31}(x) \\), what is the expression for \\( f_{16}(x) \\)?\nA. \\( ... | Given \( f(x) = \frac{1+x}{2-x} \), for \( n \in \mathbf{N} \), define \( f_{1}(x)=f(x) \) and \( f_{n+1}(x)=f\left[f_{n}(x)\right] \). If \( f_{13}(x)=f_{31}(x) \), what is the expression for \( f_{16}(x) \)?
A. \( \frac{x+1}{x} \)
B. \( \frac{x}{x+1} \)
C. \( \frac{x-1}{x} \)
D. \( \frac{x}{x-1} \) |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Takahashi has many red balls and blue balls. Now, he will place them in a row.\n\nInitially, there i... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Takahashi has many red balls and blue balls. Now, he will place them in a row.
Initially, there is no ball placed.
Takahashi, who is very ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The side $AC$ of triangle $ABC$ is extended at segment $CD = AB = 1$ . It is known that $\\angle ABC = 90^o$ , $\\angle CBD = 30^o$ . Calculate $AC$ ."
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try t... | The side $AC$ of triangle $ABC$ is extended at segment $CD = AB = 1$ . It is known that $\angle ABC = 90^o$ , $\angle CBD = 30^o$ . Calculate $AC$ . |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $A, B, C, D, E$ be integers, $B \\neq 0$ and $F=AD^{2}-BCD+B^{2}E \\neq 0$ . Prove that the number $N$ of pairs of integers $(x, y)$ such that \\[Ax^{2}+Bxy+Cx+Dy+E=0,\\] satisfies $N \\le 2 d( \\vert F \\vert )$ , where ... | Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$ . Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$ , where $d(n)$ denotes the number of positive divisors of positive integer $n$ . |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Buying 2 bottles of soda and 1 bottle of mineral water costs 7 yuan in total. Buying 4 bottles of soda and 3 bottles of mineral water costs 16 yuan in total. How much will it cost to buy 10 bottles of soda and 10 bottles of mineral wat... | Buying 2 bottles of soda and 1 bottle of mineral water costs 7 yuan in total. Buying 4 bottles of soda and 3 bottles of mineral water costs 16 yuan in total. How much will it cost to buy 10 bottles of soda and 10 bottles of mineral water?
(A) 45
(B) 46
(C) 47
(D) 55 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The budget of the Petrov family consists of the following sources of income:\n\n- Parents' salary after income tax deduction: 56000 rubles;\n- Grandmother's pension: 14300 rubles;\n- Son's scholarship: 2500 rubles.\n\nThe average month... | The budget of the Petrov family consists of the following sources of income:
- Parents' salary after income tax deduction: 56000 rubles;
- Grandmother's pension: 14300 rubles;
- Son's scholarship: 2500 rubles.
The average monthly expenses of the family include:
- Payment for communal services: 9800 rubles;
- Food ex... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \\Big(x+\\frac{1}{3}\\Big) + f \\Big(x+\\frac{1}{2}\\Big)=f(x)+ f \\Big(x+\\frac{5}{6}\\Big)}$ . Show... | Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In triangle $ABC$, the altitude $AH$ passes through the midpoint of the median $BM$. Prove that in triangle $BMC$, one of the altitudes also passes through the midpoint of one of the medians."
},
{
"from": "assistant",
"val... | In triangle $ABC$, the altitude $AH$ passes through the midpoint of the median $BM$. Prove that in triangle $BMC$, one of the altitudes also passes through the midpoint of one of the medians. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. On a $(4n + 2)\\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of ... | On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$ , what is the largest positive integer $k$ such that there must be a row or col... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of \n\n$\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }0.5 \\qquad \\mathrm{(C) \\ }1 \\qquad \\mathrm{(D) \\ }... | Define $[a,b,c]$ to mean $\frac {a+b}c$, where $c \neq 0$. What is the value of
$\left[[60,30,90],[2,1,3],[10,5,15]\right]?$
$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.5 \qquad \mathrm{(C) \ }1 \qquad \mathrm{(D) \ }1.5 \qquad \mathrm{(E) \ }2$ |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The volume of the tetrahedron \\(ABCD\\) is 5. A plane passes through the midpoints of edges \\(AD\\) and \\(BC\\) and intersects edge \\(CD\\) at point \\(M\\). The ratio of the length of segment \\(DM\\) to the length of segment \\(C... | The volume of the tetrahedron \(ABCD\) is 5. A plane passes through the midpoints of edges \(AD\) and \(BC\) and intersects edge \(CD\) at point \(M\). The ratio of the length of segment \(DM\) to the length of segment \(CM\) is \(2/3\). Calculate the area of the cross-section of the tetrahedron formed by this plane, g... |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Little Shino is interested in the fighting tournaments. Once she went to watch one of the tournament... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Little Shino is interested in the fighting tournaments. Once she went to watch one of the tournaments. There were N fighters and i^{th} figh... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Prove that the area of a triangle, whose sides are equal to the medians of a triangle with area $S$, is equal to $3S / 4$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove that the area of a triangle f... | Prove that the area of a triangle, whose sides are equal to the medians of a triangle with area $S$, is equal to $3S / 4$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( M_n \\) be the least common multiple (LCM) of the numbers \\( 1, 2, \\ldots, n \\). For which positive integers \\( n \\) does \\( M_{n-1} = M_n \\) hold? Prove your conclusion."
},
{
"from": "assistant",
"value": "... | Let \( M_n \) be the least common multiple (LCM) of the numbers \( 1, 2, \ldots, n \). For which positive integers \( n \) does \( M_{n-1} = M_n \) hold? Prove your conclusion. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In an isosceles triangle with a side length of $b$, the angle bisectors of the base angles are drawn. The segment of the line between the points where these bisectors intersect the sides is equal to $m$. Determine the base of the trian... | In an isosceles triangle with a side length of $b$, the angle bisectors of the base angles are drawn. The segment of the line between the points where these bisectors intersect the sides is equal to $m$. Determine the base of the triangle. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Prove that the product of three consecutive natural numbers is divisible by 504, given that the middle number is a perfect cube."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to prove that the produc... | Prove that the product of three consecutive natural numbers is divisible by 504, given that the middle number is a perfect cube. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nA large conference has 500 delegates attending. If each delegate knows 400 other delegates, is it always possible to select 6 delegates such that every two of them know each other?"
},
{
"from": "assistant",
"value": "<th... |
A large conference has 500 delegates attending. If each delegate knows 400 other delegates, is it always possible to select 6 delegates such that every two of them know each other? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given $\\frac{x}{y}=\\frac{10}{4}.$ If $y = 18$, what is the value of $x$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of x when y is 18, given that the ratio of x to y is 10 to 4. Let me... | Given $\frac{x}{y}=\frac{10}{4}.$ If $y = 18$, what is the value of $x$? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given convex pentagon \\(ABCDE\\), where \\(DE \\parallel AC\\), \\(\\angle ADC = 90^\\circ\\), \\(AC = 12\\), \\(CD = 6\\), \\(AC\\) bisects \\(\\angle DAB\\), and \\(\\angle BCE = 60^\\circ\\), find the minimum area of \\(\\triangle ... | Given convex pentagon \(ABCDE\), where \(DE \parallel AC\), \(\angle ADC = 90^\circ\), \(AC = 12\), \(CD = 6\), \(AC\) bisects \(\angle DAB\), and \(\angle BCE = 60^\circ\), find the minimum area of \(\triangle BCE\). |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Berland traffic is very different from traffic in other countries. The capital of Berland consists o... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Berland traffic is very different from traffic in other countries. The capital of Berland consists of n junctions and m roads. Each road con... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The focus of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a>0, b>0)\\) is \\(2c\\). The line \\(l\\) passes through points \\((a, 0)\\) and \\((0, b)\\), and the sum of the distances from point \\((1,0)\\) to line \\(l... | The focus of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a>0, b>0)\) is \(2c\). The line \(l\) passes through points \((a, 0)\) and \((0, b)\), and the sum of the distances from point \((1,0)\) to line \(l\) and from point \((-1,0)\) to line \(l\) is not less than \(\frac{4}{5}c\). Determine the range of... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. If \\( a + \\log_{3} 2017 \\), \\( a + \\log_{9} 2017 \\), \\( a + \\log_{27} 2017 \\) (\\( a \\in \\mathbb{R} \\)) form a geometric sequence, what is the common ratio?"
},
{
"from": "assistant",
"value": "<think>\nOkay, le... | If \( a + \log_{3} 2017 \), \( a + \log_{9} 2017 \), \( a + \log_{27} 2017 \) (\( a \in \mathbb{R} \)) form a geometric sequence, what is the common ratio? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The range of the function \\( f(x) = \\sqrt{x-3} + \\sqrt{12-3x} \\) is ( ).\n\n(A) \\( [1, \\sqrt{2}] \\)\n\n(B) \\( \\left[1, \\frac{3}{2}\\right] \\)\n\n(C) \\( [1, \\sqrt{3}] \\)\n\n(D) \\( [1, 2] \\)"
},
{
"from": "assista... | The range of the function \( f(x) = \sqrt{x-3} + \sqrt{12-3x} \) is ( ).
(A) \( [1, \sqrt{2}] \)
(B) \( \left[1, \frac{3}{2}\right] \)
(C) \( [1, \sqrt{3}] \)
(D) \( [1, 2] \) |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nConsider the following two sets of numbers:\n\n$$\nA=\\left\\{\\frac{3 n-4}{5 n-3}: n \\in \\mathbb{Z}\\right\\} \\quad \\text{and} \\quad B=\\left\\{\\frac{4 k-3}{7 k-6}: k \\in \\mathbb{Z}\\right\\} \\text{.}\n$$\n\nHow many elemen... |
Consider the following two sets of numbers:
$$
A=\left\{\frac{3 n-4}{5 n-3}: n \in \mathbb{Z}\right\} \quad \text{and} \quad B=\left\{\frac{4 k-3}{7 k-6}: k \in \mathbb{Z}\right\} \text{.}
$$
How many elements are in the set $A \cap B$? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Fold two opposite vertices of a rectangular sheet of paper inward so that the resulting shape is a regular hexagon. When is this task possible?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out whe... | Fold two opposite vertices of a rectangular sheet of paper inward so that the resulting shape is a regular hexagon. When is this task possible? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what v... | Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area? |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in mandarin chinese, russian and vietnamese as well. \n\nAda has an array o... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in mandarin chinese, russian and vietnamese as well.
Ada has an array of N crayons, some crayons are pointing up... |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Stepan has a set of n strings. Also, he has a favorite string s. \n\nStepan wants to do the followin... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Stepan has a set of n strings. Also, he has a favorite string s.
Stepan wants to do the following. He will take some strings of his set an... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. For the set $A = \\{1, 2, 3, \\cdots, n\\}$ and its every non-empty subset, define a unique \"alternating sum\" as follows: rearrange the subset in decreasing order, and then starting from the largest number, alternately subtract or ad... | For the set $A = \{1, 2, 3, \cdots, n\}$ and its every non-empty subset, define a unique "alternating sum" as follows: rearrange the subset in decreasing order, and then starting from the largest number, alternately subtract or add the succeeding number to obtain the result. For example, the "alternating sum" of the se... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Suppose \\(\\left(a_{1}, a_{2}, a_{3}, a_{4}\\right)\\) is a 4-term sequence of real numbers satisfying the following two conditions:\n- \\(a_{3} = a_{2} + a_{1}\\) and \\(a_{4} = a_{3} + a_{2}\\);\n- there exist real numbers \\(a, b, ... | Suppose \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) is a 4-term sequence of real numbers satisfying the following two conditions:
- \(a_{3} = a_{2} + a_{1}\) and \(a_{4} = a_{3} + a_{2}\);
- there exist real numbers \(a, b, c\) such that
\[a n^{2} + b n + c = \cos \left(a_{n}\right)\]
for all \(n \in \{1,2,3,4\}\).
Co... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a family of 4 people, if Masha's scholarship is doubled, the total income of the family will increase by 5%. If instead, the mother's salary is doubled, the total income will increase by 15%. If the father's salary is doubled, the t... | In a family of 4 people, if Masha's scholarship is doubled, the total income of the family will increase by 5%. If instead, the mother's salary is doubled, the total income will increase by 15%. If the father's salary is doubled, the total income will increase by 25%. By what percentage will the total income of the fam... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. To calculate the product of all natural numbers from 1 to 50 inclusive, it is recommended to use a computer. However, you can easily determine the number of zeros at the end of this product without the help of a computer. How many zero... | To calculate the product of all natural numbers from 1 to 50 inclusive, it is recommended to use a computer. However, you can easily determine the number of zeros at the end of this product without the help of a computer. How many zeros are there at the end? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Tea and a cake cost £4.50. Tea and an éclair cost £4. A cake and an éclair cost £6.50. What is the cost of tea, a cake, and an éclair?"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to solve this problem step... | Tea and a cake cost £4.50. Tea and an éclair cost £4. A cake and an éclair cost £6.50. What is the cost of tea, a cake, and an éclair? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( M \\) be a finite set of points in the plane. Prove that there exists a point in \\( M \\) that has at most 3 nearest points within \\( M \\)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove t... | Let \( M \) be a finite set of points in the plane. Prove that there exists a point in \( M \) that has at most 3 nearest points within \( M \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. All divisors of a certain natural number, except for one and the number itself, are listed. Some two numbers from this list differ by six times. By how much do the two largest numbers in this list differ?"
},
{
"from": "assista... | All divisors of a certain natural number, except for one and the number itself, are listed. Some two numbers from this list differ by six times. By how much do the two largest numbers in this list differ? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. When the integer \\( 300^8 \\) is written out, it has \\( d \\) digits. What is the value of \\( d \\) ?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find out how many digits the number 300^8 has when it... | When the integer \( 300^8 \) is written out, it has \( d \) digits. What is the value of \( d \) ? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Using the result of problem $\\underline{61403}$, prove the inequalities:\n\na) $\\sqrt[n]{a_{1} \\cdots a_{n}} \\leq \\frac{a_{1} + \\cdots + a_{n}}{n}$ (Cauchy's inequality).\n\nb) $\\left(\\frac{b_{1} + \\cdots + b_{n}}{n}\\right)^{... | Using the result of problem $\underline{61403}$, prove the inequalities:
a) $\sqrt[n]{a_{1} \cdots a_{n}} \leq \frac{a_{1} + \cdots + a_{n}}{n}$ (Cauchy's inequality).
b) $\left(\frac{b_{1} + \cdots + b_{n}}{n}\right)^{b_{1} + \cdots + b_{n}} \leq b_{1}^{b_{1}} \cdots b_{n}^{b_{n}}$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $BE$ and $CF$ be altitudes in triangle $ABC$ . If $AE = 24$ , $EC = 60$ , and $BF = 31$ , determine $AF$ ."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this geometry problem where BE a... | Let $BE$ and $CF$ be altitudes in triangle $ABC$ . If $AE = 24$ , $EC = 60$ , and $BF = 31$ , determine $AF$ . |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. For \\( p = 1, 2, \\cdots, 10 \\), let \\( S_{p} \\) be the sum of the first 40 terms of an arithmetic sequence with the first term \\( p \\) and the common difference \\( 2p - 1 \\). What is \\( S_{1} + S_{2} + \\cdots + S_{10} \\)?\n... | For \( p = 1, 2, \cdots, 10 \), let \( S_{p} \) be the sum of the first 40 terms of an arithmetic sequence with the first term \( p \) and the common difference \( 2p - 1 \). What is \( S_{1} + S_{2} + \cdots + S_{10} \)?
(A) 80000
(B) 80200
(C) 80400
(D) 80600
(E) 80800 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The capacities of three containers $A$, $B$, and $C$, each of which is shaped like a cube, are in the ratio $1: 8: 27$, and the volumes of water initially filled in them are in the ratio $1:2:3$. After transferring some water from cont... | The capacities of three containers $A$, $B$, and $C$, each of which is shaped like a cube, are in the ratio $1: 8: 27$, and the volumes of water initially filled in them are in the ratio $1:2:3$. After transferring some water from container $A$ to container $B$ and from container $B$ to container $C$, each container en... |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef has $K$ chocolates and he wants to distribute them to $N$ people (numbered $1$ through $N$). Th... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef has $K$ chocolates and he wants to distribute them to $N$ people (numbered $1$ through $N$). These people are standing in a line in suc... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. 10 different positive numbers are given. In what order should they be denoted as \\(a_{1}, a_{2}, \\ldots, a_{10}\\) so that the sum \\(a_{1} + 2a_{2} + 3a_{3} + \\ldots + 10a_{10}\\) is maximized?"
},
{
"from": "assistant",
... | 10 different positive numbers are given. In what order should they be denoted as \(a_{1}, a_{2}, \ldots, a_{10}\) so that the sum \(a_{1} + 2a_{2} + 3a_{3} + \ldots + 10a_{10}\) is maximized? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Express the number 80 as the sum of two prime numbers. In how many ways can this be done? Note that the number one is not a prime number."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out how many ... | Express the number 80 as the sum of two prime numbers. In how many ways can this be done? Note that the number one is not a prime number. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given the prime numbers \\( p \\) and \\( q \\) such that \\( q^5 - 2p^2 = 1 \\), find the value of \\( p + q \\)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: given prime numbers p and q... | Given the prime numbers \( p \) and \( q \) such that \( q^5 - 2p^2 = 1 \), find the value of \( p + q \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given that the integer part of the real number \\(\\frac{2+\\sqrt{2}}{2-\\sqrt{2}}\\) is \\(a\\), and the decimal part is \\(1 - b\\), find the value of \\(\\frac{(b-1) \\cdot (5-b)}{\\sqrt{a^{2}-3^{2}}}\\)."
},
{
"from": "assi... | Given that the integer part of the real number \(\frac{2+\sqrt{2}}{2-\sqrt{2}}\) is \(a\), and the decimal part is \(1 - b\), find the value of \(\frac{(b-1) \cdot (5-b)}{\sqrt{a^{2}-3^{2}}}\). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Suppose the prosecutor from the previous problem made the following two statements in court:\n\n1) Either $\\mathrm{X}$ is guilty, or $\\mathrm{Y}$ is guilty.\n2) $\\mathrm{X}$ is not guilty.\n\nWhat conclusion would you reach based on... | Suppose the prosecutor from the previous problem made the following two statements in court:
1) Either $\mathrm{X}$ is guilty, or $\mathrm{Y}$ is guilty.
2) $\mathrm{X}$ is not guilty.
What conclusion would you reach based on these statements? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. A rectangle has an area of $16$ and a perimeter of $18$ ; determine the length of the diagonal of the rectangle.\n\n*2015 CCA Math Bonanza Individual Round #8*"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I hav... | A rectangle has an area of $16$ and a perimeter of $18$ ; determine the length of the diagonal of the rectangle.
*2015 CCA Math Bonanza Individual Round #8* |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out the value of b here. So, we have two equations: 2a + 1 = 1 and b - a = 1. I n... | We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( a, b, c, d \\) be real numbers such that \\( a^{2}+b^{2}+c^{2}+d^{2}=1 \\). Determine the minimum value of \\( (a-b)(b-c)(c-d)(d-a) \\) and determine all values of \\( (a, b, c, d) \\) such that the minimum value is achieved."
... | Let \( a, b, c, d \) be real numbers such that \( a^{2}+b^{2}+c^{2}+d^{2}=1 \). Determine the minimum value of \( (a-b)(b-c)(c-d)(d-a) \) and determine all values of \( (a, b, c, d) \) such that the minimum value is achieved. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Two springs with stiffnesses of 3 kN/m and 6 kN/m are connected in series. How much work is required to stretch this system by 5 cm?"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is about two sp... | Two springs with stiffnesses of 3 kN/m and 6 kN/m are connected in series. How much work is required to stretch this system by 5 cm? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. $2n+1$ students are positioned such that the distances between any two of them are all different. At a given moment, each of them sprays the closest student with a water gun.\n\na) Show that there exists a pair of students who spray ea... | $2n+1$ students are positioned such that the distances between any two of them are all different. At a given moment, each of them sprays the closest student with a water gun.
a) Show that there exists a pair of students who spray each other.
b) Show that at least one of the students is not targeted. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. A polyhedron has faces that are all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?"
},
{... | A polyhedron has faces that are all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Solve the following equation for real numbers: $\\log _{a} x=x$, where $a=x^{\\log _{4} x}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to solve this equation step by step. The problem is to solve for re... | Solve the following equation for real numbers: $\log _{a} x=x$, where $a=x^{\log _{4} x}$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $\\Gamma$ be a circle with center $O$ and $P$ a point outside the circle. Let $A \\in \\Gamma$ such that the line segment $PA$ is tangent to $\\Gamma$, and let $B$ be the point diametrically opposite to $A$. Let $\\Delta$ be a line... | Let $\Gamma$ be a circle with center $O$ and $P$ a point outside the circle. Let $A \in \Gamma$ such that the line segment $PA$ is tangent to $\Gamma$, and let $B$ be the point diametrically opposite to $A$. Let $\Delta$ be a line passing through $P$ and intersecting $\Gamma$ at points $C$ and $D$. The line parallel to... |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Last summer Peter was at his granny's in the country, when a wolf attacked sheep in the nearby fores... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Last summer Peter was at his granny's in the country, when a wolf attacked sheep in the nearby forest. Now he fears to walk through the fore... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nLet \\( f(x) \\) be a function satisfying the following conditions:\n1. If \\( x > y \\) and \\( f(x) + x \\geq w \\geq f(y) + y \\), then there exists a real number \\( z \\in [y, x] \\) such that \\( f(z) = w - z \\);\n2. The equat... |
Let \( f(x) \) be a function satisfying the following conditions:
1. If \( x > y \) and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \) such that \( f(z) = w - z \);
2. The equation \( f(x) = 0 \) has at least one solution, and there is a solution of this equation that is less t... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given real numbers \\( a_{1}, a_{2}, \\cdots, a_{2016} \\) satisfying \\( 9 a_{i} > 11 a_{i+1}^{2} \\) for \\( i = 1, 2, \\cdots, 2015 \\). Find the maximum value of \\( \\left( a_{1} - a_{2}^{2} \\right) \\left( a_{2} - a_{3}^{2} \\ri... | Given real numbers \( a_{1}, a_{2}, \cdots, a_{2016} \) satisfying \( 9 a_{i} > 11 a_{i+1}^{2} \) for \( i = 1, 2, \cdots, 2015 \). Find the maximum value of \( \left( a_{1} - a_{2}^{2} \right) \left( a_{2} - a_{3}^{2} \right) \cdots \left( a_{2015} - a_{2016}^{2} \right) \left( a_{2016} - a_{1}^{2} \right) \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. We have three piles of 51, 49, and 5 pebbles. At each step, we can either merge two piles or divide a pile into two equal parts. Is it possible that after a certain number of steps, we end up with 105 piles of one pebble each?"
},
... | We have three piles of 51, 49, and 5 pebbles. At each step, we can either merge two piles or divide a pile into two equal parts. Is it possible that after a certain number of steps, we end up with 105 piles of one pebble each? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a $10 \\times 10$ grid, place 1 white piece in any 91 squares. Each time, take a white piece from the grid and place a black piece in an empty square. Then again, take a white piece and place a black piece. Continue this process unt... | In a $10 \times 10$ grid, place 1 white piece in any 91 squares. Each time, take a white piece from the grid and place a black piece in an empty square. Then again, take a white piece and place a black piece. Continue this process until there are no more white pieces left. Prove that there must be a moment when there a... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. How many positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}\\ ?$\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 7\\qquad \\textbf{(C) } 12\\qquad \\textbf{(D) } 65\\qquad \\textbf{(E) } 125$"
},
{
... | How many positive integers $n$ satisfy the following condition:
$(130n)^{50} > n^{100} > 2^{200}\ ?$
$\textbf{(A) } 0\qquad \textbf{(B) } 7\qquad \textbf{(C) } 12\qquad \textbf{(D) } 65\qquad \textbf{(E) } 125$ |
[
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef's phone shows a Battery Low notification if the battery level is 15 \\% or less.\n\nGiven that ... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef's phone shows a Battery Low notification if the battery level is 15 \% or less.
Given that the battery level of Chef's phone is X \%, ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In the Cartesian coordinate system $xOy$, given points $M(-1,2)$ and $N(1,4)$, point $P$ moves along the x-axis. When the angle $\\angle MPN$ reaches its maximum value, the x-coordinate of point $P$ is ______."
},
{
"from": "as... | In the Cartesian coordinate system $xOy$, given points $M(-1,2)$ and $N(1,4)$, point $P$ moves along the x-axis. When the angle $\angle MPN$ reaches its maximum value, the x-coordinate of point $P$ is ______. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given the ellipse \\(\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\) with \\(a > b > 0\\), let points \\(A\\) and \\(B\\) be the left and right vertices of the ellipse, respectively. Point \\(P\\) is another point on the ellipse dif... | Given the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with \(a > b > 0\), let points \(A\) and \(B\) be the left and right vertices of the ellipse, respectively. Point \(P\) is another point on the ellipse different from \(A\) and \(B\) such that \(|AP| = |OA|\). Prove that the slope \(k\) of the line \(O... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$?\n$\\textbf{(A) } 4900 \\qquad \\textbf{(B) } 4950\\qquad \\textbf{(C) } 5000\\qquad \\textbf{(D) } 5050 \\qquad \\textbf{(E) } ... | Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\] How many integers $n$ are there such that $P(n)\leq 0$?
$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$ |
[
{
"from": "user",
"value": "Introduction\nYou are given a random integer generator with the following implementation\n\nThe first invocation always returns 1.\nThe second invocation returns a random integer between 1 and 2.\nThe third invocation returns a random integer between 1 and 3.\nThe nth invocation ... | Introduction
You are given a random integer generator with the following implementation
The first invocation always returns 1.
The second invocation returns a random integer between 1 and 2.
The third invocation returns a random integer between 1 and 3.
The nth invocation returns a random integer between 1 and n, incl... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. A plane passing through the vertex of a cone intersects the base along a chord, the length of which is equal to the radius of the base. Determine the ratio of the volumes of the resulting parts of the cone."
},
{
"from": "assis... | A plane passing through the vertex of a cone intersects the base along a chord, the length of which is equal to the radius of the base. Determine the ratio of the volumes of the resulting parts of the cone. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( K(n) \\) be the number of triples in a given system. Prove that:\na) \\( K(n) > \\frac{n}{6} - 1 \\);\n\nb) \\( K(n) < \\frac{2n}{9} \\)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove these ... | Let \( K(n) \) be the number of triples in a given system. Prove that:
a) \( K(n) > \frac{n}{6} - 1 \);
b) \( K(n) < \frac{2n}{9} \). |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Is it possible to wrap a unit cube in a cloth of size $3 \\times 3$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out if a 3x3 cloth can wrap a unit cube. Let me start by understanding the problem... | Is it possible to wrap a unit cube in a cloth of size $3 \times 3$? |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Consider the sequence \\( a_1, a_2, a_3, \\cdots a_n, \\cdots \\) which satisfies \\( a_1 = a_2 = 1, a_3 = 2 \\), and for any natural number \\( n \\), we have \\( a_n \\cdot a_{n+1} \\cdot a_{n+2} \\neq 1 \\). Additionally, \\( a_n \\... | Consider the sequence \( a_1, a_2, a_3, \cdots a_n, \cdots \) which satisfies \( a_1 = a_2 = 1, a_3 = 2 \), and for any natural number \( n \), we have \( a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1 \). Additionally, \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). Determine the value... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the envelope of a family of straight lines that form a triangle with a right angle and have an area of $a^{2} / 2$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the envelope of a family of stra... | Find the envelope of a family of straight lines that form a triangle with a right angle and have an area of $a^{2} / 2$. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Calculate the volumes of solids generated by rotating the region bounded by the graphs of the functions about the x-axis.\n\n$$\ny = \\sin^2(x), x = \\frac{\\pi}{2}, y = 0\n$$\n"
},
{
"from": "assistant",
"value": "<think>\... | Calculate the volumes of solids generated by rotating the region bounded by the graphs of the functions about the x-axis.
$$
y = \sin^2(x), x = \frac{\pi}{2}, y = 0
$$
|
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. The sum of the numbers on the opposite faces of a die always equals 7."
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out why the sum of the numbers on opposite faces of a die always equals 7. Hmm,... | The sum of the numbers on the opposite faces of a die always equals 7. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $\\xi$ and $\\eta$ be two random variables with $\\mathrm{E}[\\xi] = \\mathrm{E}[\\eta] = 0$ and $\\mathrm{D}[\\xi] = \\mathrm{D}[\\eta] = 1$, and the correlation coefficient $\\rho = \\rho(\\xi, \\eta)$. Show that\n\n$$\n\\mathrm{... | Let $\xi$ and $\eta$ be two random variables with $\mathrm{E}[\xi] = \mathrm{E}[\eta] = 0$ and $\mathrm{D}[\xi] = \mathrm{D}[\eta] = 1$, and the correlation coefficient $\rho = \rho(\xi, \eta)$. Show that
$$
\mathrm{E} \max \left(\xi^{2}, \eta^{2}\right) \leq 1 + \sqrt{1 - \rho^{2}}.
$$ |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the smallest positive integer $N$ such that $N / 3$, $N / 4$, $N / 5$, $N / 6$, and $N / 7$ are all integers?\n(a) 420\n(b) 350\n(c) 210\n(d) 300\n(e) 280"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tr... | What is the smallest positive integer $N$ such that $N / 3$, $N / 4$, $N / 5$, $N / 6$, and $N / 7$ are all integers?
(a) 420
(b) 350
(c) 210
(d) 300
(e) 280 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In an isosceles trapezoid that is inscribed around a circle, the ratio of the parallel sides is $k$. Find the angle at the base."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem about an isosceles ... | In an isosceles trapezoid that is inscribed around a circle, the ratio of the parallel sides is $k$. Find the angle at the base. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. In trapezoid \\(ABCD\\), the base \\(AD\\) is equal to 16, the sum of the diagonals \\(AC\\) and \\(BD\\) is 36, and the angle \\(\\angle CAD\\) is \\(60^\\circ\\). The ratio of the areas of triangles \\(AOD\\) and \\(BOC\\), where \\(... | In trapezoid \(ABCD\), the base \(AD\) is equal to 16, the sum of the diagonals \(AC\) and \(BD\) is 36, and the angle \(\angle CAD\) is \(60^\circ\). The ratio of the areas of triangles \(AOD\) and \(BOC\), where \(O\) is the point of intersection of the diagonals, is 4. Find the area of the trapezoid. |
[
{
"from": "user",
"value": "The Catalan numbers (OEIS) are a sequence of natural numbers often appearing in combinatorics.\nThe nth Catalan number is the number of Dyck words (balanced strings of parenthesis or brackets such as [[][]]; formally defined as a string using two characters a and b such that any ... | The Catalan numbers (OEIS) are a sequence of natural numbers often appearing in combinatorics.
The nth Catalan number is the number of Dyck words (balanced strings of parenthesis or brackets such as [[][]]; formally defined as a string using two characters a and b such that any substring starting from the beginning has... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nThere are 40 identical gas cylinders with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the co... |
There are 40 identical gas cylinders with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the connection, the gas pressure in the connected cylinders becomes equal to the arithmetic ... |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Solve the equations \\(2x^3 - 5x^2 + 6x - 2 = 0\\) and \\(6x^3 - 3x^2 - 2x + 1 = 0\\), given that they have a common root."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve these two cubic equations: \\... | Solve the equations \(2x^3 - 5x^2 + 6x - 2 = 0\) and \(6x^3 - 3x^2 - 2x + 1 = 0\), given that they have a common root. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.