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values | source stringclasses 9
values | test_cases stringlengths 33 44.2M ⌀ | starter_code stringlengths 0 956 ⌀ | response stringlengths 1.25k 120k ⌀ |
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[
{
"from": "user",
"value": "Real numbers $x, y$ satisfy $x^{2}+x y+y^{2}=1, F(x, y)=x^{3} y+x y^{3}$.\n(1) Prove: For any $x, y \\in \\mathbf{R}$ that meet the requirements, $F(x, y) \\geqslant-2$.\n(2) Find the maximum possible value of $F(x, y)$."
},
{
"from": "assistant",
"value": "<think>\nO... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "The crowd wasn't that big. It can better be described as what?\nA: field\nB: alone\nC: small group\nD: solitude\nE: solitary"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tackle this question step by step. The sentence given is \"The crowd wasn't that big. It ... | The crowd wasn't that big. It can better be described as what?
A: field
B: alone
C: small group
D: solitude
E: solitary | Okay, let's tackle this question step by step. The sentence given is "The crowd wasn't that big. It can better be described as what?" with options A to E. I need to figure out the correct answer by understanding the context and the meanings of each option.
First, the key point here is the contrast between the original... | The sentence contrasts the initial description of a "crowd" (implying a large group) by stating it "wasn't that big," requiring a term for a smaller gathering.
- **A: field** → Incorrect. A "field" is a physical space, not a group of people.
- **B: alone** → Incorrect. "Alone" describes a single individual, not a g... | C | puzzle | riddle_sense | null | null | <think>
Okay, let's tackle this question step by step. The sentence given is "The crowd wasn't that big. It can better be described as what?" with options A to E. I need to figure out the correct answer by understanding the context and the meanings of each option.
First, the key point here is the contrast between the ... |
[
{
"from": "human",
"value": "What are the remainders when the polynomial \\( f(x) \\) from problem number \\(\\underline{61052}\\) is divided by polynomials of the form \\( x - x_{i} \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the remainders when the polynomial f(x... | What are the remainders when the polynomial \( f(x) \) from problem number \(\underline{61052}\) is divided by polynomials of the form \( x - x_{i} \)? | Okay, so I need to figure out the remainders when the polynomial f(x) from problem number 61052 is divided by polynomials of the form x - x_i. Hmm, let's start by recalling some basic polynomial division concepts. When you divide a polynomial f(x) by a linear divisor of the form x - a, the remainder is simply f(a). Thi... | The remainders when the polynomial \( f(x) \) is divided by polynomials of the form \( x - x_i \) are given by the **Remainder Theorem**, which states that the remainder of dividing \( f(x) \) by \( x - a \) is \( f(a) \).
**Solution:**
For each \( x_i \), the remainder when \( f(x) \) is divided by \( x - x_i \) i... | 1. **Apply the Remainder Theorem** (also known as Bézout's Theorem):
According to the Remainder Theorem, for a given polynomial \( f(x) \) and a value \( x_i \), the remainder of the division of \( f(x) \) by \( x - x_i \) is \( f(x_i) \). Mathematically, if you divide \( f(x) \) by \( x - x_i \), you can write:
... | math | numina_math | null | null | <think>
Okay, so I need to figure out the remainders when the polynomial f(x) from problem number 61052 is divided by polynomials of the form x - x_i. Hmm, let's start by recalling some basic polynomial division concepts. When you divide a polynomial f(x) by a linear divisor of the form x - a, the remainder is simply f... |
[
{
"from": "human",
"value": "Person A and Person B take turns placing pieces (bishops) on a chessboard. Person A goes first and places one white bishop each turn, while Person B places one black bishop each turn. Each piece must be placed on an empty square, and it can be placed on either a white or black s... | Person A and Person B take turns placing pieces (bishops) on a chessboard. Person A goes first and places one white bishop each turn, while Person B places one black bishop each turn. Each piece must be placed on an empty square, and it can be placed on either a white or black square. The placement of a bishop must ens... | Okay, so there's this game where two people take turns placing bishops on a chessboard. Person A is white and goes first, then Person B is black. Each turn, they place one bishop of their color on an empty square. The key rule is that each new bishop must be placed such that it can't be captured by any existing bishop.... | Under optimal strategies, Person B (the second player) will win. Here's the reasoning:
1. **Game Structure**: The chessboard's symmetry and the bishops' movement rules allow the game to decompose into two independent subgames: one on white squares and one on black squares. Each player alternately places bishops on eit... |
To solve this problem, we need to determine who will win the game under optimal play strategies. We will use strategy and symmetry in our solution.
1. **Describe the Setup**:
- The game is played on a grid of squares on an international chessboard.
- Player 甲 (A) plays first and places a white elephant on an ... | math | numina_math | null | null | <think>
Okay, so there's this game where two people take turns placing bishops on a chessboard. Person A is white and goes first, then Person B is black. Each turn, they place one bishop of their color on an empty square. The key rule is that each new bishop must be placed such that it can't be captured by any existing... |
[
{
"from": "human",
"value": "Prove that a composite number \\( n \\) always has a divisor greater than 1 but not greater than \\( \\sqrt{n} \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that a composite number n always has a divisor greater than 1 but not greater than the... | Prove that a composite number \( n \) always has a divisor greater than 1 but not greater than \( \sqrt{n} \). | Okay, so I need to prove that a composite number n always has a divisor greater than 1 but not greater than the square root of n. Hmm, let me start by recalling some basic number theory concepts.
First, a composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, if n i... | To prove that a composite number \( n \) always has a divisor greater than 1 but not greater than \( \sqrt{n} \), we proceed as follows:
**Proof by Contradiction:**
1. **Assumption:** Suppose \( n \) is composite, so \( n = a \cdot b \) where \( 1 < a, b < n \). Assume, for contradiction, that all divisors of \( n \) ... |
We need to prove that every composite number \( n \) always has a divisor greater than 1 but not greater than \( \sqrt{n} \).
1. **Define Properties of Composite Numbers:**
A composite number \( n \) is defined as a natural number greater than 1 that is not prime. This means that \( n \) has positive divisors othe... | math | numina_math | null | null | <think>
Okay, so I need to prove that a composite number n always has a divisor greater than 1 but not greater than the square root of n. Hmm, let me start by recalling some basic number theory concepts.
First, a composite number is a positive integer that has at least one positive divisor other than 1 and itself. So... |
[
{
"from": "human",
"value": "There are 28 identical coins arranged in the shape of a triangle. It is known that the total mass of any trio of coins that touch each other in pairs is 10 g. Find the total mass of all 18 coins on the border of the triangle."
},
{
"from": "gpt",
"value": "<think>\nO... | There are 28 identical coins arranged in the shape of a triangle. It is known that the total mass of any trio of coins that touch each other in pairs is 10 g. Find the total mass of all 18 coins on the border of the triangle. | Okay, let me try to figure out this problem. So, there are 28 identical coins arranged in a triangle. The triangle shape probably means they are arranged in rows, where each row has one more coin than the row above. Like, the first row has 1 coin, the second has 2, and so on until the seventh row, since 1+2+3+4+5+6+7 =... | The total mass of all 18 coins on the border of the triangle is **60 grams**.
**Step-by-Step Explanation:**
1. **Understanding the Arrangement:**
The 28 coins form a triangular grid with 7 rows. Each row \( i \) contains \( i \) coins. The border coins include the three edges: left, right, and bottom, totaling 1... | 1. **Understanding the Problem:**
- We have 28 coins arranged in a triangular shape.
- The total mass of any three coins that touch each other pairwise is 10 grams.
- We need to find the total mass of the 18 coins on the boundary of the triangle.
2. **Visualizing the Problem:**
- Coloring the coins can hel... | math | numina_math | null | null | <think>
Okay, let me try to figure out this problem. So, there are 28 identical coins arranged in a triangle. The triangle shape probably means they are arranged in rows, where each row has one more coin than the row above. Like, the first row has 1 coin, the second has 2, and so on until the seventh row, since 1+2+3+4... |
[
{
"from": "human",
"value": "Let $\\alpha > 0$ be a real number. Compute the limit of the sequence $\\{x_n\\}_{n\\geq 1}$ defined by $$ x_n=\\begin{cases} \\sum \\limits_{k=1}^n \\sinh \\left(\\frac{k}{n^2}\\right),& \\text{when}\\ n>\\frac{1}{\\alpha} 0,& \\text{when}\\ n\\leq \\frac{1}{\\alpha}\\end... | Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$ x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha} 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases} $$ | Okay, let's try to figure out this limit problem. So we have a sequence {x_n} where for n > 1/α, x_n is the sum from k=1 to n of sinh(k/n²), and for n ≤ 1/α, it's just 0. We need to find the limit as n approaches infinity of this sequence. Since α is a positive real number, 1/α is some finite number, so as n grows, eve... | To compute the limit of the sequence \(\{x_n\}_{n \geq 1}\) defined by
\[ x_n = \begin{cases}
\sum_{k=1}^n \sinh\left(\frac{k}{n^2}\right), & \text{when } n > \frac{1}{\alpha} \\
0, & \text{when } n \leq \frac{1}{\alpha}
\end{cases} \]
we focus on the case where \(n > \frac{1}{\alpha}\) since the limit as \(n\) appr... | 1. Given the sequence \( \{x_n\}_{n \geq 1} \) defined by:
\[
x_n = \begin{cases}
\sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right), & \text{when}\ n > \frac{1}{\alpha} \\
0, & \text{when}\ n \leq \frac{1}{\alpha}
\end{cases}
\]
we need to compute the limit of \( x_n \) as \( n \to \infty \).
... | math | numina_math | null | null | <think>
Okay, let's try to figure out this limit problem. So we have a sequence {x_n} where for n > 1/α, x_n is the sum from k=1 to n of sinh(k/n²), and for n ≤ 1/α, it's just 0. We need to find the limit as n approaches infinity of this sequence. Since α is a positive real number, 1/α is some finite number, so as n gr... |
[
{
"from": "user",
"value": "In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Show that the points \\( B, C, I, I_{A} \\) are concyclic, where \\( I \\) is the incenter of triangle \\( ABC \\) and \\( I_{A} \\) is the excenter opposite to vertex \\( A \\).\n\nShow that \\( S \\) is the center of the circle passing through \\( B, C, I, I_{A} \\). This circ... | Show that the points \( B, C, I, I_{A} \) are concyclic, where \( I \) is the incenter of triangle \( ABC \) and \( I_{A} \) is the excenter opposite to vertex \( A \).
Show that \( S \) is the center of the circle passing through \( B, C, I, I_{A} \). This circle is called the Antarctic Circle relative to \( A \). | Okay, so I need to show that points B, C, I, and I_A are concyclic, meaning they all lie on the same circle. Then, I also have to prove that point S is the center of this circle, called the Antarctic Circle relative to A. Let me start by recalling some properties of incenters and excenters in a triangle.
First, in tri... | To show that the points \( B, C, I, I_{A} \) are concyclic and that \( S \) is the center of the circle passing through these points, we proceed as follows:
1. **Concyclicity of \( B, C, I, I_{A} \)**:
- The incenter \( I \) and the excenter \( I_{A} \) lie on the internal angle bisector of \(\angle BAC\).
- In ... |
1. **Definition and Construction:**
- Let $I$ be the incenter of triangle $ABC$, which is the center of the inscribed circle.
- Let $I_A$ be the excenter relative to vertex $A$, which is the center of the excircle opposite to $A$.
2. **Angle Consideration:**
- The excenter $I_A$ is constructed by the externa... | math | numina_math | null | null | <think>
Okay, so I need to show that points B, C, I, and I_A are concyclic, meaning they all lie on the same circle. Then, I also have to prove that point S is the center of this circle, called the Antarctic Circle relative to A. Let me start by recalling some properties of incenters and excenters in a triangle.
First... |
[
{
"from": "user",
"value": "Let $x$ be a real number between $0$ and $\\tfrac{\\pi}2$ such that \\[\\dfrac{\\sin^4(x)}{42}+\\dfrac{\\cos^4(x)}{75} = \\dfrac{1}{117}.\\] Find $\\tan(x)$ ."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of tan(x) where x is b... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "6. Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with all terms being positive. If $2 a_{4}+a_{3}-2 a_{2}-a_{1}=8$, then the minimum value of $2 a_{8}+a_{7}$ is $-2 x-2$ $\\qquad$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to tackle this pro... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "At the start of a 5-hour trip, the odometer in Jill's car indicates that her car had already been driven \\( 13831 \\, \\text{km} \\). The integer 13831 is a palindrome, because it is the same when read forwards or backwards. At the end of the 5-hour trip, the odometer reading w... | At the start of a 5-hour trip, the odometer in Jill's car indicates that her car had already been driven \( 13831 \, \text{km} \). The integer 13831 is a palindrome, because it is the same when read forwards or backwards. At the end of the 5-hour trip, the odometer reading was another palindrome. If Jill never drove fa... | Okay, so let's see. The problem is about Jill's car odometer showing a palindrome number at the start of a 5-hour trip, which is 13831 km. Then, after the trip, the odometer is another palindrome. We need to find her greatest possible average speed, given that she never drove faster than 80 km/h. The options are given,... | Jill's odometer started at 13831 km, a palindrome. After a 5-hour trip, the odometer showed another palindrome. We need to find the greatest possible average speed, given she never drove faster than 80 km/h. The maximum distance she could drive is 5 * 80 = 400 km, so the odometer could read up to 13831 + 400 = 14231 km... |
1. **Determine the maximum distance Jill could have driven:**
Since it is given that Jill never drove faster than $80 \, \text{km/h}$ and the trip lasted for 5 hours, the total maximum distance that she could have driven can be calculated as:
\[
80 \, \text{km/h} \times 5 \, \text{hours} = 400 \, \text{km}
... | math | numina_math | null | null | <think>
Okay, so let's see. The problem is about Jill's car odometer showing a palindrome number at the start of a 5-hour trip, which is 13831 km. Then, after the trip, the odometer is another palindrome. We need to find her greatest possible average speed, given that she never drove faster than 80 km/h. The options ar... |
[
{
"from": "user",
"value": "7.5. The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms, and if instead Anya gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How m... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Aizuwakamatsu City is known as the \"City of History\". About 400 years ago, the skeleton of the castle town was created by Gamo Ujisato, but after that, it became the central city of the Aizu clan 230,000 stones, whose ancestor was Hoshina Masayuki, the half-brother of Tokugawa... | Aizuwakamatsu City is known as the "City of History". About 400 years ago, the skeleton of the castle town was created by Gamo Ujisato, but after that, it became the central city of the Aizu clan 230,000 stones, whose ancestor was Hoshina Masayuki, the half-brother of Tokugawa's third shogun Iemitsu. Developed. Many to... | Okay, let's tackle this problem. The goal is to find the minimum number of lanterns required along the streets connecting historic sites such that all sites are connected via streets with lanterns. The key points here are that the streets form a graph, and we need to select a subset of edges (street segments) that form... | To solve this problem, we need to determine the minimum number of lanterns required along the streets connecting historic sites such that all sites are connected via streets with lanterns. The key is to minimize the number of lanterns while ensuring all sites are reachable.
### Approach
1. **Problem Analysis**:
- T... | null | code | code_contests | {"inputs": ["4\n4\n0,1,1500\n0,2,2000\n1,2,600\n1,3,500\n0", "4\n4\n0,1,1500\n0,2,2000\n1,2,600\n2,3,500\n0", "4\n4\n0,1,1500\n0,2,2000\n1,2,600\n2,3,400\n0", "4\n4\n0,1,0500\n0,2,2000\n0,2,600\n2,3,400\n0", "4\n4\n0,1,0400\n0002,2,0\n0,2,600\n2,3,400\n0", "4\n4\n0,1,1500\n0,2,2000\n1,2,600\n2,3,600\n0", "4\n4\n0,1,060... | <think>
Okay, let's tackle this problem. The goal is to find the minimum number of lanterns required along the streets connecting historic sites such that all sites are connected via streets with lanterns. The key points here are that the streets form a graph, and we need to select a subset of edges (street segments) t... | |
[
{
"from": "user",
"value": "If $x=\\sqrt2+\\sqrt3+\\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of |a + b + c + d| where the polynomial x⁴ + a x³... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Two spheres touch the plane of triangle \\( A B C \\) at points \\( A \\) and \\( B \\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \\( \\sqrt{481} \\). The center of a third sphere with ... | Two spheres touch the plane of triangle \( A B C \) at points \( A \) and \( B \) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \( \sqrt{481} \). The center of a third sphere with radius 9 is located at point \( C \), and this sph... | Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the problem and understand all the given information. So, we have triangle ABC lying on a plane. There are two spheres that touch this plane at points A and B, and they're on opposite sides of the plane. The sum of their radii is 1... | Given triangle \(ABC\) with two spheres touching the plane at points \(A\) and \(B\) on opposite sides, and a third sphere centered at \(C\) with radius 9, we need to find the radius of the circumcircle of triangle \(ABC\).
1. **Spheres touching the plane**:
- The sum of the radii \(r_1\) and \(r_2\) of the spheres... |
1. **Understanding the Problem:**
Given are three spheres tangent to the plane of triangle \(ABC\) at points \(A\), \(B\), and \(C\) respectively. The distance between the centers of the first two spheres is given, and their respective radii add up to a known value. The center of the third sphere is at point \(C\) ... | math | numina_math | null | null | <think>
Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the problem and understand all the given information. So, we have triangle ABC lying on a plane. There are two spheres that touch this plane at points A and B, and they're on opposite sides of the plane. The sum of their ra... |
[
{
"from": "human",
"value": "Let $h_1, h_2, h_3$ be the lengths of the three altitudes of a triangle, and let $r$ be the radius of the inscribed circle of this triangle. Show that\n\n$$\n\\frac{1}{h_1} + \\frac{1}{h_2} + \\frac{1}{h_3} = \\frac{1}{r}\n$$"
},
{
"from": "gpt",
"value": "<think>\nA... | Let $h_1, h_2, h_3$ be the lengths of the three altitudes of a triangle, and let $r$ be the radius of the inscribed circle of this triangle. Show that
$$
\frac{1}{h_1} + \frac{1}{h_2} + \frac{1}{h_3} = \frac{1}{r}
$$ | Alright, so I need to show that the sum of the reciprocals of the three altitudes of a triangle equals the reciprocal of the inradius. Let me start by recalling some basic formulas related to triangles, altitudes, and the inradius.
First, in a triangle, the altitude corresponding to a side is given by the formula \( ... | To demonstrate that \(\frac{1}{h_1} + \frac{1}{h_2} + \frac{1}{h_3} = \frac{1}{r}\) for a triangle with altitudes \(h_1, h_2, h_3\) and inradius \(r\), follow these steps:
1. **Express Altitudes Using Area and Sides**:
For sides \(a, b, c\) and area \(\Delta\), the altitudes are:
\[
h_a = \frac{2\Delta}{a}, \... |
We are given the three lengths of the altitudes of a triangle denoted by $h_1, h_2,$ and $h_3$, and the radius of the inscribed circle, $r$. The goal is to show that:
$$\frac{1}{h_1} + \frac{1}{h_2} + \frac{1}{h_3} = \frac{1}{r}$$
1. **Define the sides and the incenter**:
Label the sides of the triangle as $a$, $... | math | numina_math | null | null | <think>
Alright, so I need to show that the sum of the reciprocals of the three altitudes of a triangle equals the reciprocal of the inradius. Let me start by recalling some basic formulas related to triangles, altitudes, and the inradius.
First, in a triangle, the altitude corresponding to a side is given by the for... |
[
{
"from": "human",
"value": "Find \\( x \\) and \\( y \\), if\n\n\\[ \n\\sqrt[5]{119287 - 48682 \\sqrt{6}} = x + y \\sqrt{6} \n\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find real numbers x and y such that the fifth root of this expression, 119287 minus 48682 times the square... | Find \( x \) and \( y \), if
\[
\sqrt[5]{119287 - 48682 \sqrt{6}} = x + y \sqrt{6}
\] | Okay, so I need to find real numbers x and y such that the fifth root of this expression, 119287 minus 48682 times the square root of 6, equals x plus y times the square root of 6. Hmm, fifth roots can be tricky, especially when dealing with expressions that include radicals. Let me think about how to approach this.
F... | To find \( x \) and \( y \) such that
\[
\sqrt[5]{119287 - 48682 \sqrt{6}} = x + y \sqrt{6},
\]
we start by considering the fifth power of \( x + y \sqrt{6} \). Expanding \( (x + y \sqrt{6})^5 \) using the binomial theorem, we separate the rational parts and the coefficients of \( \sqrt{6} \):
\[
(x + y \sqrt{6})^5... |
1. Start with the given equation:
\[
\sqrt[5]{119287 - 48682 \sqrt{6}} = x + y \sqrt{6}.
\]
2. Begin by approximating some values using a calculator:
\[
\sqrt[5]{119287 - 48682 \sqrt{6}} \approx 2.101121,
\]
\[
\sqrt{6} \approx 2.44949.
\]
3. Formulate the equation from the approximations:... | math | numina_math | null | null | <think>
Okay, so I need to find real numbers x and y such that the fifth root of this expression, 119287 minus 48682 times the square root of 6, equals x plus y times the square root of 6. Hmm, fifth roots can be tricky, especially when dealing with expressions that include radicals. Let me think about how to approach ... |
[
{
"from": "user",
"value": "Task 1. Determine the value of the numerical expression\n\n$$\n8004^{2}-8003^{2}+8002^{2}-8001^{2}+\\ldots+4^{2}-3^{2}+2^{2}-1^{2}\n$$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the value of this big numerical expression: 8004 squared ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "## Task 6\n\nIn a pioneer competition in the circle, 26 students from a school participated. Exactly half of them managed to win a medal.\n\nHow many students from this school received a medal?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so let me try to figur... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Given an empty array A of infinite length and a positive integer K you have to process \nQ queries of type: \n- ! l r x: Fill the value x in all the empty places in the array from index l to r (both inclusive).\n- ? l r: Print the number of prime factors of K which is also a... | Given an empty array A of infinite length and a positive integer K you have to process
Q queries of type:
- ! l r x: Fill the value x in all the empty places in the array from index l to r (both inclusive).
- ? l r: Print the number of prime factors of K which is also a prime factor of at least one number in A[l]... | Okay, I need to solve this programming problem. Let's read the problem statement carefully.
So, the problem is about handling two types of queries on an array that's initially empty (but has infinite length, though in practice we can limit our processing to the ranges given). The queries are either to fill a range wit... | To solve this problem, we need to efficiently handle two types of queries on an initially empty array of infinite length. The queries involve filling empty positions in a specified range with a given value and determining the number of prime factors of a given integer K that are present in the prime factors of any numb... | null | code | apps | {"inputs": [["20 5", "? 1 5", "! 3 5 4", "? 1 5", "! 1 4 15", "? 1 5"]], "outputs": [["0", "1", "2"]]} | <think>
Okay, I need to solve this programming problem. Let's read the problem statement carefully.
So, the problem is about handling two types of queries on an array that's initially empty (but has infinite length, though in practice we can limit our processing to the ranges given). The queries are either to fill a r... | |
[
{
"from": "user",
"value": "Example 19 (Problem 22 of the 1988 All-Union Mathematical Olympiad) Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$. Try to find the minimum value of the following expression:\n$$\nS=\\frac{x y}{z}+\\frac{y z}{x}+\\frac{z x}{y} .\n$$"
},
{
"from": "assistant"... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Consider the set $E = \\{5, 6, 7, 8, 9\\}$ . For any partition ${A, B}$ of $E$ , with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$ . Let $N$ be the largest prime number amonh these... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Find the largest positive integer \\( n \\) such that \\( n! \\) ends with exactly 100 zeros."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest positive integer n where n factorial ends with exactly 100 zeros. Hmm, let me think. I remember t... | Find the largest positive integer \( n \) such that \( n! \) ends with exactly 100 zeros. | Okay, so I need to find the largest positive integer n where n factorial ends with exactly 100 zeros. Hmm, let me think. I remember that the number of trailing zeros in a factorial is determined by the number of times the number 10 is a factor in that factorial. Since 10 is 2 multiplied by 5, the number of trailing zer... | To find the largest positive integer \( n \) such that \( n! \) ends with exactly 100 zeros, we need to determine the number of trailing zeros in \( n! \). The number of trailing zeros is given by the sum of the integer divisions of \( n \) by 5, 25, 125, etc., until the quotient is zero. This is because each multiple ... |
To determine the largest positive integer $n$ such that $n!$ ends with exactly 100 zeros, we need to consider the number of trailing zeros in the factorial of a number.
1. **Understanding Trailing Zeros in Factorials:**
Trailing zeros in a number are created by factors of 10. Since \(10 = 2 \times 5\), we need bo... | math | numina_math | null | null | <think>
Okay, so I need to find the largest positive integer n where n factorial ends with exactly 100 zeros. Hmm, let me think. I remember that the number of trailing zeros in a factorial is determined by the number of times the number 10 is a factor in that factorial. Since 10 is 2 multiplied by 5, the number of trai... |
[
{
"from": "human",
"value": "Xiaoyuan has a basket of peaches. The first time, he ate more than half of the peaches by 1. The second time, he ate less than half of the remaining peaches by 1. Now there are 4 peaches left in the basket. How many peaches were originally in the basket?"
},
{
"from": "g... | Xiaoyuan has a basket of peaches. The first time, he ate more than half of the peaches by 1. The second time, he ate less than half of the remaining peaches by 1. Now there are 4 peaches left in the basket. How many peaches were originally in the basket? | Okay, let's try to figure out how many peaches Xiaoyuan originally had. Hmm, so the problem says that he ate more than half by 1 the first time, then less than half of the remaining by 1 the second time, and now there are 4 left. I need to find the original number.
Let me start by breaking down each step. Let's denote... | Xiaoyuan originally had **14 peaches** in the basket.
**Step-by-Step Explanation:**
1. **Define Variables:** Let \( N \) be the original number of peaches.
2. **First Eating:**
- Ate \( \frac{N}{2} + 1 \) peaches (more than half by 1).
- Remaining peaches:
\[
R_1 = N - \left(\frac{N}{2} + 1... |
1. Let's denote the original number of peaches in the basket as \( x \).
2. From the problem statement, the first time XiaoYuan eats half of the total peaches plus 1 more, which can be mathematically described as:
\[
\text{Peaches eaten in first time} = \frac{x}{2} + 1
\]
3. Thus, the number of peaches left... | math | numina_math | null | null | <think>
Okay, let's try to figure out how many peaches Xiaoyuan originally had. Hmm, so the problem says that he ate more than half by 1 the first time, then less than half of the remaining by 1 the second time, and now there are 4 left. I need to find the original number.
Let me start by breaking down each step. Let'... |
[
{
"from": "user",
"value": "Example 11. Find the number of consecutive zeros at the end of 1987!. \n\n untranslated text:\n将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 \n\n translated text:\nExample 11. Find the number of consecutive zeros at the end of 1987!. \n\nNote: The note at the end is not part of the original... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "6.14. Find the derivative of the function\n\n$$\nf(x)=\\left\\{\\begin{array}{l}\n\\operatorname{arctg}\\left(x^{2} \\cdot \\sin (1 /(5 x))\\right), \\quad x \\neq 0, \\\\\n0, x=0\n\\end{array}\\right.\n$$\n\nand compute its value at the point $x_{0}=0$."
},
{
"from": "as... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "26. For each integer $n \\geqslant 2$, determine the minimum value of \n$$\na_{0}+a_{1}+\\cdots+a_{n}\n$$\nsatisfying the conditions\n$$\na_{0}=1, a_{i} \\leqslant a_{i+1}+a_{i+2}, i=0,1, \\cdots, n-2\n$$\nwhere $a_{0}, a_{1}, \\cdots, a_{n}$ are non-negative numbers."
},
{
... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Some languages like Chinese, Japanese, and Thai do not have spaces between words. However, most natural languages processing tasks like part-of-speech tagging require texts that have segmented words. A simple and reasonably effective algorithm to segment a sentence into its comp... | Some languages like Chinese, Japanese, and Thai do not have spaces between words. However, most natural languages processing tasks like part-of-speech tagging require texts that have segmented words. A simple and reasonably effective algorithm to segment a sentence into its component words is called "MaxMatch".
## Max... | Okay, I need to implement the MaxMatch algorithm in Python. Let me think about how this works.
So the problem is to split a given sentence into words using the MaxMatch approach. The algorithm starts at the beginning of the sentence and finds the longest possible valid word from the current position. If no valid word... | To solve this problem, we need to segment a spaceless, alphanumeric, lowercased string into valid words using the MaxMatch algorithm. This algorithm starts from the beginning of the string and finds the longest valid word, then recursively processes the remaining substring until the entire string is segmented.
### App... | null | code | apps | {"fn_name": "max_match", "inputs": [[""]], "outputs": [[[]]]} |
def max_match(sentence):
| <think>
Okay, I need to implement the MaxMatch algorithm in Python. Let me think about how this works.
So the problem is to split a given sentence into words using the MaxMatch approach. The algorithm starts at the beginning of the sentence and finds the longest possible valid word from the current position. If no va... |
[
{
"from": "user",
"value": "Given an equilateral triangular prism \\( S-ABC \\) with height \\( SO = 3 \\) and a base edge length of 6. A perpendicular is drawn from point \\( A \\) to the opposite face \\( SBC \\), with the foot of the perpendicular at \\( O' \\). On \\( AO' \\), a point \\( P \\) is chose... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Given that \\( x, y, z \\) are positive numbers, and satisfy the system of equations:\n\n\\[\n\\begin{cases}\nx^2 + y^2 + xy = 3, \\\\\ny^2 + z^2 + yz = 4, \\\\\nz^2 + x^2 + zx = 7\n\\end{cases}\n\\]\n\nfind \\( x + y + z \\)."
},
{
"from": "gpt",
"value": "<think>\n... | Given that \( x, y, z \) are positive numbers, and satisfy the system of equations:
\[
\begin{cases}
x^2 + y^2 + xy = 3, \\
y^2 + z^2 + yz = 4, \\
z^2 + x^2 + zx = 7
\end{cases}
\]
find \( x + y + z \). | Okay, let's try to figure out how to solve this system of equations. We have three equations with three variables: x, y, z, all positive numbers. The equations are:
1. \(x^2 + y^2 + xy = 3\)
2. \(y^2 + z^2 + yz = 4\)
3. \(z^2 + x^2 + zx = 7\)
We need to find \(x + y + z\). Hmm, since all variables are positive, maybe... | Given the system of equations:
\[
\begin{cases}
x^2 + y^2 + xy = 3, \\
y^2 + z^2 + yz = 4, \\
z^2 + x^2 + zx = 7
\end{cases}
\]
we need to find \( x + y + z \).
First, we recognize that each equation resembles the Law of Cosines for a triangle with an angle of 120°. This geometric interpretation helps us understand ... | ### Problem:
已知 $x, y, z$ 是正数, 且满足
\[
\begin{cases}
x^{2}+y^{2}+x y=3, \\
y^{2}+z^{2}+y z=4, \\
z^{2}+x^{2}+z x=7,
\end{cases}
\]
求 $x+y+z$.
1. **Geometric Interpretation**:
Suppose the points $P, A, B, C, E, D$ are arranged to form some triangles such that the given algebraic relations can be interpreted as dista... | math | numina_math | null | null | <think>
Okay, let's try to figure out how to solve this system of equations. We have three equations with three variables: x, y, z, all positive numbers. The equations are:
1. \(x^2 + y^2 + xy = 3\)
2. \(y^2 + z^2 + yz = 4\)
3. \(z^2 + x^2 + zx = 7\)
We need to find \(x + y + z\). Hmm, since all variables are positiv... |
[
{
"from": "user",
"value": "47.1. Find the sum of all three-digit numbers divisible by 7.\n\n$(9-10$ grade $)$\n\n6"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the sum of all three-digit numbers that are divisible by 7. Hmm, let's see. First, I should figure out what th... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two min... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Brown, Jones, and Smith are suspects in a crime. One of them committed the crime. During the investigation, each of them made two statements:\n\nBrown: 1. I am not the criminal. 2. Jones is not the criminal.\n\nJones: 1. Brown is not the criminal. 2. The criminal is Smith.\n\nSm... | Brown, Jones, and Smith are suspects in a crime. One of them committed the crime. During the investigation, each of them made two statements:
Brown: 1. I am not the criminal. 2. Jones is not the criminal.
Jones: 1. Brown is not the criminal. 2. The criminal is Smith.
Smith: 1. The criminal is Brown. 2. I am not the ... | Okay, let's try to figure out who the criminal is here. We have three suspects: Brown, Jones, and Smith. Each made two statements. The key points are that one of them lied twice, another told the truth both times, and the third lied once and told the truth once. The criminal is one of them.
First, I need to analyze ea... | **Solution:**
To determine the criminal, we analyze each suspect's statements under the given conditions:
1. **Assume Brown is the criminal:**
- **Brown's statements:**
1. "I am not the criminal." → **Lie**.
2. "Jones is not the criminal." → **Truth** (since Brown is guilty).
*Brown: 1 lie, 1 trut... |
1. **Initial Analysis:**
We have three individuals: Brown, Jones, and Smith. Each made two statements:
- **Brown:**
1. I am not the criminal.
2. Jones is not the criminal.
- **Jones:**
1. Brown is not the criminal.
2. The criminal is Smith.
- **Smith:**
1. The criminal... | math | numina_math | null | null | <think>
Okay, let's try to figure out who the criminal is here. We have three suspects: Brown, Jones, and Smith. Each made two statements. The key points are that one of them lied twice, another told the truth both times, and the third lied once and told the truth once. The criminal is one of them.
First, I need to an... |
[
{
"from": "human",
"value": "Consider 2n rows of the seats in a bus. n rows of the seats on the left and n rows of the seats on the right. Each row can be filled by two people. So the total capacity of the bus is 4n.\n\nConsider that m (m ≤ 4n) people occupy the seats in the bus. The passengers entering the... | Consider 2n rows of the seats in a bus. n rows of the seats on the left and n rows of the seats on the right. Each row can be filled by two people. So the total capacity of the bus is 4n.
Consider that m (m ≤ 4n) people occupy the seats in the bus. The passengers entering the bus are numbered from 1 to m (in the order... | Okay, let's see. I need to find the order in which passengers exit the bus based on their seating. The problem gives me n and m, and I have to output the exit order of the m passengers. Hmm.
First, I need to understand the seating pattern and the exit order. Let's break it down step by step.
So, the bus has 2n rows. ... | To solve this problem, we need to determine the order in which passengers exit a bus based on their seating arrangement. The seating is filled in a specific order, and the exit order follows a different pattern.
### Approach
1. **Seating Arrangement**: Passengers fill window seats first, starting from the left window... | null | code | code_contests | {"inputs": ["2 7\n", "9 36\n", "100 199\n", "10 10\n", "10 39\n", "1 1\n", "100 1\n", "100 200\n", "77 13\n", "100 103\n", "10 40\n", "3 9\n", "100 399\n", "1 4\n", "77 280\n", "77 1\n", "100 13\n", "10 1\n", "100 201\n", "100 400\n", "100 77\n", "77 53\n", "100 300\n", "10 17\n", "10 9\n", "2 1\n", "59 13\n", "100 107... | <think>
Okay, let's see. I need to find the order in which passengers exit the bus based on their seating. The problem gives me n and m, and I have to output the exit order of the m passengers. Hmm.
First, I need to understand the seating pattern and the exit order. Let's break it down step by step.
So, the bus has 2... | |
[
{
"from": "user",
"value": "Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to t... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Miss Spelling has enough sheets of paper to give each pupil in her class 3 sheets and have 31 sheets left over. Alternatively, she could give each pupil 4 sheets and have 8 sheets left over. How many sheets of paper does she have?"
},
{
"from": "gpt",
"value": "<thin... | Miss Spelling has enough sheets of paper to give each pupil in her class 3 sheets and have 31 sheets left over. Alternatively, she could give each pupil 4 sheets and have 8 sheets left over. How many sheets of paper does she have? | Okay, let's see. So, the problem is about Miss Spelling having some sheets of paper. She can give each pupil 3 sheets and have 31 left, or give each pupil 4 sheets and have 8 left. We need to find out how many sheets she has in total. Hmm, let me break this down step by step.
First, let's denote the number of pupils a... | **Solution:**
Let's denote:
- \( T \) = total number of sheets
- \( p \) = number of pupils
**Step 1: Set up equations based on the problem:**
1. If each pupil gets 3 sheets, 31 sheets remain:
\( T = 3p + 31 \).
2. If each pupil gets 4 sheets, 8 sheets remain:
\( T = 4p + 8 \).
**Step 2: Equate the two e... | 1. Define the number of pupils in the class as \( x \).
2. The problem states that she has enough sheets to give each pupil in her class 3 sheets and have 31 sheets left over. Therefore, the total number of sheets she has can be expressed as:
\[
3x + 31
\]
3. Alternatively, if she gives each pupil 4 sheets a... | math | numina_math | null | null | <think>
Okay, let's see. So, the problem is about Miss Spelling having some sheets of paper. She can give each pupil 3 sheets and have 31 left, or give each pupil 4 sheets and have 8 left. We need to find out how many sheets she has in total. Hmm, let me break this down step by step.
First, let's denote the number of ... |
[
{
"from": "user",
"value": "3. From the eight natural numbers $1 \\sim 8$, the number of ways to choose three numbers such that there are no consecutive natural numbers is $\\qquad$ kinds."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out how many ways there are to choo... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "5. Let $N=2019^{2}-1$. How many positive factors of $N^{2}$ do not divide $N$ ?"
},
{
"from": "assistant",
"value": "<think>\nAlright, let's tackle this problem. So, we have N = 2019² - 1, and we need to find how many positive factors of N² do not divide N. Hmm, okay.... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Let \\( ABCDEF \\) be a convex hexagon such that \\( AB = BC \\), \\( CD = DE \\), and \\( EF = FA \\). Prove that \\(\\frac{BC}{BE} + \\frac{DE}{DA} + \\frac{FA}{FC} \\geq \\frac{3}{2}\\), and determine the conditions under which the equality holds."
},
{
"from": "assist... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Koa the Koala and her best friend want to play a game.\n\nThe game starts with an array $a$ of length $n$ consisting of non-negative integers. Koa and her best friend move in turns and each have initially a score equal to $0$. Koa starts.\n\nLet's describe a move in the game: D... | Koa the Koala and her best friend want to play a game.
The game starts with an array $a$ of length $n$ consisting of non-negative integers. Koa and her best friend move in turns and each have initially a score equal to $0$. Koa starts.
Let's describe a move in the game: During his move, a player chooses any element ... | Okay, let's see. I need to solve this problem where Koa and her friend play a game by taking elements from an array, and their scores are the XOR of the elements they pick. The winner is the one with the highest score. If they play optimally, what's the outcome for each test case?
Hmm. So the key here is to model the ... | To solve this problem, we need to determine the outcome of a game where Koa and her friend take turns picking elements from an array, with each player's score being updated using the XOR operation. The winner is the player with the highest score after all elements are picked, considering both play optimally.
### Appro... | null | code | apps | {
"inputs": [
"3\n3\n1 2 2\n3\n2 2 3\n5\n0 0 0 2 2\n",
"4\n5\n4 1 5 1 3\n4\n1 0 1 6\n1\n0\n2\n5 4\n"
],
"outputs": [
"WIN\nLOSE\nDRAW\n",
"WIN\nWIN\nDRAW\nWIN\n"
]
} | <think>
Okay, let's see. I need to solve this problem where Koa and her friend play a game by taking elements from an array, and their scores are the XOR of the elements they pick. The winner is the one with the highest score. If they play optimally, what's the outcome for each test case?
Hmm. So the key here is to mo... | |
[
{
"from": "human",
"value": "In the Bus of Characters there are n rows of seat, each having 2 seats. The width of both seats in the i-th row is w_i centimeters. All integers w_i are distinct.\n\nInitially the bus is empty. On each of 2n stops one passenger enters the bus. There are two types of passengers: ... | In the Bus of Characters there are n rows of seat, each having 2 seats. The width of both seats in the i-th row is w_i centimeters. All integers w_i are distinct.
Initially the bus is empty. On each of 2n stops one passenger enters the bus. There are two types of passengers:
* an introvert always chooses a row whe... | Okay, let's tackle this problem. So, the task is to determine which row each passenger takes when they enter the bus. Hmm. Let's see the problem statement again to make sure I understand it correctly.
The bus has n rows, each with two seats. All the w_i are distinct. Passengers come in 2n stops, each being introvert (... | To solve this problem, we need to determine which row each passenger will take when they enter the bus, based on their personality type (introvert or extrovert). The solution involves efficiently managing the selection of seats using priority queues to handle the constraints and requirements of each passenger type.
##... | null | code | code_contests | {"inputs": ["6\n10 8 9 11 13 5\n010010011101\n", "2\n3 1\n0011\n", "2\n1000000000 1\n0101\n", "2\n1000000000 999999999\n0011\n", "2\n1 1000000\n0011\n", "10\n24 53 10 99 83 9 15 62 33 47\n00100000000111111111\n", "1\n1000000\n01\n", "1\n1\n01\n", "2\n1100000000 999999999\n0011\n", "2\n1 1001000\n0011\n", "10\n24 53 10 ... | <think>
Okay, let's tackle this problem. So, the task is to determine which row each passenger takes when they enter the bus. Hmm. Let's see the problem statement again to make sure I understand it correctly.
The bus has n rows, each with two seats. All the w_i are distinct. Passengers come in 2n stops, each being int... | |
[
{
"from": "user",
"value": "Two circles with radius one are drawn in the coordinate plane, one with center \\((0,1)\\) and the other with center \\((2, y)\\), for some real number \\(y\\) between 0 and 1. A third circle is drawn so as to be tangent to both of the other two circles as well as the \\(x\\)-axi... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three... | In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three sides each of length 5. If the length of... | Okay, so I need to solve this geometry problem about a cyclic hexagon and a chord that splits it into two quadrilaterals. Let me try to visualize and break it down step by step. The problem states:
In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length ... | Given a cyclic hexagon with three consecutive sides of length 3 and the other three sides of length 5, we need to find the length of a chord that splits the hexagon into two quadrilaterals, each with three sides of the respective lengths.
1. **Hexagon Structure and Central Angles**:
- Label the vertices as \(A, B,... |
Given a regular hexagon inscribed in a circle, we are provided with sides of lengths $3$ and $5$. We need to determine the arc length corresponding to these sides when projected onto the circle divided into two quadrilaterals.
1. **Label setup**:
- Let the vertices of the hexagon be labeled $A, B, C, D, E, F$ cons... | math | numina_math | null | null | <think>
Okay, so I need to solve this geometry problem about a cyclic hexagon and a chord that splits it into two quadrilaterals. Let me try to visualize and break it down step by step. The problem states:
In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of... |
[
{
"from": "human",
"value": "In one month, three Wednesdays fell on even dates. On which day will the second Sunday fall in this month?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to figure out this problem step by step. So, the question says that in a certain month, there are three ... | In one month, three Wednesdays fell on even dates. On which day will the second Sunday fall in this month? | Okay, let's try to figure out this problem step by step. So, the question says that in a certain month, there are three Wednesdays that fall on even dates. We need to determine on which day the second Sunday of that month will fall. Hmm, let's break this down.
First, let's clarify what "even dates" mean. I think it re... | The problem states that in a month, three Wednesdays fall on even dates. We need to determine the date (numerical day) of the second Sunday.
### Key Steps:
1. **Identify Possible Wednesdays**:
- For three Wednesdays to be even, they must be spaced 14 days apart (since adding 7 days alternates parity). The only sequ... |
1. Let's determine the possible dates for Wednesdays that fall on even days of the month. The month can have different starting days, but we need to meet the condition that there are three Wednesdays on even days within a single calendar month.
2. Considering the condition, the earliest possible Wednesday on an ev... | math | numina_math | null | null | <think>
Okay, let's try to figure out this problem step by step. So, the question says that in a certain month, there are three Wednesdays that fall on even dates. We need to determine on which day the second Sunday of that month will fall. Hmm, let's break this down.
First, let's clarify what "even dates" mean. I thi... |
[
{
"from": "human",
"value": "Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies $a_{n+1}=\\frac{3^{n+1} a_{n}}{a_{n}+3^{n+1}}, \\, a_{1}=3$, find the general term of the sequence $\\left\\{a_{n}\\right\\}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to figure out the general te... | Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=\frac{3^{n+1} a_{n}}{a_{n}+3^{n+1}}, \, a_{1}=3$, find the general term of the sequence $\left\{a_{n}\right\}$. | Okay, let's try to figure out the general term for this sequence. The problem says that the sequence {a_n} is defined by the recursive formula a_{n+1} = (3^{n+1} a_n) / (a_n + 3^{n+1}) with the initial term a_1 = 3. Hmm, recursive sequences can sometimes be tricky, especially when dealing with fractions and exponents. ... | Given the sequence $\left\{a_{n}\right\}$ defined by the recursive formula $a_{n+1} = \frac{3^{n+1} a_{n}}{a_{n} + 3^{n+1}}$ with the initial term $a_{1} = 3$, we aim to find the general term of the sequence.
First, we compute the first few terms to identify a pattern:
- $a_1 = 3$
- $a_2 = \frac{9 \cdot 3}{3 + 9} = \f... |
To find the general term \(a_n\) of the sequence \(\left\{a_{n}\right\}\) given by the recurrence relation:
\[a_{n+1} = \frac{3^{n+1} a_{n}}{a_{n} + 3^{n+1}}, \quad a_{1} = 3,\]
we first rewrite the recurrence in a more convenient form by working with reciprocals. Here's the process:
1. **Express the Reciprocal of ... | math | numina_math | null | null | <think>
Okay, let's try to figure out the general term for this sequence. The problem says that the sequence {a_n} is defined by the recursive formula a_{n+1} = (3^{n+1} a_n) / (a_n + 3^{n+1}) with the initial term a_1 = 3. Hmm, recursive sequences can sometimes be tricky, especially when dealing with fractions and exp... |
[
{
"from": "human",
"value": "For which values of the parameter \\( p \\) does the inequality\n\n$$\n\\left(1+\\frac{1}{\\sin x}\\right)^{3} \\geq \\frac{p}{\\tan^2 x}\n$$\n\nhold for any \\( 0 < x < \\frac{\\pi}{2} \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all values o... | For which values of the parameter \( p \) does the inequality
$$
\left(1+\frac{1}{\sin x}\right)^{3} \geq \frac{p}{\tan^2 x}
$$
hold for any \( 0 < x < \frac{\pi}{2} \)? | Okay, so I need to find all values of the parameter \( p \) such that the inequality
\[
\left(1 + \frac{1}{\sin x}\right)^3 \geq \frac{p}{\tan^2 x}
\]
holds for any \( x \) in the interval \( 0 < x < \frac{\pi}{2} \). Hmm, let me start by understanding the problem step by step.
First, the inequality has to hold for... | To determine the values of the parameter \( p \) for which the inequality
\[
\left(1 + \frac{1}{\sin x}\right)^3 \geq \frac{p}{\tan^2 x}
\]
holds for any \( 0 < x < \frac{\pi}{2} \), we start by rewriting the inequality in terms of \( \sin x \) and \( \cos x \). Let \( t = \sin x \) where \( 0 < t < 1 \). The inequal... |
Given the inequality:
\[
\left(1+\frac{1}{\sin x}\right)^{3} \geq \frac{p}{\tan^2 x}
\]
for any \(0 < x < \frac{\pi}{2}\), we desire to find all possible values of the parameter \(p\).
1. **Initial Considerations**:
The left-hand side and the reciprocal of the tangent squared on the right-hand side are both alwa... | math | numina_math | null | null | <think>
Okay, so I need to find all values of the parameter \( p \) such that the inequality
\[
\left(1 + \frac{1}{\sin x}\right)^3 \geq \frac{p}{\tan^2 x}
\]
holds for any \( x \) in the interval \( 0 < x < \frac{\pi}{2} \). Hmm, let me start by understanding the problem step by step.
First, the inequality has to ... |
[
{
"from": "user",
"value": "6. (1988 Mathematics Retest Question for Science Experimental Class Admission) Let $S=[\\sqrt{1}]+[\\sqrt{2}]+\\cdots+[\\sqrt{1988}]$, find $[\\sqrt{S}]$.\n\n保留源文本的换行和格式，翻译结果如下：\n\n6. (1988 Mathematics Retest Question for Science Experimental Class Admission) Let $S=[\\sqrt{1}]+[... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "9. Among the eight vertices of a regular octagon, three vertices are chosen at random. The probability that these three points form the vertices of a right triangle is $\\qquad$ .\n\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the probabilit... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Let \\( z \\) be a complex number with a modulus of 1. Then the maximum value of \\(\\left|\\frac{z+\\mathrm{i}}{z+2}\\right|\\) is \\(\\ \\ \\ \\ \\ \\ \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the maximum value of the expression |... | Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \). | Okay, let's see. I need to find the maximum value of the expression |(z + i)/(z + 2)| where z is a complex number with modulus 1. Hmm, so z lies on the unit circle in the complex plane. Alright, let me start by recalling some complex number properties.
First, since |z| = 1, z can be written as e^{iθ} where θ is the ar... | Given a complex number \( z \) with modulus 1, we need to find the maximum value of \( \left| \frac{z + i}{z + 2} \right| \).
1. **Express the modulus ratio**:
\[
\left| \frac{z + i}{z + 2} \right| = \frac{|z + i|}{|z + 2|}
\]
Since \( |z| = 1 \), \( z \) lies on the unit circle.
2. **Parametrize \( z \)*... |
1. Given that \( z \) is a complex number such that \( |z| = 1 \). We need to find the maximum value of \( \left| \frac{z + \mathrm{i}}{z + 2} \right| \).
2. Assume \( z \) is of the form \( z = \cos \theta + \mathrm{i} \sin \theta \) where \( \theta \) is a real number.
3. Substitute \( z = \cos \theta + \mathrm{i}... | math | numina_math | null | null | <think>
Okay, let's see. I need to find the maximum value of the expression |(z + i)/(z + 2)| where z is a complex number with modulus 1. Hmm, so z lies on the unit circle in the complex plane. Alright, let me start by recalling some complex number properties.
First, since |z| = 1, z can be written as e^{iθ} where θ i... |
[
{
"from": "user",
"value": "Let \\( P \\) be a regular 2006-sided polygon. A diagonal of \\( P \\) is called a \"good edge\" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \\( P \\) is also considered a \"good edge\".\n\nGiven that 2003... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "There are real numbers \\(x_1, x_2, \\ldots, x_{100}\\) such that \\(x_1^3 + x_2 = x_2^3 + x_3 = \\ldots = x_{100}^3 + x_{101} = x_{101}^3 + x_1\\). Prove that all these numbers are equal to each other."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove... | There are real numbers \(x_1, x_2, \ldots, x_{100}\) such that \(x_1^3 + x_2 = x_2^3 + x_3 = \ldots = x_{100}^3 + x_{101} = x_{101}^3 + x_1\). Prove that all these numbers are equal to each other. | Okay, so I need to prove that all these numbers x₁, x₂, ..., x₁₀₁ are equal to each other given that each consecutive pair satisfies the equation x_i³ + x_{i+1} = x_{i+1}³ + x_{i+2}, and this continues in a cycle with the last one connecting back to x₁. Hmm, let me try to break this down step by step.
First, let's res... | Given real numbers \(x_1, x_2, \ldots, x_{101}\) such that \(x_1^3 + x_2 = x_2^3 + x_3 = \ldots = x_{100}^3 + x_{101} = x_{101}^3 + x_1\), we need to prove that all these numbers are equal.
First, we note that each consecutive pair satisfies the equation \(x_i^3 + x_{i+1} = C\) for some constant \(C\). This forms a cy... |
Given the sequence of real numbers \( x_1, x_2, \ldots, x_{100} \) such that the following equality holds for every \( i \) (with indices taken modulo 100):
\[
x_i^3 + x_{i+1} = x_{i+1}^3 + x_{i+2}
\]
we aim to prove that all these numbers are equal.
1. **Assume for contradiction that \( x_2 > x_1 \)**:
If \( x... | math | numina_math | null | null | <think>
Okay, so I need to prove that all these numbers x₁, x₂, ..., x₁₀₁ are equal to each other given that each consecutive pair satisfies the equation x_i³ + x_{i+1} = x_{i+1}³ + x_{i+2}, and this continues in a cycle with the last one connecting back to x₁. Hmm, let me try to break this down step by step.
First, l... |
[
{
"from": "human",
"value": "Point $C$ is a midpoint of $AB$ . Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$ . Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$ . ... | Point $C$ is a midpoint of $AB$ . Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$ . Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$ . Point $Q$ is a midpoint of arc $BD$ o... | Okay, so I need to prove that PQ is perpendicular to CD. Let me try to visualize the problem first. We have a line segment AB with midpoint C. Then there's circle o1 passing through A and C, and circle o2 passing through B and C. These two circles intersect at points C and D. So D is another intersection point besides ... | To prove that \(PQ \perp CD\), we start by considering the given configuration: point \(C\) is the midpoint of \(AB\), circle \(o_1\) passes through \(A\) and \(C\), circle \(o_2\) passes through \(B\) and \(C\), and both circles intersect at points \(C\) and \(D\). Points \(P\) and \(Q\) are the midpoints of arcs \(AD... | 1. **Understanding the Problem:**
- Point \( C \) is the midpoint of segment \( AB \).
- Circle \( o_1 \) passes through points \( A \) and \( C \).
- Circle \( o_2 \) passes through points \( B \) and \( C \).
- Circles \( o_1 \) and \( o_2 \) intersect at points \( C \) and \( D \).
- Point \( P \) is ... | math | numina_math | null | null | <think>
Okay, so I need to prove that PQ is perpendicular to CD. Let me try to visualize the problem first. We have a line segment AB with midpoint C. Then there's circle o1 passing through A and C, and circle o2 passing through B and C. These two circles intersect at points C and D. So D is another intersection point ... |
[
{
"from": "human",
"value": "Observe the following equations:\n$$\n\\begin{array}{l}\n1+3=4=2 \\times 2 \\\\\n1+3+5=9=3 \\times 3 \\\\\n1+3+5+7=16=4 \\times 4 .\n\\end{array}\n$$\n\nCalculate: $1+3+5+\\cdots+2019=$ $\\qquad$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to calc... | Observe the following equations:
$$
\begin{array}{l}
1+3=4=2 \times 2 \\
1+3+5=9=3 \times 3 \\
1+3+5+7=16=4 \times 4 .
\end{array}
$$
Calculate: $1+3+5+\cdots+2019=$ $\qquad$ | Okay, let's see. I need to calculate the sum of the odd numbers starting from 1 up to 2019. The examples given show that adding the first few odd numbers results in perfect squares. Like 1+3=4 which is 2x2, then 1+3+5=9=3x3, and 1+3+5+7=16=4x4. So there seems to be a pattern here where the sum of the first n odd number... | The sum of the first \( n \) odd numbers is \( n^2 \). To find the sum \( 1 + 3 + 5 + \cdots + 2019 \):
1. **Determine \( n \):**
The \( n \)-th odd number is \( 2n - 1 \). Setting \( 2n - 1 = 2019 \):
\[
2n = 2020 \implies n = 1010.
\]
2. **Compute the sum:**
The sum of the first 1010 odd number... |
1. **Identify the Pattern:**
The problem gives a sequence of summations of odd numbers and their corresponding squared results:
\[
\begin{array}{l}
1+3=4=2 \times 2 \\
1+3+5=9=3 \times 3 \\
1+3+5+7=16=4 \times 4.
\end{array}
\]
From the given examples, we observe that the sum of the first \... | math | numina_math | null | null | <think>
Okay, let's see. I need to calculate the sum of the odd numbers starting from 1 up to 2019. The examples given show that adding the first few odd numbers results in perfect squares. Like 1+3=4 which is 2x2, then 1+3+5=9=3x3, and 1+3+5+7=16=4x4. So there seems to be a pattern here where the sum of the first n od... |
[
{
"from": "user",
"value": "Jack Sparrow needed to distribute 150 piastres across 10 purses. After placing a certain amount of piastres in the first purse, he placed more piastres in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "You are given $n$ integers $a_1, a_2, \\ldots, a_n$. Find the maximum value of $max(a_l, a_{l + 1}, \\ldots, a_r) \\cdot min(a_l, a_{l + 1}, \\ldots, a_r)$ over all pairs $(l, r)$ of integers for which $1 \\le l < r \\le n$.\n\n\n-----Input-----\n\nThe first line contains a sing... | You are given $n$ integers $a_1, a_2, \ldots, a_n$. Find the maximum value of $max(a_l, a_{l + 1}, \ldots, a_r) \cdot min(a_l, a_{l + 1}, \ldots, a_r)$ over all pairs $(l, r)$ of integers for which $1 \le l < r \le n$.
-----Input-----
The first line contains a single integer $t$ ($1 \le t \le 10000$) — the number o... | Okay, I need to solve this problem where for each test case, given an array of integers, I have to find the maximum value of the product of the maximum and minimum in any subarray of length at least 2. Hmm. Let's think about how to approach this efficiently because the constraints are tight.
So the problem says that f... | To solve this problem, we need to find the maximum value of the product of the maximum and minimum values of all possible subarrays of length at least 2. The key insight is that the maximum product is achieved by considering consecutive elements in the array, as any subarray's product can be represented by the product ... | null | code | taco | {"inputs": ["4\n3\n2 4 3\n4\n3 2 3 1\n2\n69 69\n6\n719313 273225 402638 473783 804745 323328\n", "1\n2\n140537 269100\n", "1\n2\n411662 754535\n", "1\n2\n140537 269100\n", "1\n2\n411662 754535\n", "1\n2\n140537 91097\n", "1\n2\n441125 754535\n", "4\n3\n2 2 3\n4\n3 2 3 1\n2\n69 69\n6\n719313 273225 402638 473783 804745 ... | <think>
Okay, I need to solve this problem where for each test case, given an array of integers, I have to find the maximum value of the product of the maximum and minimum in any subarray of length at least 2. Hmm. Let's think about how to approach this efficiently because the constraints are tight.
So the problem say... | |
[
{
"from": "user",
"value": "In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have?"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The probl... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Example. Solve the integral equation\n\n$$\n\\int_{0}^{x}(x-t) \\varphi(t) d t=x^{2}\n$$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this integral equation: the integral from 0 to x of (x - t) times φ(t) dt equals x squared. Hmm, integral eq... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "20. Currently, the largest hydropower station in the world that has been built is $\\qquad$\nA. Gezhouba Hydropower Station\nB. Sanmenxia Hydropower Station\nC. Itaipu Hydropower Station\nD. Three Gorges Hydropower Station"
},
{
"from": "assistant",
"value": "<think>\... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Let \\( a_1, a_2, \\cdots \\) be a non-negative sequence that satisfies the condition\n\\[ a_{n+m} \\leq a_n + a_m \\quad \\text{for} \\quad m, n \\in \\mathbb{N}. \\]\nProve that for any \\( n \\geq m \\), we have\n\\[ a_n \\leq m a_1 + \\left( \\frac{n}{m} - 1 \\right) a_m. \\... | Let \( a_1, a_2, \cdots \) be a non-negative sequence that satisfies the condition
\[ a_{n+m} \leq a_n + a_m \quad \text{for} \quad m, n \in \mathbb{N}. \]
Prove that for any \( n \geq m \), we have
\[ a_n \leq m a_1 + \left( \frac{n}{m} - 1 \right) a_m. \] | Okay, so I have this problem here about a non-negative sequence \( a_1, a_2, \cdots \) that satisfies the condition \( a_{n+m} \leq a_n + a_m \) for all natural numbers \( m \) and \( n \). I need to prove that for any \( n \geq m \), the inequality \( a_n \leq m a_1 + \left( \frac{n}{m} - 1 \right) a_m \) holds. Hmm, ... | To prove that for any \( n \geq m \), the inequality \( a_n \leq m a_1 + \left( \frac{n}{m} - 1 \right) a_m \) holds for a non-negative subadditive sequence \( \{a_n\} \), follow these steps:
1. **Decompose \( n \) into multiples of \( m \):**
Write \( n = qm + r \), where \( q = \lfloor \frac{n}{m} \rfloor \) and ... |
1. **Given Conditions**:
- The sequence $a_{1}, a_{2}, \ldots$ satisfies
\[
a_{n+m} \leq a_{n} + a_{m} \quad \text{for all } m, n \in \mathbb{N}.
\]
- We need to prove that for any $n \geq m$:
\[
a_{n} \leq m a_{1} + \left( \frac{n}{m} - 1 \right) a_{m}.
\]
2. **Dividing $n$**:
... | math | numina_math | null | null | <think>
Okay, so I have this problem here about a non-negative sequence \( a_1, a_2, \cdots \) that satisfies the condition \( a_{n+m} \leq a_n + a_m \) for all natural numbers \( m \) and \( n \). I need to prove that for any \( n \geq m \), the inequality \( a_n \leq m a_1 + \left( \frac{n}{m} - 1 \right) a_m \) hold... |
[
{
"from": "human",
"value": "Find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from it to the sides of the triangle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the point inside a given triangle that maximizes the p... | Find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from it to the sides of the triangle. | Okay, so I need to find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from that point to the sides of the triangle. Hmm, let's see. I remember that in geometry, there are various centers of a triangle like the centroid, incenter, circumcenter, and orthocenter.... | To find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from it to the sides of the triangle, we start by considering the problem in a coordinate system for a right-angled triangle. We then generalize the approach using calculus and Lagrange multipliers.
1. **C... |
To find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from it to the sides of the triangle, we need to find the centroid of the triangle.
Here's a step-by-step proof showing that the centroid is the correct solution:
1. **Define coordinates:**
Let the v... | math | numina_math | null | null | <think>
Okay, so I need to find the point inside a given triangle that maximizes the product of the lengths of the perpendiculars dropped from that point to the sides of the triangle. Hmm, let's see. I remember that in geometry, there are various centers of a triangle like the centroid, incenter, circumcenter, and orth... |
[
{
"from": "user",
"value": "5th Putnam 1942 Problem B4 A particle moves in a circle through the origin under the influence of a force a/r k towards the origin (where r is its distance from the origin). Find k."
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to solve this problem... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$."
},
{
"fro... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Problem 3. Let $n$ be a non-zero natural number and $A=\\{1,2, \\ldots, n\\}$. Determine the number of increasing functions $f: A \\rightarrow A$ with the property that $|f(x)-f(y)| \\leq|x-y|$ for any $x, y \\in A$."
},
{
"from": "assistant",
"value": "<think>\nOkay,... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\\circ}$ left. He then moves $3$ uni... | Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left... | Okay, let's try to figure out this problem step by step. So, Andy the Ant starts at (-20, 20) facing east. He moves 1 unit east, then turns left (which would be north), then moves 2 units north, turns left (west), moves 3 units west, turns left (south), and so on, increasing the distance by 1 unit each time and always ... | Andy the Ant starts at the point \((-20, 20)\) facing east. He moves in a pattern where he increases his distance by 1 unit each time and always turns left after each move. The directions cycle every four moves: east, north, west, south.
To determine Andy's position after the 2020th left turn, we note that each cycle... | To solve this problem, we need to understand the pattern of Andy's movements and how his position changes with each move. Andy starts at $(-20, 20)$ and moves in a spiral pattern, increasing the distance he moves by $1$ unit after each turn, and always turning left.
1. **Initial Moves and Pattern Recognition:**
- *... | math | numina_math | null | null | <think>
Okay, let's try to figure out this problem step by step. So, Andy the Ant starts at (-20, 20) facing east. He moves 1 unit east, then turns left (which would be north), then moves 2 units north, turns left (west), moves 3 units west, turns left (south), and so on, increasing the distance by 1 unit each time and... |
[
{
"from": "user",
"value": "Problem 10.2. It is known that $\\frac{a+b}{a-b}=3$. Find the value of the expression $\\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tackle this problem. So, we have the equation (a + b)/(a - b) = 3, and we need to find t... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Given an isosceles triangle \\( \\triangle ABC \\) with base angles \\( \\angle ABC = \\angle ACB = 50^\\circ \\), points \\( D \\) and \\( E \\) lie on \\( BC \\) and \\( AC \\) respectively. Lines \\( AD \\) and \\( BE \\) intersect at point \\( P \\). Given \\( \\angle ABE = ... | Given an isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \), points \( D \) and \( E \) lie on \( BC \) and \( AC \) respectively. Lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), find \( \angle BED \... | Okay, let's try to figure out this geometry problem. Hmm, we have an isosceles triangle ABC with base angles at B and C, each 50 degrees. So, that means the vertex angle at A must be 180 - 50 - 50 = 80 degrees. Got that. Now, points D and E are on BC and AC respectively. Lines AD and BE intersect at point P. We're give... | To solve the problem, we start by analyzing the given isosceles triangle \( \triangle ABC \) with \( \angle ABC = \angle ACB = 50^\circ \) and \( \angle BAC = 80^\circ \). Points \( D \) and \( E \) are on \( BC \) and \( AC \), respectively, such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). We aim... | ### Problem:
Given isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \). Points \( D \) and \( E \) are on \( BC \) and \( AC \) respectively, such that lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), ... | math | numina_math | null | null | <think>
Okay, let's try to figure out this geometry problem. Hmm, we have an isosceles triangle ABC with base angles at B and C, each 50 degrees. So, that means the vertex angle at A must be 180 - 50 - 50 = 80 degrees. Got that. Now, points D and E are on BC and AC respectively. Lines AD and BE intersect at point P. We... |
[
{
"from": "user",
"value": "In the square $A B C D$, the point $P$ was taken inside such that $A P: B P: C P=1: 2: 3$. What is the measure of the angle $A P B$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the measure of angle APB in square ABCD where point P is inside t... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Petya and Vasya calculated that if they walk at a speed of 4 km per hour to the neighboring village, which is 4 kilometers away, they will be 10 minutes late for the football match held there for the district championship. How should they proceed to arrive at the match on time an... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "If \\( m \\) and \\( n \\) are positive integers that satisfy the equation \\( 3 m^{3}=5 n^{5} \\), the smallest possible value for \\( m+n \\) is:\n(A) 900\n(B) 450\n(C) 720\n(D) 810\n(E) 630"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smalle... | If \( m \) and \( n \) are positive integers that satisfy the equation \( 3 m^{3}=5 n^{5} \), the smallest possible value for \( m+n \) is:
(A) 900
(B) 450
(C) 720
(D) 810
(E) 630 | Okay, so I need to find the smallest possible value of m + n where m and n are positive integers that satisfy the equation 3m³ = 5n⁵. The answer choices are given, so maybe I can check them, but probably I need to derive it step by step. Let me think.
First, the equation is 3m³ = 5n⁵. Both m and n are positive integer... | Given the equation \(3m^3 = 5n^5\) where \(m\) and \(n\) are positive integers, we need to find the smallest possible value of \(m + n\).
1. **Prime Factorization Analysis**:
- The equation implies that the prime factors of both sides must match.
- \(m\) must be divisible by 5, and \(n\) must be divisible by 3.
... |
Given the equation:
\[
3 m^{3} = 5 n^{5}
\]
where \( m \) and \( n \) are positive integers, we need to find the smallest possible value of \( m + n \).
1. **Determine divisibility of \( m \) and \( n \) by 3 and 5:**
Since \( 3 m^{3} \) is a multiple of 3, \( 5 n^{5} \) must also be a multiple of 3. Given tha... | math | numina_math | null | null | <think>
Okay, so I need to find the smallest possible value of m + n where m and n are positive integers that satisfy the equation 3m³ = 5n⁵. The answer choices are given, so maybe I can check them, but probably I need to derive it step by step. Let me think.
First, the equation is 3m³ = 5n⁵. Both m and n are positive... |
[
{
"from": "human",
"value": "Given point \\( P \\) is any point inside \\( \\triangle ABC \\). Extend \\( AP \\), \\( BP \\), and \\( CP \\) to intersect \\( BC \\), \\( AC \\), and \\( AB \\) at points \\( D \\), \\( E \\), and \\( F \\) respectively. Prove that:\n\n\\[\n\\frac{S_{\\triangle DEF}}{S_{\\tri... | Given point \( P \) is any point inside \( \triangle ABC \). Extend \( AP \), \( BP \), and \( CP \) to intersect \( BC \), \( AC \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively. Prove that:
\[
\frac{S_{\triangle DEF}}{S_{\triangle ABC}} = \frac{2 PD \cdot PE \cdot PF}{PA \cdot PB \cdot PC}
\] | Okay, so I need to prove that the ratio of the area of triangle DEF to the area of triangle ABC is equal to 2 times the product of PD, PE, PF divided by the product of PA, PB, PC. Hmm, let me start by understanding the problem setup.
We have triangle ABC with a point P inside it. Then, we extend AP, BP, and CP to meet... | Given point \( P \) inside \( \triangle ABC \), we extend \( AP \), \( BP \), and \( CP \) to intersect \( BC \), \( AC \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively. We need to prove that:
\[
\frac{S_{\triangle DEF}}{S_{\triangle ABC}} = \frac{2 PD \cdot PE \cdot PF}{PA \cdot PB \cdot PC}
\]
... |
Given: Point \( P \) is an interior point of \(\triangle ABC\). The extensions of \( AP \), \( BP \), and \( CP \) intersect \( BC \), \( AC \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively.
We aim to prove:
\[
\frac{S_{\triangle DEF}}{S_{\triangle ABC}} = \frac{2 PD \cdot PE \cdot PF}{PA \cdot P... | math | numina_math | null | null | <think>
Okay, so I need to prove that the ratio of the area of triangle DEF to the area of triangle ABC is equal to 2 times the product of PD, PE, PF divided by the product of PA, PB, PC. Hmm, let me start by understanding the problem setup.
We have triangle ABC with a point P inside it. Then, we extend AP, BP, and CP... |
[
{
"from": "user",
"value": "5 (1133). For what value of $a$ does the sum of the squares of the roots of the quadratic trinomial $x^{2}-(a-2) x-a-1$ take the smallest value?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of 'a' that makes the sum of the squares of... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "2. Let $\\theta$ be the largest interior angle of a triangle, and satisfies $\\sin 8 \\theta=\\sin 2 \\theta$. Then the set of possible values of $\\theta$ is $\\qquad$ (express using the roster method)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "## Task Condition\n\nFind the $n$-th order derivative.\n\n$y=\\frac{x}{2(3 x+2)}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the nth order derivative of the function y = x / [2(3x + 2)]. Hmm, derivatives can get tricky when you go to higher ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "A new airplane SuperPuperJet has an infinite number of rows, numbered with positive integers starting with 1 from cockpit to tail. There are six seats in each row, denoted with letters from 'a' to 'f'. Seats 'a', 'b' and 'c' are located to the left of an aisle (if one looks in t... | A new airplane SuperPuperJet has an infinite number of rows, numbered with positive integers starting with 1 from cockpit to tail. There are six seats in each row, denoted with letters from 'a' to 'f'. Seats 'a', 'b' and 'c' are located to the left of an aisle (if one looks in the direction of the cockpit), while seats... | Okay, let's try to figure out how to solve this problem. So, the problem is about calculating the time Vasya has to wait until he gets his lunch based on his seat in the airplane. The flight attendants are serving the rows in a particular pattern, and each seat in a row has a specific serving order.
First, I need to ... | To solve this problem, we need to determine how many seconds Vasya will have to wait before getting his lunch based on his seat in the airplane. The flight attendants serve the rows in a specific pattern, and each seat in a row has a particular serving order.
### Approach
1. **Understand the Serving Pattern**: The fl... | null | code | codeforces | {"inputs": ["1f", "2d", "4a", "5e", "2c", "1b", "1000000000000000000d", "999999999999999997a", "1c", "1d", "1e", "1a", "2a", "2b", "2e", "2f", "3a", "3b", "3c", "3d", "3e", "3f", "4b", "4c", "4d", "4e", "4f", "999999997a", "999999997b", "999999997c", "999999997d", "999999997e", "999999997f", "999999998a", "999999998b",... | <think>
Okay, let's try to figure out how to solve this problem. So, the problem is about calculating the time Vasya has to wait until he gets his lunch based on his seat in the airplane. The flight attendants are serving the rows in a particular pattern, and each seat in a row has a specific serving order.
First, I ... | |
[
{
"from": "user",
"value": "How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \\leq 343$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the number of ordered triplets (a, b, c) of positive integers such that 30a + 50b + 70c ≤ 343. Hmm, ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Everybody in MIT knows Krishna Sundar is a cool, young and energetic DP programmer, whereas Vaidyanathan Pk is a young and romantic programmer. According to Krishna, energy is needed to code. But Vaidhy thinks that, to be a coder, you need to be romantic.\nWhenever the math pro... | Everybody in MIT knows Krishna Sundar is a cool, young and energetic DP programmer, whereas Vaidyanathan Pk is a young and romantic programmer. According to Krishna, energy is needed to code. But Vaidhy thinks that, to be a coder, you need to be romantic.
Whenever the math professor asks any questions, it is Vaidhy wh... | Okay, I need to solve this problem. Let's read the problem statement carefully.
The problem is about calculating the sum of F(s_i, 1) for all valid substrings s_i of a given string. Then take this sum modulo m. The substrings must not have leading zeros and must be palindromes. Then for each such substring, compute F(... | To solve this problem, we need to compute the sum of a specific function applied to all valid palindromic substrings of a given string, modulo a given value. The valid substrings are those that do not have leading zeros (unless they are a single '0') and are palindromes. The function F(n, k) is defined recursively, but... | null | code | taco | {"inputs": ["100\n3072007449697535427577656670667399744791454166143162875272767573698264\n94\n6332123442545362943151236787214819292310292619380\n2\n476\n62\n254072198579081329812150489412744714202261\n80\n031785682263140008460422\n10\n791418349255137555396480992856042480311019319028103011832280846952676332\n8\n38764552... | <think>
Okay, I need to solve this problem. Let's read the problem statement carefully.
The problem is about calculating the sum of F(s_i, 1) for all valid substrings s_i of a given string. Then take this sum modulo m. The substrings must not have leading zeros and must be palindromes. Then for each such substring, co... | |
[
{
"from": "user",
"value": "1.53. Given a triangle $A B C$ such that $A B=15 \\text{ cm}, B C=12 \\text{ cm}$ and $A C=18 \\text{ cm}$. In what ratio does the center of the inscribed circle divide the angle bisector of angle $C$?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to f... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Let \\( p, q, r \\) be the three sides of triangle \\( PQR \\). If \\( p^{4} + q^{4} + r^{4} = 2r^{2}(p^{2} + q^{2}) \\), find \\( a \\), where \\( a = \\cos^{2} R \\) and \\( R \\) denotes the angle opposite \\( r \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, s... | Let \( p, q, r \) be the three sides of triangle \( PQR \). If \( p^{4} + q^{4} + r^{4} = 2r^{2}(p^{2} + q^{2}) \), find \( a \), where \( a = \cos^{2} R \) and \( R \) denotes the angle opposite \( r \). | Okay, so I need to find the value of \( a = \cos^2 R \) given that in triangle PQR, the sides opposite angles P, Q, R are p, q, r respectively, and the equation \( p^4 + q^4 + r^4 = 2r^2(p^2 + q^2) \) holds. Let me start by recalling some trigonometric identities and theorems related to triangles, especially the Law of... | Given a triangle \( PQR \) with sides \( p, q, r \) opposite angles \( P, Q, R \) respectively, we need to find \( a = \cos^2 R \) given the equation \( p^4 + q^4 + r^4 = 2r^2(p^2 + q^2) \).
First, we use the Law of Cosines, which states:
\[ r^2 = p^2 + q^2 - 2pq \cos R \]
We substitute \( r^2 \) from the Law of Cosi... |
Given:
\[
p^4 + q^4 + r^4 = 2r^2(p^2 + q^2)
\]
We need to find \( a \), where \( a = \cos^2 R \) and \( R \) is the angle opposite the side \( r \).
1. Using the Law of Cosines, we have:
\[
\cos R = \frac{p^2 + q^2 - r^2}{2pq}
\]
2. Squaring both sides to find \( \cos^2 R \):
\[
a = \cos^2 R = \left( \frac{p^2 ... | math | numina_math | null | null | <think>
Okay, so I need to find the value of \( a = \cos^2 R \) given that in triangle PQR, the sides opposite angles P, Q, R are p, q, r respectively, and the equation \( p^4 + q^4 + r^4 = 2r^2(p^2 + q^2) \) holds. Let me start by recalling some trigonometric identities and theorems related to triangles, especially th... |
[
{
"from": "human",
"value": "Given a triangle \\( ABC \\) with \\( |AB| < |BC| \\). Prove that for any point \\( M \\) on the median drawn from vertex \\( B \\), \\( \\angle BAM > \\angle BCM \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that in triangle ABC where AB is s... | Given a triangle \( ABC \) with \( |AB| < |BC| \). Prove that for any point \( M \) on the median drawn from vertex \( B \), \( \angle BAM > \angle BCM \). | Okay, so I need to prove that in triangle ABC where AB is shorter than BC, for any point M on the median from vertex B, the angle BAM is greater than angle BCM. Hmm, let's start by understanding the problem.
First, let me visualize the triangle. ABC, with AB < BC. So, vertex B is connected to A and C, with BC longer t... | To prove that in triangle \( ABC \) with \( |AB| < |BC| \), for any point \( M \) on the median from \( B \), \( \angle BAM > \angle BCM \), we proceed as follows:
**Step 1: Coordinate Setup**
Place \( B \) at the origin \((0,0)\), \( C \) at \((2c,0)\), and \( A \) at \((0,a)\) with \( a < 2c \) (since \( |AB| = a < ... |
1. Let us consider the triangle \( ABC \) with \( |AB| < |BC| \).
2. Let \( M \) be an arbitrary point on the median from vertex \( B \), dividing the triangle into two sections.
3. By the properties of medians, point \( M \) divides the triangle into two regions \( \triangle BAM \) and \( \triangle BCM \).
4. Firs... | math | numina_math | null | null | <think>
Okay, so I need to prove that in triangle ABC where AB is shorter than BC, for any point M on the median from vertex B, the angle BAM is greater than angle BCM. Hmm, let's start by understanding the problem.
First, let me visualize the triangle. ABC, with AB < BC. So, vertex B is connected to A and C, with BC ... |
[
{
"from": "human",
"value": "Translate the curve \\( y = \\log_{2} x \\) rightward by 1 unit along the x-axis and then downward by 2 units along the y-axis to obtain curve \\( C \\). Among the following curves, which one is symmetric to \\( C \\) with respect to the line \\( x + y = 0 \\)?\n\nA. \\( y = 2^{... | Translate the curve \( y = \log_{2} x \) rightward by 1 unit along the x-axis and then downward by 2 units along the y-axis to obtain curve \( C \). Among the following curves, which one is symmetric to \( C \) with respect to the line \( x + y = 0 \)?
A. \( y = 2^{x+2} - 1 \)
B. \( y = -2^{x+2} - 1 \)
C. \( y = -2^... | Okay, let's try to figure out this problem step by step. The question is about transforming a logarithmic curve and then finding its symmetric counterpart with respect to the line x + y = 0. Hmm, symmetry with respect to a line usually means reflection, right? Let me start by understanding each part of the problem.
Fi... | To determine which curve is symmetric to \( C \) with respect to the line \( x + y = 0 \), follow these steps:
1. **Transform the original curve**:
- Start with \( y = \log_{2} x \).
- Shift right by 1 unit: \( y = \log_{2}(x - 1) \).
- Shift down by 2 units: \( y = \log_{2}(x - 1) - 2 \). This is curve \( C ... |
1. **Translation along $x$-axis:**
- First, translate the curve \( y = \log_{2} x \) 1 unit to the right along the $x$-axis.
- This gives the new equation:
\[
y = \log_{2} (x - 1)
\]
2. **Translation along $y$-axis:**
- Next, translate the curve 2 units down along the $y$-axis.
- This modi... | math | numina_math | null | null | <think>
Okay, let's try to figure out this problem step by step. The question is about transforming a logarithmic curve and then finding its symmetric counterpart with respect to the line x + y = 0. Hmm, symmetry with respect to a line usually means reflection, right? Let me start by understanding each part of the prob... |
[
{
"from": "human",
"value": "A recently found Ancient Prophesy is believed to contain the exact Apocalypse date. The prophesy is a string that only consists of digits and characters \"-\".\n\nWe'll say that some date is mentioned in the Prophesy if there is a substring in the Prophesy that is the date's rec... | A recently found Ancient Prophesy is believed to contain the exact Apocalypse date. The prophesy is a string that only consists of digits and characters "-".
We'll say that some date is mentioned in the Prophesy if there is a substring in the Prophesy that is the date's record in the format "dd-mm-yyyy". We'll say tha... | Okay, I need to solve this problem. Let's read the problem statement carefully.
So, the task is to find the correct date (in the format dd-mm-yyyy) that appears the most times as a substring in the prophesy string. The correct date must have a year between 2013 and 2015, valid month (1-12), and valid day for that mont... | To solve this problem, we need to determine the date of the Apocalypse mentioned most frequently in a given string formatted as "dd-mm-yyyy". The valid dates must fall between the years 2013 and 2015, have valid months (1-12), and valid days according to the month. The solution involves identifying all valid dates in t... | null | code | code_contests | {"inputs": ["777-444---21-12-2013-12-2013-12-2013---444-777\n", "12-12-201312-12-201312-12-201313--12-201313--12-201313--12-201313--12-201313--12-201313--12-201313--12-201313--12-2013\n", "01--01--2013-12-2013-01--01--2013\n", "01-04-201425-08-201386-04-201525-10-2014878-04-20102-06-201501-04-2014-08-20159533-45-00-121... | <think>
Okay, I need to solve this problem. Let's read the problem statement carefully.
So, the task is to find the correct date (in the format dd-mm-yyyy) that appears the most times as a substring in the prophesy string. The correct date must have a year between 2013 and 2015, valid month (1-12), and valid day for t... | |
[
{
"from": "human",
"value": "Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset of them (such that each subset is equally likely) and puts them in a bag. You then draw three marbles from the bag with replacement. The colors you see are red, blue, red. What is the... | Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset of them (such that each subset is equally likely) and puts them in a bag. You then draw three marbles from the bag with replacement. The colors you see are red, blue, red. What is the probability that the only marbles in the... | Okay, so Jessica has three marbles: red, green, and blue. She randomly selects a non-empty subset of them, and each subset is equally likely. Then, I draw three marbles with replacement, seeing red, blue, red. I need to find the probability that the only marbles in the bag are red and blue. Hmm, let's break this down s... | To determine the probability that the bag contains only red and blue marbles given the observed draws (red, blue, red), we apply Bayes' theorem.
**Step 1: Identify Possible Subsets**
Jessica selects a non-empty subset from {Red, Green, Blue}. There are 7 equally likely subsets:
1. {Red}, 2. {Green}, 3. {Blue}, 4. {... | :
1. **List Possible Sets:**
Jessica has three marbles: one red, one green, and one blue. She selects a non-empty subset at random. Thus, the possible non-empty subsets are:
\[
\{ \text{red} \}, \{ \text{green} \}, \{ \text{blue} \}, \{ \text{red, green} \}, \{ \text{red, blue} \}, \{ \text{green, blue} \}, ... | math | numina_math | null | null | <think>
Okay, so Jessica has three marbles: red, green, and blue. She randomly selects a non-empty subset of them, and each subset is equally likely. Then, I draw three marbles with replacement, seeing red, blue, red. I need to find the probability that the only marbles in the bag are red and blue. Hmm, let's break thi... |
[
{
"from": "human",
"value": "Let \\(ABC\\) be a triangle such that \\(\\frac{|BC|}{|AB| - |BC|} = \\frac{|AB| + |BC|}{|AC|}\\). Determine the ratio \\(\\angle A : \\angle C\\)."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this geometry problem here: In triangle ABC, the ratio of BC... | Let \(ABC\) be a triangle such that \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\). Determine the ratio \(\angle A : \angle C\). | Alright, so I have this geometry problem here: In triangle ABC, the ratio of BC to (AB minus BC) is equal to the ratio of (AB plus BC) to AC. I need to find the ratio of angle A to angle C. Hmm, okay. Let me try to unpack this step by step.
First, let me write down what's given. The equation is (|BC|)/(|AB| - |BC|) = ... | Given a triangle \(ABC\) with the condition \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\), we need to determine the ratio \(\angle A : \angle C\).
First, we denote the sides as follows:
- \(BC = a\)
- \(AB = c\)
- \(AC = b\)
The given equation translates to:
\[
\frac{a}{c - a} = \frac{c + a}{b}
\]
Cross-mul... |
1. **Define the side lengths**:
Let $|BC| = a$, $|AC| = b$, and $|AB| = c$.
2. **Given condition transformation**:
The given condition is:
\[
\frac{a}{c - a} = \frac{c + a}{b}
\]
Cross-multiplying to clear the fractions:
\[
a \cdot b = (c - a) \cdot (c + a)
\]
Simplifying the r... | math | numina_math | null | null | <think>
Alright, so I have this geometry problem here: In triangle ABC, the ratio of BC to (AB minus BC) is equal to the ratio of (AB plus BC) to AC. I need to find the ratio of angle A to angle C. Hmm, okay. Let me try to unpack this step by step.
First, let me write down what's given. The equation is (|BC|)/(|AB| - ... |
[
{
"from": "user",
"value": "Suppose $a$, $b$, and $c$ are relatively prime integers such that \\[\\frac{a}{b+c} = 2\\qquad\\text{and}\\qquad \\frac{b}{a+c} = 3.\\] What is $|c|$?"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I have this problem here where I need to find the absolute ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \\(a\\) and \\(b\\) from the blackboard and write \\(ab + 1\\). After performing this operation 127 times, only one number is left. Let \\(A\\) be the maximum possible value of this r... | Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\). | Okay, so I need to figure out the last digit of the maximum possible number left after performing this operation 127 times starting with 128 ones. Let me try to break this down step by step.
First, the problem says we have 128 ones on a blackboard. Each step involves erasing two numbers, say a and b, and replacing the... | To determine the last digit of the maximum possible value \( A \) left after performing 127 operations on 128 ones, where each operation replaces two numbers \( a \) and \( b \) with \( ab + 1 \), we need to analyze the process step by step.
1. **Initial Step**: Start with 128 ones. Each pair of ones (1, 1) is combine... |
Given:
- There are 128 ones initially written on a blackboard.
- We can erase any two numbers $a$ and $b$ from the board and write $ab + 1$ instead.
- This process is repeated 127 times until only one number remains on the board.
We are to find the last digit of the maximum possible value of the number left after the... | math | numina_math | null | null | <think>
Okay, so I need to figure out the last digit of the maximum possible number left after performing this operation 127 times starting with 128 ones. Let me try to break this down step by step.
First, the problem says we have 128 ones on a blackboard. Each step involves erasing two numbers, say a and b, and repla... |
[
{
"from": "human",
"value": "\\( f(x) \\) is an odd function, \\( g(x) \\) is an even function, and \\( f(x) + g(x) + f(x) g(x) = \\left(e^x + \\sin x\\right) \\cdot \\cos x \\). Determine the number of zeros of \\( f(x) \\) in the interval \\([0, 4\\pi)\\)."
},
{
"from": "gpt",
"value": "<think... | \( f(x) \) is an odd function, \( g(x) \) is an even function, and \( f(x) + g(x) + f(x) g(x) = \left(e^x + \sin x\right) \cdot \cos x \). Determine the number of zeros of \( f(x) \) in the interval \([0, 4\pi)\). | Okay, let's try to tackle this problem. So we have an odd function f(x) and an even function g(x), and their combination plus their product equals (e^x + sinx) times cosx. We need to find the number of zeros of f(x) in the interval [0, 4π). Hmm, alright, let's start by recalling what odd and even functions are.
An odd... | To determine the number of zeros of the odd function \( f(x) \) in the interval \([0, 4\pi)\), we start with the given equation:
\[
f(x) + g(x) + f(x)g(x) = (e^x + \sin x) \cos x
\]
where \( f(x) \) is odd and \( g(x) \) is even. By substituting \( x \) with \( -x \) and using the properties of odd and even functions... |
1. **Given properties of \( f(x) \) and \( g(x) \):**
- \( f(x) \) is an odd function: \( f(-x) = -f(x) \).
- \( g(x) \) is an even function: \( g(-x) = g(x) \).
2. **Given equation:**
\[
f(x) + g(x) + f(x)g(x) = \left( e^x + \sin x \right) \cos x
\]
3. **Evaluate the function at \( -x \):**
\[
... | math | numina_math | null | null | <think>
Okay, let's try to tackle this problem. So we have an odd function f(x) and an even function g(x), and their combination plus their product equals (e^x + sinx) times cosx. We need to find the number of zeros of f(x) in the interval [0, 4π). Hmm, alright, let's start by recalling what odd and even functions are.... |
[
{
"from": "user",
"value": "3. A two-digit number has the following property: if it is added to the number with the same digits but in reverse order, the result is a perfect square.\n\nFind all such two-digit numbers."
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to find all t... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Find the volume $V$ of a truncated cone with height $h$ and base radii $R$ and $r$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the volume of a truncated cone, also known as a frustum, with height h and base radii R and r. Hmm, let me start by rec... | Find the volume $V$ of a truncated cone with height $h$ and base radii $R$ and $r$. | Okay, so I need to find the volume of a truncated cone, also known as a frustum, with height h and base radii R and r. Hmm, let me start by recalling what a truncated cone is. It's a cone with the top cut off by a plane parallel to the base, right? So, originally, there was a full cone, and then we sliced off the top p... | To find the volume \( V \) of a truncated cone (frustum) with height \( h \) and base radii \( R \) and \( r \), we can use the following steps:
1. **Consider the original and removed cones**:
- The original cone has height \( H \) and radius \( R \).
- The removed smaller cone has height \( H - h \) and radius ... | We are given a frustum of a cone with height $h$ and radii of the bases $R$ and $r$. We need to find the volume $V$ of the frustum.
1. **Volume Formula for a Frustum of a Cone**: The volume \( V \) of a frustum of a cone with height \(h\), larger base radius \(R\), and smaller base radius \(r\) can be found using the ... | math | numina_math | null | null | <think>
Okay, so I need to find the volume of a truncated cone, also known as a frustum, with height h and base radii R and r. Hmm, let me start by recalling what a truncated cone is. It's a cone with the top cut off by a plane parallel to the base, right? So, originally, there was a full cone, and then we sliced off t... |
[
{
"from": "user",
"value": "5.1. A random number sensor outputs a number $a$ - one of the natural numbers $1,2, \\ldots, 100$ (with equal probability). For this value of $a$, we find the maximum possible value $M$ of the function\n\n$$\nf(x)=\\frac{700}{x^{2}-2 x+2 a}\n$$\n\nThe probability that $M>10$ is $... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Given a binary number in the form of string, the task is to check whether the decimal representation of the given binary number is divisible by 10 or not. The number could be very large and may not fit even in long long int.\nExample 1:\nInput:\nbin = \"1010\"\nOutput:\n1\nExpla... | Given a binary number in the form of string, the task is to check whether the decimal representation of the given binary number is divisible by 10 or not. The number could be very large and may not fit even in long long int.
Example 1:
Input:
bin = "1010"
Output:
1
Explanation:
The decimal representation of given
stri... | Okay, I need to write a Python function called isDivisibleBy10 that takes a binary string and checks if its decimal equivalent is divisible by 10. The challenge is that the binary number can be very large, so converting it directly to an integer isn't feasible because it might be way too big to handle. So I need a smar... | To solve this problem, we need to determine if the decimal representation of a given binary string is divisible by 10. Direct conversion of large binary numbers to decimal is impractical due to potential size constraints. Instead, we can use modular arithmetic to efficiently check divisibility by 10 without handling th... | null | code | taco | {"inputs": ["bin = \"1010\"", "bin = \"10\""], "outputs": ["1", "0"]} | #User function Template for python3
class Solution:
def isDivisibleBy10(self,bin):
#code here | <think>
Okay, I need to write a Python function called isDivisibleBy10 that takes a binary string and checks if its decimal equivalent is divisible by 10. The challenge is that the binary number can be very large, so converting it directly to an integer isn't feasible because it might be way too big to handle. So I nee... |
[
{
"from": "user",
"value": "12. $f(x)=a x^{2}+b x+c, a, b, c \\in \\mathbf{R}$, and when $|x| \\leqslant 1$, $|f(x)| \\leqslant 1$, then when $|x| \\leqslant 1$, the maximum value of $|a x+b|$ is $\\qquad$ ."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to tackle this problem ste... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$."
},
... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "2. Let $x, y$ be real numbers such that\n\n$$\nx^{2}+y^{2}=2 \\text{ and }\\left(x^{3}+y^{3}\\right)^{2}=8\n$$\n\nCalculate the value of the expression $x^{4}+y^{4}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of \\( x^4 + y^4 \\) ... | null | null | null | null | null | null | null | null | null |
[
{
"from": "human",
"value": "Let \\( e, f, g \\) be straight lines passing through a point but not lying in the same plane. The projection of \\( e \\) onto the plane of \\( f \\) and \\( g \\), that of \\( f \\) onto the plane of \\( g \\) and \\( e \\), and that of \\( g \\) onto the plane of \\( e \\) an... | Let \( e, f, g \) be straight lines passing through a point but not lying in the same plane. The projection of \( e \) onto the plane of \( f \) and \( g \), that of \( f \) onto the plane of \( g \) and \( e \), and that of \( g \) onto the plane of \( e \) and \( f \) are denoted by \( e_{1}, f_{1}, g_{1} \) respecti... | Okay, let's try to tackle this geometry problem. It seems a bit complicated at first glance, but maybe if I break it down step by step, it'll make more sense. So, we have three lines e, f, g all passing through a single point, but they aren't all in the same plane. That means they form a sort of 3D coordinate system, r... | **Step-by-Step Explanation and Answer:**
To determine if \( FG \) is perpendicular to \( e_1 \), we analyze the geometric constructions and projections involved:
1. **Setup and Projections:**
- Lines \( e, f, g \) pass through the origin and are non-coplanar.
- \( e_1 \) is the projection of \( e \) onto the pl... | 1. **Introduction and Initial Assumptions:**
- Let $e, f, g$ be lines passing through a point $O$ but not lying in the same plane.
- Let the projections of $e$ onto the planes formed by $f$ and $g$, $f$ onto the planes formed by $g$ and $e$, and $g$ onto the planes formed by $e$ and $f$ be represented by $e_1, f_... | math | numina_math | null | null | <think>
Okay, let's try to tackle this geometry problem. It seems a bit complicated at first glance, but maybe if I break it down step by step, it'll make more sense. So, we have three lines e, f, g all passing through a single point, but they aren't all in the same plane. That means they form a sort of 3D coordinate s... |
[
{
"from": "human",
"value": "Point \\( O \\) is located on side \\( AC \\) of triangle \\( ABC \\) such that \\( CO : CA = 2 : 3 \\). When this triangle is rotated by a certain angle around point \\( O \\), vertex \\( B \\) moves to vertex \\( C \\), and vertex \\( A \\) moves to point \\( D \\), which lies... | Point \( O \) is located on side \( AC \) of triangle \( ABC \) such that \( CO : CA = 2 : 3 \). When this triangle is rotated by a certain angle around point \( O \), vertex \( B \) moves to vertex \( C \), and vertex \( A \) moves to point \( D \), which lies on side \( AB \). Find the ratio of the areas of triangles... | Okay, let's see. I need to find the ratio of the areas of triangles BOD and ABC. First, let me try to understand the problem step by step.
We have triangle ABC. Point O is on side AC such that CO:CA is 2:3. So, that means CO is two parts and CA is three parts, right? Wait, no. If the ratio CO:CA is 2:3, then CO is tw... | Point \( O \) is located on side \( AC \) of triangle \( ABC \) such that \( CO : CA = 2 : 3 \). When the triangle is rotated around point \( O \), vertex \( B \) moves to vertex \( C \), and vertex \( A \) moves to point \( D \) on side \( AB \). We need to find the ratio of the areas of triangles \( BOD \) and \( ABC... |
1. **Given Information:**
The problem states that point \( O \) is on side \( AC \) of triangle \( ABC \) such that \( CO:CA = 2:3 \). It also states that when triangle \( ABC \) is rotated around point \( O \), vertex \( B \) moves to \( C \), and vertex \( A \) moves to point \( D \) on side \( AB \). We need... | math | numina_math | null | null | <think>
Okay, let's see. I need to find the ratio of the areas of triangles BOD and ABC. First, let me try to understand the problem step by step.
We have triangle ABC. Point O is on side AC such that CO:CA is 2:3. So, that means CO is two parts and CA is three parts, right? Wait, no. If the ratio CO:CA is 2:3, then ... |
[
{
"from": "human",
"value": "What are the last 8 digits of the product \\( 11 \\times 101 \\times 1001 \\times 10001 \\times 1000001 \\times 111 \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the last 8 digits of the product 11 × 101 × 1001 × 10001 × 1000001 × 111. Hmm, let... | What are the last 8 digits of the product \( 11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111 \)? | Okay, so I need to find the last 8 digits of the product 11 × 101 × 1001 × 10001 × 1000001 × 111. Hmm, let's start by understanding what the problem is asking. The last 8 digits of a number are equivalent to the number modulo 10^8, right? Because any number mod 10^8 gives its last 8 digits. So essentially, I need to co... | The last 8 digits of the product \( 11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111 \) can be determined by calculating the product modulo \( 10^8 \). Here's the step-by-step process:
1. **Multiply sequentially, reducing modulo \( 10^8 \) at each step:**
- \( 11 \times 101 = 1111 \)
- \( 1111 \t... |
To find the last 8 digits of the product \( 11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111 \), we will perform step-by-step calculations and utilize properties of multiplication.
1. Start with the product in its original form:
\[
11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111... | math | numina_math | null | null | <think>
Okay, so I need to find the last 8 digits of the product 11 × 101 × 1001 × 10001 × 1000001 × 111. Hmm, let's start by understanding what the problem is asking. The last 8 digits of a number are equivalent to the number modulo 10^8, right? Because any number mod 10^8 gives its last 8 digits. So essentially, I ne... |
[
{
"from": "user",
"value": "3.2. Let's say that number A hides number B if you can erase several digits from A to get B (for example, the number 123 hides the numbers 1, 2, 3, 12, 13, and 23). Find the smallest natural number that hides the numbers 121, 221, 321, ..., 1921, 2021, 2121.\n\nNatural numbers ar... | null | null | null | null | null | null | null | null | null |
[
{
"from": "user",
"value": "Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the smallest integer k greater than 1 such that there exist k distinct primes wh... | null | null | null | null | null | null | null | null | null |
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