instruction stringlengths 21 6.8k | input stringclasses 1
value | output stringlengths 815 71.5k | system stringclasses 1
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|---|---|---|---|---|
Given the equations \(16x^2 + kx + 9 = 0\) and \(x^2 + 15x + k = 0\), find the value of \(k\). | <think>
Okay, let's tackle this problem. We have two quadratic equations: 16x² + kx + 9 = 0 and x² +15x + k = 0. We need to find the value of k. Hmm, I wonder if these equations have a common solution or something? Maybe they share a root? That might be the case here. Let me think.
If two quadratic equations have a co... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 776188,
"category": "math",
"source": "OpenMathReasoning"
} | |
A permutation p is an ordered group of numbers p1, p2, ..., pn, consisting of n distinct positive integers, each is no more than n. We'll define number n as the length of permutation p1, p2, ..., pn.
Simon has a positive integer n and a non-negative integer k, such that 2k ≤ n. Help him find permutation a of length 2n... | <think>
Okay, let's see. The problem is to find a permutation of length 2n such that the sum of the absolute differences of pairs minus the absolute value of the sum of the differences equals 2k. Hmm. Let's try to understand what the formula means.
Wait, the formula given in the problem is the sum of |a_i - a_{i+1}| f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3679377,
"category": "code",
"source": "OpenCodeReasoning"
} | |
How many different integers can be obtained from the sum of 3 different numbers from the set $\{5, 9, 13, 17, \ldots, 41\}$? | <think>
Okay, so I have this problem here: I need to find how many different integers can be obtained from the sum of three different numbers from the set {5, 9, 13, 17, …, 41}. Hmm, let's see. First, I should figure out what exactly this set is. The numbers start at 5 and each subsequent number increases by 4. Let me ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1426264,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a list of bombs. The range of a bomb is defined as the area where its effect can be felt. This area is in the shape of a circle with the center as the location of the bomb. The bombs are represented by a 0-indexed 2D integer array bombs where bombs[i] = [xi, yi, ri]. xi and yi denote the X-coordinate and ... | <think>
Okay, let's see. I need to solve this problem where I have a list of bombs, each with their coordinates and radius. When I detonate one, it can trigger other bombs in its radius, and those can trigger more bombs in their radii, and so on. The goal is to find the maximum number of bombs that can be detonated by ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1213070,
"category": "code",
"source": "leetcode"
} | |
You are given three strings s, t and p consisting of lowercase Latin letters. You may perform any number (possibly, zero) operations on these strings.
During each operation you choose any character from p, erase it from p and insert it into string s (you may insert this character anywhere you want: in the beginning of... | <think>
Okay, let's see. I need to solve this problem where given three strings s, t, p, I have to determine if it's possible to turn s into t by adding characters from p (removing them from p each time). Hmm.
First, let's think about the conditions that must be met.
So the main idea is that after adding some charact... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3032142,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Find all integer solutions \((x, y, z)\) to the equation \(x^4 + 2x^2 + 1 = (101y + 10z)^2\). | <think>
Okay, so I need to solve this equation: x⁴ + 2x² + 1 = (101y + 10z)². Hmm, let's see. The left side looks like a quadratic in terms of x². Maybe I can factor it or recognize it as a perfect square. Let me check: x⁴ + 2x² + 1. If I let t = x², then it becomes t² + 2t + 1, which factors into (t + 1)². So substitu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2463196,
"category": "math",
"source": "OpenMathReasoning"
} | |
Classify the equilibrium points of the flow associated with the differential equations:
\[
\begin{cases}
x' = x - \pi y \\
y' = \pi x + y
\end{cases}
\] | <think>
Okay, so I need to classify the equilibrium points of the given system of differential equations. Let me start by recalling what equilibrium points are. They are the points where both derivatives x' and y' are zero. So, first step is to find those points by solving the system of equations:
x' = x - πy = 0
y'... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1203917,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider the function \( f_n(x) = \cos^2\left(\frac{n}{2} \arctan x\right) \). Find a rational function \( F_n(x, y) \) in terms of \( x \) and \( y = \sqrt{x^2 + 1} \) such that \( f_n(x) = F_n(x, y) \). For \( n = 4 \), express \( F_4(x, y) \) in its simplest form and calculate the sum of the coefficients of the nume... | <think>
Okay, let's see. I need to find a rational function F_n(x, y) where y is sqrt(x² + 1), such that f_n(x) = cos²(n/2 * arctan x) equals F_n(x, y). Then, specifically for n=4, simplify F_4(x, y) and find the sum of the coefficients of the numerator and denominator.
First, let me recall some trigonometric identiti... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2017502,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let $f(x) = ax^7 + bx^3 + cx - 5.$ If $f(-7) = 7,$ then find $f(7).$ | <think>
Okay, let me try to figure out this problem. So, the function given is f(x) = ax^7 + bx^3 + cx - 5. They told us that f(-7) = 7, and we need to find f(7). Hmm, interesting.
First, let's recall that for polynomials, if there are only odd-powered terms, then the function is odd. An odd function has the property... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3355051,
"category": "math",
"source": "OpenMathReasoning"
} | |
You have matrix a of size n × n. Let's number the rows of the matrix from 1 to n from top to bottom, let's number the columns from 1 to n from left to right. Let's use aij to represent the element on the intersection of the i-th row and the j-th column.
Matrix a meets the following two conditions:
* for any numbe... | <think>
Okay, I need to solve this programming problem. Let's read the question carefully.
So, the problem is about matrices. We have a matrix a of size n x n, and we need to determine if there exists some integer k ≥ 1 such that when we raise matrix a to the k-th power (matrix exponentiation), the resulting matrix (a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1128894,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a Python function to find the minimum number of swaps needed to convert a binary string to have alternating characters.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def min_swap_to_alternate(self, s):
```
Please place the solution code in the following for... | <think>
Okay, I need to solve this problem where I have to find the minimum number of swaps required to convert a binary string into one with alternating characters. The function is part of a class called Solution, and the method is min_swap_to_alternate which takes self and s as parameters.
Hmm. Let's think about th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2917594,
"category": "code",
"source": "opc-sft-stage2"
} | |
Study the continuity of the function $f(x) = xE(x)$ over its domain, where $E(x)$ is the integer part of $x$. | <think>
Okay, let's see. I need to study the continuity of the function f(x) = x times E(x), where E(x) is the integer part of x. Hmm, right, the integer part function, also known as the floor function, which gives the greatest integer less than or equal to x. So, for example, E(2.3) is 2, E(-1.5) is -2, right?
First,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1075744,
"category": "math",
"source": "OpenMathReasoning"
} | |
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$? | <think>
Okay, so I need to find the radius r of a sphere in which a rectangular box P is inscribed. The surface area of the box is 384, and the sum of all its edges is 112. Hmm, let's start by recalling some properties of a rectangular box inscribed in a sphere.
First, if a rectangular box is inscribed in a sphere, t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2149034,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given the definition: For \( x \in \mathbf{R} \), if and only if \( n \leqslant x < n+1 \) where \( n \in \mathbf{N}^{*} \), then \([x] = n\). Find the solution set of the inequality \( 4[x]^{2} - 36[x] + 45 \leqslant 0 \). | <think>
Okay, let's see. I need to solve the inequality 4[x]² - 36[x] + 45 ≤ 0, where [x] is defined such that for any real number x, if n ≤ x < n+1 with n being a positive integer, then [x] = n. So [x] is basically the floor function, right? It takes x and gives the greatest integer less than or equal to x, but since ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2127523,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Express the integral \( I = \int \frac{dx}{\sqrt{1-x^2}} \) as a power series of the form \( I = f(x) \sqrt{1-x^2} \). | <think>
Okay, so I need to express the integral of 1 over the square root of (1 - x squared) dx as a power series that looks like f(x) times sqrt(1 - x squared). Hmm, let's start by recalling some calculus concepts. The integral of 1/sqrt(1 - x²) is a standard one, right? It's arcsin(x) + C. But the problem wants it as... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2734391,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the number of functions \( f \colon \mathbb{Z} \to \{1,2,3,...,228\} \) such that:
1. \( f(0) = 1 \),
2. \( 229 \mid f(a+b) - f(a)f(b) \) for all integers \( a \) and \( b \),
3. \( f(c+16d) = f(c) \) for all integers \( c \) and \( d \). | <think>
Okay, let's try to tackle this problem step by step. So, we need to find the number of functions \( f \colon \mathbb{Z} \to \{1,2,3,...,228\} \) that satisfy three conditions. Let me list them again to make sure I have them right:
1. \( f(0) = 1 \)
2. For all integers \( a \) and \( b \), \( 229 \) divides \( ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2925437,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( P(n) \) be the product of the first \( n \) positive integers, and \( Q(m) \) be the product of the first \( m \) even positive integers. Find the smallest integer \( k \) such that \( P(k) > Q(m) \) for all \( m \geq 2 \). | <think>
Okay, let's see. The problem is asking for the smallest integer k such that the product of the first k positive integers (which is just k factorial, right?) is greater than the product of the first m even positive integers for all m ≥ 2. Hmm. So I need to find the smallest k where P(k) = k! and Q(m) = product o... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3219391,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the maximum value of
\[
\frac{a^{2}b^{2}c^{2}}{(a^2+ab+b^2)(b^2+bc+c^2)(c^2+ca+a^2)}
\]
when \(a + b + c = 0\) and \(a, b, c\) are real numbers. | <think>
Okay, so I need to find the maximum value of this expression:
\[
\frac{a^{2}b^{2}c^{2}}{(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}
\]
given that \( a + b + c = 0 \) and \( a, b, c \) are real numbers. Hmm, interesting. Let's start by understanding the problem step by step.
First, the constraint is \( ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 301953,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to find the largest prime number that is less than or equal to n.
Solve the problem starting with the provided function header.
Function header:
```
def largest_Prime(n):
```
Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to write a Python function called largest_Prime that finds the largest prime number less than or equal to n. Hmm, how to approach this?
First, I remember that a prime number is a number greater than 1 that has no divisors other than 1 and itself. So the largest prime <=n could be n itself if n is ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3586367,
"category": "code",
"source": "opc-sft-stage2"
} | |
What is the largest possible number of distinct positive integers such that no four of them have a sum divisible by 4? | <think>
Okay, so I need to figure out the largest number of distinct positive integers where no four of them add up to a multiple of 4. Hmm, let's start by understanding the problem better.
First, the problem is about selecting as many distinct positive integers as possible, but with the condition that if I pick any ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2900030,
"category": "math",
"source": "OpenMathReasoning"
} | |
For \(a, b, c > 0\), find the minimum value of \(\left\lfloor \frac{a+b}{c} \right\rfloor + \left\lfloor \frac{b+c}{a} \right\rfloor + \left\lfloor \frac{c+a}{b} \right\rfloor\). | <think>
Okay, so I need to find the minimum value of the expression \(\left\lfloor \frac{a+b}{c} \right\rfloor + \left\lfloor \frac{b+c}{a} \right\rfloor + \left\lfloor \frac{c+a}{b} \right\rfloor\) where \(a\), \(b\), and \(c\) are all positive real numbers. Hmm, floor functions can be tricky because they depend on th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2683899,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to find the number of ways to decode a message encoded with a specific encoding scheme. The encoding scheme uses 'A' to 'Z' corresponding to '1' to '26' to encode each letter of the message. The function should return the number of ways the encoded message can be decoded.
Solve the problem star... | <think>
Okay, I need to solve this problem where I have to find the number of ways to decode a message that's encoded using numbers 1 to 26 for letters A to Z. The function is part of a class called Solution, and the method is decodeVariations which takes a string s as input. Hmm, right, so for example, if the input is... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1475194,
"category": "code",
"source": "opc-sft-stage2"
} | |
For a given integer \( N \), let \( p \) be the smallest prime such that there exist \( N \) distinct integers \( k_1, k_2, \ldots, k_N \in (0, p) \) satisfying
\[
\sum_{j=1}^{k_i} j^{-1} \equiv 0 \pmod{p}, \quad i = 1, \ldots, N.
\]
Calculate the sum of all possible values of \( p \) for \( N = 5 \). | <think>
Okay, let's try to tackle this problem. So, we need to find the smallest prime p such that there are 5 distinct integers k₁, k₂, ..., k₅ in the interval (0, p) where the sum of the reciprocals of the integers from 1 to k_i modulo p is congruent to 0. Then, we have to calculate the sum of all possible values of ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1513576,
"category": "math",
"source": "OpenMathReasoning"
} | |
Manao is trying to open a rather challenging lock. The lock has n buttons on it and to open it, you should press the buttons in a certain order to open the lock. When you push some button, it either stays pressed into the lock (that means that you've guessed correctly and pushed the button that goes next in the sequenc... | <think>
Okay, I need to solve this problem where Manao is trying to open a lock with n buttons, and I have to find the maximum number of button presses he might need in the worst case, using the optimal strategy. Let's think about this.
The problem description says that when you press a button, if it's the correct nex... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3011374,
"category": "code",
"source": "OpenCodeReasoning"
} | |
The executive chef is trying to bring some competitive spirit into his kitchen. He wants to split the chefs into two teams based on their age - he'll form the young and the old team. To make it fair, he will split them evenly or give the young team one person advantage when there is an odd number of chefs. Ages of all ... | <think>
Okay, let's tackle this problem. Hmm, the goal is to process each chef as they arrive and, after each arrival, output the absolute difference between the young and old team's ratings. The splitting is based on age, sorted, and split into two teams. But how exactly?
So, the problem says that after each chef arr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2929081,
"category": "code",
"source": "OpenCodeReasoning"
} | |
A curve in the $XY$ plane is given by the parametric equations $x=t^2+t+1$, $y=t^2-t+1$, where the parameter $t$ varies over all non-negative real numbers. Find the number of straight lines passing through the point $(1,1)$ which are tangent to the curve. | <think>
Okay, so I need to find the number of straight lines that pass through the point (1,1) and are tangent to the curve defined by the parametric equations x = t² + t + 1 and y = t² - t + 1, where t is any non-negative real number. Hmm, let's break this down step by step.
First, I remember that a tangent line to a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2074756,
"category": "math",
"source": "OpenMathReasoning"
} | |
In triangle \(ABC\) with circumcircle \((O)\), let \(AO\) intersect \(BC\) at \(D\). The line through \(D\) parallel to \(AB\) intersects \(OC\) at \(E\), and the line through \(D\) parallel to \(AC\) intersects \(OB\) at \(F\). If \(\angle EAF = 2\theta\), find the value of \(\theta\) given that \(\angle BAC = 60^\cir... | <think>
Okay, let me try to solve this geometry problem. Hmm, so we have triangle ABC with a circumcircle (O). AO intersects BC at D. Then there are lines through D parallel to AB and AC, intersecting OC and OB at E and F respectively. We need to find the value of θ where angle EAF is 2θ, given that angle BAC is 60 deg... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3645054,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider the arithmetic sequence $1$, $4$, $7$, $10$, $13$, $\ldots$. Find the $15^{\text{th}}$ term in the sequence. | <think>
Okay, so I have this arithmetic sequence here: 1, 4, 7, 10, 13, and so on. The problem is asking for the 15th term in this sequence. Let me think about how to approach this.
First, I remember that in an arithmetic sequence, each term is obtained by adding a common difference to the previous term. Let me verify... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3423331,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(a, b, c\) be three positive real numbers and \(\zeta > 3\). Find the maximum value of the expression
\[
\frac{a}{\sqrt{a^2 + \zeta bc}} + \frac{b}{\sqrt{b^2 + \zeta ac}} + \frac{c}{\sqrt{c^2 + \zeta ab}}.
\] | <think>
Okay, so I need to find the maximum value of this expression: (a/sqrt(a² + ζ bc)) + (b/sqrt(b² + ζ ac)) + (c/sqrt(c² + ζ ab)), where a, b, c are positive real numbers and ζ is greater than 3. Hmm, let me think.
First, the problem states that ζ > 3, which probably is a key part here. The expression looks symmet... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2497459,
"category": "math",
"source": "OpenMathReasoning"
} | |
Is it possible to rearrange the numbers \(1, 2, 3, \ldots, n\) so that adjacent numbers in the sequence differ by either 3 or 5 if a) \(n = 25\); b) \(n = 1000\)? | <think>
Okay, so I have this problem where I need to determine if it's possible to rearrange the numbers from 1 to n (specifically for n=25 and n=1000) such that any two adjacent numbers in the sequence differ by either 3 or 5. Hmm, interesting. Let me try to figure this out step by step.
First, let me think about wha... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1346142,
"category": "math",
"source": "NuminaMath-CoT"
} | |
For a fixed natural number \( k \), does there exist a Bidiophantine \( n \)-gon (where \( n \geq 3 \)) such that each side or diagonal is equal to \( k \)? If it exists, determine all such \( n \) for \( k = 3 \). | <think>
Okay, let me try to work through this problem step by step. So the question is about whether, for a given natural number k, there exists a "Bidiophantine n-gon" where each side or diagonal is equal to k. And then specifically, if k is 3, determine all such n. Hmm, first, I need to make sure I understand what a ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2285406,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all continuous functions \( f: [0,1] \to \mathbb{R} \) such that \(\int_0^1 f(x) \, dx \ge \frac{1}{3} + \int_0^1 f^2(x^2) \, dx\). | <think>
Okay, so I need to find all continuous functions f from [0,1] to the real numbers that satisfy the integral inequality: the integral of f(x) from 0 to 1 is greater than or equal to 1/3 plus the integral of f squared of x squared from 0 to 1. Hmm, let's write that down to make sure I have it right.
The problem ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1372990,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation:
\[ f(xf(x+y)) = f(yf(x)) + x^2. \] | <think>
Okay, so I need to find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(xf(x+y)) = f(yf(x)) + x^2 \). Hmm, functional equations can be tricky, but let me start by trying some standard approaches. Maybe plugging in specific values for x and y to simplify the equation.
First, let'... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 430919,
"category": "math",
"source": "OpenMathReasoning"
} | |
Seryozha has a very changeable character. This time he refused to leave the room to Dima and his girlfriend (her hame is Inna, by the way). However, the two lovebirds can always find a way to communicate. Today they are writing text messages to each other.
Dima and Inna are using a secret code in their text messages. ... | <think>
Okay, I need to solve this programming problem. Let's read the problem statement carefully.
So, the task is to determine if a given text message could have been generated by Dima's encoding method. The encoding steps are: insert a heart (<3) before each word and after the last word. Then, add random characters... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3790640,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a python function to check if a given string contains any palindrome substrings of length greater than 2.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def find_palindrome_substrings(self, s):
```
Please place the solution code in the following format:
```p... | <think>
Okay, I need to write a Python function called find_palindrome_substrings as part of the Solution class. The goal is to check if the given string s contains any palindrome substrings of length greater than 2. Hmm, right. So the function should return True if there's at least one such substring, otherwise False.... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2737387,
"category": "code",
"source": "opc-sft-stage2"
} | |
Given an integer array nums and an integer k, return the k most frequent elements. You may return the answer in any order.
Write Python code to solve the problem. Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, so I need to solve this problem where I have to find the k most frequent elements in an integer array. Hmm, let me think about how to approach this.
First, the problem says, given an array of integers nums and an integer k, return the k most frequent elements. And the answer can be in any order. Alright... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2956404,
"category": "code",
"source": "leetcode"
} | |
The sum of two positive numbers multiplied by the first of the two is 84. The difference of the two multiplied by the second is 16. Find the largest and smallest possible sums for the two numbers. | <think>
Okay, let's try to solve this problem. Hmm, so we have two positive numbers. Let me call them x and y, where x is the first number and y is the second. The problem says two things:
1. The sum of the two numbers multiplied by the first is 84. So that translates to x*(x + y) = 84.
2. The difference of the two mu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2941346,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve over positive integers: $\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}+\frac{1}{xyz}=1$ | <think>
Okay, so I need to solve this equation: 1/(xy) + 1/(yz) + 1/(xz) + 1/(xyz) = 1, where x, y, z are positive integers. Hmm, let's see. Alright, let's start by trying to simplify the equation. Maybe I can factor something out or find a common denominator. Let's look at each term:
The terms are 1/(xy), 1/(yz), 1/(... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1061942,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a complex number \( z \) such that \( 2 < |z| < 4 \), find the probability that \( 2 < |\Re(z)| + |\Im(z)| < 4 \). | <think>
Okay, let's see. The problem is about finding the probability that a complex number z, which lies in the annulus where the modulus is between 2 and 4, also satisfies that the sum of the absolute values of its real and imaginary parts is between 2 and 4. Hmm, probability here must mean the ratio of the areas of ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3621696,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given positive real numbers \(a, b, c\), define the expression
\[
S = \frac{a(a+b)(b+c)}{a^3+b^2c} + \frac{b(b+c)(c+a)}{b^3+c^2a} + \frac{c(c+a)(a+b)}{c^3+a^2b}.
\]
Calculate the minimum value of \(S\). | <think>
Okay, let's see. I need to find the minimum value of this expression S. It's a sum of three fractions, each with variables a, b, c in both the numerator and denominator. The problem states that a, b, c are positive real numbers, so they can't be zero, but they can be any positive value. The goal is to find the ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2928408,
"category": "math",
"source": "OpenMathReasoning"
} | |
Several consecutive integers sum to 2003. Given that the number of these integers is an odd number greater than 3, determine the smallest of them. | <think>
Okay, so I need to solve this problem where several consecutive integers add up to 2003. The number of integers has to be an odd number greater than 3. And I have to find the smallest of these integers. Hmm, let's think about how to approach this.
First, let me recall that the sum of consecutive integers can b... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 545803,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the polynomial $p(x),$ with real coefficients, such that $p(2) = 5$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\]for all real numbers $x$ and $y.$ | <think>
Okay, so I need to find a polynomial p(x) with real coefficients that satisfies two conditions: first, when x is 2, p(2) equals 5, and second, for all real numbers x and y, the product p(x)p(y) equals p(x) + p(y) + p(xy) minus 2. Hmm, functional equations with polynomials can be tricky, but maybe I can start by... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3335929,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( N \) be the number of complex numbers \( z \) with the properties that \( |z| = 1 \) and \( z^{6!} - z^{5!} \) is a real number. Find the remainder when \( N \) is divided by 1000. | <think>
Okay, let's tackle this problem. So, we need to find the number of complex numbers z such that |z| = 1 and z^(6!) - z^(5!) is a real number. Then, we have to find this number modulo 1000. Hmm, okay.
First, since |z| = 1, z must lie on the unit circle in the complex plane. That means we can represent z as e^(iθ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 977827,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function that implements the Extended Euclidean Algorithm to find the greatest common divisor (GCD) of two integers and the coefficients of Bézout's identity.
Bézout's identity states that for any integers a and b, there exist integers x and y such that:
a*x + b*y = gcd(a, b)
Solve the problem startin... | <think>
Okay, I need to write a Python function called extended_euclidean in the Solution class. The function should return the GCD of two integers a and b, along with the coefficients x and y from Bézout's identity. Hmm, right, the Extended Euclidean Algorithm does this. Let's think about how that works.
First, I rem... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3288012,
"category": "code",
"source": "opc-sft-stage2"
} | |
Find all functions \( f \) from the real numbers to the real numbers such that
\[ f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1 \]
for all \( x, y \) in the real numbers. | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy the equation:
f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1
for all real numbers x and y. Alright, let me start by trying to understand what kind of function this might be. Functional equations often require substitutio... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1622064,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( X = (C[0,1], \parallel \cdot \parallel) \) with \( \parallel x \parallel = \max_{t \in [0,1]} |x(t)| \) and \( U: X \to X \) defined by \( U(x)(s) = \int_{0}^{1} stx(t) \, dt \) for \( s \in [0,1] \). Compute or prove whether the following statements are true or false:
a) \(\parallel U \parallel = ?\) | <think>
Okay, let me try to figure out the norm of the operator U. So, the problem is about the operator U defined on the space C[0,1] with the sup norm. The operator U takes a function x(t) and maps it to the function U(x)(s) which is the integral from 0 to 1 of s*t*x(t) dt. And we need to find the operator norm of U,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2526488,
"category": "math",
"source": "OpenMathReasoning"
} | |
While some participants are busy with their events, some are busy checking out the various Food Stalls. Now, it so happens that some of the participants are very health conscious but they also want to check out the various food stalls. So, they decide to eat only a specific amount of proteins, vitamins, and cholester... | <think>
Okay, so I need to solve this problem where Alex wants to eat exactly certain amounts of proteins, vitamins, and cholesterol by choosing some food items. Each food can be either taken once or not at all. And I have to determine for each test case if it's possible. Hmm.
Let me start by understanding the problem... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 237815,
"category": "code",
"source": "OpenCodeReasoning"
} | |
For each of the following properties, list an example of a c-triangle $[a,b,c]$ satisfying that property.
a. $\operatorname{ccos}(\angle A)=0$
b. $\operatorname{ccos}(\angle A)=2$
c. $\operatorname{ccos}(\angle A)=i$
d. $\operatorname{sqar}[a,b,c]=36$
e. $\operatorname{sqar}[a,b,c]=-36$
f. $\operatorname{sq... | <think>
Okay, let's tackle this problem step by step. So, there are six parts here, all related to c-triangles, which I assume stands for complex triangles. The first three parts (a, b, c) ask for examples where the ccos (complex cosine?) of angle A is 0, 2, and i, respectively. The next three parts (d, e, f) are about... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 641877,
"category": "math",
"source": "OpenMathReasoning"
} | |
In a square \( ABCD \), the point \( P \) is inside such that the distances to vertices \( A, B, \) and \( C \) are \( 2, 3, \) and \( 4 \) respectively. Construct the square and calculate its area. | <think>
Alright, so I have this geometry problem here: In a square ABCD, there's a point P inside such that the distances to vertices A, B, and C are 2, 3, and 4 respectively. I need to construct the square and calculate its area. Hmm, okay. Let's try to figure this out step by step.
First, let me visualize the square... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1646889,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find the last two digits of $2012^{2011^{2010}}$. | <think>
Okay, so I need to find the last two digits of 2012 raised to the power of 2011 raised to 2010. That is, compute 2012^(2011^2010) modulo 100. Hmm, right. Since we're dealing with the last two digits, modulo 100 makes sense. But exponentiating such huge numbers directly is impossible, so I need to use properties... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3190204,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given the sequence defined by $a_{1}=1$ and $a_{n+1}=\frac{\sqrt{1+a_{n}^{2}}-1}{a_{n}}$, find $\lim_{n\to\infty}(2^{n}a_{n})$. | <think>
Okay, let's see. I have this sequence where a₁ is 1, and each next term is defined by aₙ₊₁ = [√(1 + aₙ²) - 1]/aₙ. I need to find the limit of 2ⁿ aₙ as n approaches infinity. Hmm, that sounds a bit tricky, but maybe I can find a pattern or simplify the recursion somehow.
First, let me try to compute the first f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3241252,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a triangle $\triangle ABC$ with circumcircle $\Omega$, excircle $\Gamma$ tangent to segment $BC$ at $T$, and $A$-mixtillinear incircle $\omega$ tangent to $AB$ and $AC$ at $D$ and $E$ respectively. Let $N$ be the midpoint of the arc $BAC$, and $G$ be the intersection of $AT$ and $\Omega$. If the circles $(NDE)$ a... | <think>
Okay, so I have this geometry problem here. Let me try to parse it step by step. The problem involves a triangle ABC with its circumcircle Ω. There's an excircle Γ tangent to BC at T. Then there's the A-mixtilinear incircle ω, which is tangent to AB and AC at D and E respectively. N is the midpoint of the arc B... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3231385,
"category": "math",
"source": "OpenMathReasoning"
} | |
HTML Generator Sample Page
Yesterday night while walking under the street lights, We saw a drunk man. He was taking random steps. But there was something unique with his steps. He would either take a step North or South or East or West as if he was walking on a rectangular grid.
... | <think>
Okay, let's tackle this problem. So the task is to calculate the probability that after N steps, the drunk man never visited the same location more than once. Hmm, right. The input gives the probabilities for each direction (E, W, N, S) and we need to compute this probability for each test case.
First, let me ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 570500,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find
\[\frac{a}{p} + \frac{b}{q}... | <think>
Okay, let's see. I need to solve this geometry problem where a plane intersects the x, y, and z-axes at points A, B, and C respectively, and passes through a fixed point (a, b, c). Then, the sphere passing through A, B, C, and O (the origin) has its center at (p, q, r). I need to find the value of a/p + b/q + c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2724405,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(ABC\) be a triangle with \(A'\) as the midpoint of \(BC\). Let \(1\) be the incircle of \(ABC\), and \(D, E, F\) be the points of contact of \(1\) with \(BC, CA, AB\) respectively. Let \(I\) be the center of \(1\), \(A''\) be the second point of intersection of \(DI\) with \(1\), \(U\) be the intersection of \(DI... | <think>
Okay, so I need to solve this geometry problem. Let me first parse all the given information and see what we're dealing with here.
We have triangle ABC, with A' being the midpoint of BC. The incircle of ABC touches the sides BC, CA, AB at points D, E, F respectively. I is the incenter. Then, A'' is the second ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1370513,
"category": "math",
"source": "OpenMathReasoning"
} | |
What does the graph of $y^3 = x^3 + 9$ look like? Where are its stationary points? | <think>
Okay, so I need to figure out what the graph of the equation y³ = x³ + 9 looks like and find its stationary points. Hmm, let's start by recalling what stationary points are. They're points where the derivative (slope) of the function is zero, right? So, I need to find dy/dx and set it to zero to find those poin... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 761582,
"category": "math",
"source": "OpenMathReasoning"
} | |
Some young pioneers arrive at the camp, and each young pioneer knows between 50 to 100 other young pioneers. Prove that it is possible to give each young pioneer a hat of a certain color, with no more than 1331 different colors in total, such that for each young pioneer, among all the people they know, there are at lea... | <think>
Okay, so I need to figure out how to assign hats to these young pioneers such that each one has a hat color, and the total number of colors doesn't exceed 1331. Also, for each pioneer, the people they know must have at least 20 different hat colors. Each pioneer knows between 50 to 100 others. Hmm, let me start... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2805947,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \( n \) be a non-negative integer that is not a square number. Suppose the first \( m \) decimal places of the decimal part of \( \sqrt{n} \) are all equal to \( u \). If \( u = 0 \) or \( u = 9 \), find the minimum value of \( \sqrt{n} - \frac{10^m}{2} \) for \( m = 3 \). | <think>
Okay, let's try to solve this problem. So, we need to find a non-square integer n such that the first m=3 decimal places of the square root of n are all equal to u, where u is either 0 or 9. Then, we have to find the minimum value of sqrt(n) - (10^m)/2. Since m is 3, 10^m/2 would be 1000/2 = 500. Wait, but sqrt... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2364155,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve the equation $\log 10^{\tan{x}} = \tan{x}$. | <think>
Okay, so I need to solve the equation log(10^{tanx}) = tanx. Hmm, let's start by recalling some logarithm properties. I remember that log(a^b) is equal to b*log(a). Since the logarithm here is base 10, because it's just written as log without any base specified, right? So applying that property, log(10^{tanx}) ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2357191,
"category": "math",
"source": "OpenMathReasoning"
} | |
Iroha has a sequence of N strings S_1, S_2, ..., S_N. The length of each string is L.
She will concatenate all of the strings in some order, to produce a long string.
Among all strings that she can produce in this way, find the lexicographically smallest one.
Here, a string s=s_1s_2s_3...s_n is lexicographically sma... | <think>
Okay, so the problem is to find the lexicographically smallest string by concatenating all the given strings in some order. Hmm, right. Let's see. All the strings are of the same length, L. So the key is to determine the order in which to arrange them so that when you put them together, the result is the smalle... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2625937,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Find the nth term for the sequence 1, 2, 10, 67, 467, 3268, 22876. | <think>
Okay, let's see. I need to find the nth term for this sequence: 1, 2, 10, 67, 467, 3268, 22876. Hmm, it's not immediately obvious what the pattern is. Let me start by looking at the differences between the terms to see if there's a pattern there.
First, the terms are:
a₁ = 1
a₂ = 2
a₃ = 10
a₄ = 67
a₅ = 467
a₆ ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2910974,
"category": "math",
"source": "OpenMathReasoning"
} | |
A merchant accidentally mixed first-grade candies (priced at 3 rubles per pound) with second-grade candies (priced at 2 rubles per pound). At what price should this mixture be sold to yield the same total amount, given that the initial total cost of all first-grade candies was equal to the total cost of all second-grad... | <think>
Okay, let's see. So the problem is about a merchant who mixed first-grade candies with second-grade candies. The first-grade ones are 3 rubles per pound, and the second-grade are 2 rubles per pound. The total cost of all the first-grade candies was equal to the total cost of all the second-grade candies. The qu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2336007,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Chef has $N$ decimal strings $S_{1}, S_{2}, \ldots, S_{N}$ with him (that is, each of these strings contains only characters '0' through '9'). The characters in all strings are indexed starting from $1$. Chef wants to form a single long string $W$ by *interleaving* all these strings.
A string $W$ is said to be formed ... | <think>
Okay, I need to solve this problem where I have to interleave multiple strings to form a single string W with the minimal possible cost. The cost is the sum of the squares of the differences between consecutive characters. Hmm, how do I approach this?
First, let's understand the problem. We have N strings, eac... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3838267,
"category": "code",
"source": "OpenCodeReasoning"
} | |
In a quadrilateral \(ABCD\) with an incircle centered at \(O\), the tangency points of the incircle to the sides \(AB\), \(BC\), \(CD\), and \(DA\) are \(E\), \(F\), \(G\), and \(H\) respectively. Let \(P\) be the point of concurrency of the lines \(BG\), \(DF\), and \(AC\). If the lengths of the segments \(AE\), \(EB\... | <think>
Okay, let's see. I need to find the length of segment OP in this quadrilateral ABCD that has an incircle. The incircle touches each side at points E, F, G, H. Then, lines BG, DF, and AC all meet at point P. All those tangent segment lengths (AE, EB, BF, FC, etc.) are integers, and the area is 120. Hmm.
First, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 985353,
"category": "math",
"source": "OpenMathReasoning"
} | |
Parametrize the boundary curve \( C = \partial S \) of the surface \( S: \frac{x^2}{9} + \frac{y^2}{9} + \frac{z^2}{16} = 1 \) where \( z \leq 0 \). | <think>
Okay, let's see. The problem is asking me to parametrize the boundary curve C of the surface S, where S is defined by the equation x²/9 + y²/9 + z²/16 = 1 with z ≤ 0. So, first, I need to understand what this surface looks like and what the boundary of it would be.
Wait, the equation x²/9 + y²/9 + z²/16 = 1. T... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 126887,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the sum \( S = \frac{C^{12}_{12}}{11 \cdot 12} + \frac{C^{12}_{13}}{12 \cdot 13} + \cdots + \frac{C^{12}_{2015}}{2014 \cdot 2015} + \frac{C^{12}_{2016}}{2015 \cdot 2016} \). | <think>
Okay, so I need to evaluate this sum S that's given by a series of terms from k=12 to k=2016. Each term is a combination C(12, k) divided by (k-1)*k. Let me write that down more clearly:
S = Σ [C(12, k) / (k(k-1))] from k=12 to k=2016.
Hmm, first, I should recall what C(n, k) is. That's the combination formul... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3476461,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the integral $\int \frac{x}{1+\cos^2 x} \, dx$. | <think>
Okay, so I need to evaluate the integral of x divided by (1 + cos²x) dx. Hmm, that looks a bit tricky. Let me think about how to approach this.
First, I remember that integrals involving trigonometric functions can sometimes be simplified using substitutions or identities. The denominator here is 1 + cos²x. M... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 678221,
"category": "math",
"source": "OpenMathReasoning"
} | |
Prove that if the polynomials \( P(x) \), \( Q(x) \), and \( R(x) \) of degree three with real coefficients satisfy the inequalities \( P(x) \leq Q(x) \leq R(x) \) for all real \( x \), and there is at least one point \( x_0 \in \mathbb{R} \) such that \( P(x_0) = R(x_0) \), then there exists a number \( k \in [0, 1] \... | <think>
Okay, so I need to prove that if three cubic polynomials P(x), Q(x), and R(x) with real coefficients satisfy P(x) ≤ Q(x) ≤ R(x) for all real x, and there's at least one point x₀ where P(x₀) = R(x₀), then Q(x) is a convex combination of P(x) and R(x). That is, Q(x) ≡ kP(x) + (1 - k)R(x) for some k in [0,1]. Then... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2348251,
"category": "math",
"source": "NuminaMath-CoT"
} | |
In the plane, let $AB_1C_1D_1$ and $AB_2C_2D_2$ be two squares having a common vertex $A$ (vertices are cited in the same sense). If $B$, $C$, and $D$ are respectively the midpoints of the segments $[B_1B_2]$, $[C_1C_2]$, and $[D_1D_2]$, find the ratio of the area of the quadrilateral $ABCD$ to the area of square $AB_1... | <think>
Okay, let's see. I need to solve this geometry problem where there are two squares, AB₁C₁D₁ and AB₂C₂D₂, sharing a common vertex A. The vertices are cited in the same sense, which probably means they're both oriented the same way, like both clockwise or both counterclockwise. Then, B, C, D are midpoints of the ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2661358,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to generate all permutations of length n from a given list.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def perm_gen_lex(self, input_list, n):
```
Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to write a Python function called perm_gen_lex which is a method of the Solution class. The function takes an input_list and an integer n, and it should generate all permutations of length n from that list. Hmm, let's think about how to approach this.
Wait, the function header says perm_gen_lex. L... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1792269,
"category": "code",
"source": "opc-sft-stage2"
} | |
In a triangle, one side is three times smaller than the sum of the other two sides. Prove that the angle opposite to this side is the smallest angle of the triangle. | <think>
Okay, so I need to prove that in a triangle where one side is three times smaller than the sum of the other two sides, the angle opposite that side is the smallest angle. Hmm, let's start by understanding the problem.
First, let's denote the triangle as ABC, with sides a, b, and c opposite angles A, B, and C r... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 997236,
"category": "math",
"source": "NuminaMath-CoT"
} | |
How many m-digit systems can be formed using the digits 1, 2, and 3 such that no two adjacent digits are the same? | <think>
Okay, so I need to figure out how many m-digit numbers can be formed using the digits 1, 2, and 3 with the condition that no two adjacent digits are the same. Let me break this down step by step.
First, let's consider a simple case. Suppose m is 1. Then, there are 3 possible numbers: 1, 2, or 3. That makes sen... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1572306,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the integral \(\int_{-1}^{1} h(x) \, dx\) where \(h(x) = \begin{cases}
\frac{1}{x^2} & \text{if } x \in [-1,1] \setminus \{0\} \\
0 & \text{if } x = 0
\end{cases}\). | <think>
Okay, so I have this integral to solve: the integral from -1 to 1 of h(x) dx, where h(x) is defined as 1/x² when x is in [-1,1] but not zero, and 0 when x is zero. Hmm, let me think about this step by step.
First, I remember that integrals over intervals where the function has points of discontinuity might be ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 328216,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider a ring $R$ with unity, where $R$ is an $m$-ring. If $R$ is a Boolean ring, find the smallest possible value of $m$ such that $m$ is even and $m-1$ has no prime factor of the form $2^p - 1$. | <think>
Okay, let's try to figure out this problem. So, we need to find the smallest even integer m such that R is an m-ring, R is a Boolean ring with unity, and m-1 has no prime factor of the form 2^p - 1. Hmm, okay. Let me start by recalling what each of these terms means.
First, a Boolean ring is a ring where every... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1798090,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(a, b, c\) be positive real numbers. Find the minimum value of the expression:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{ab + bc + ca}{a^2b + b^2c + c^2a}
\] | <think>
Okay, so I need to find the minimum value of the expression (1/a + 1/b + 1/c) + (ab + bc + ca)/(a²b + b²c + c²a), where a, b, c are positive real numbers. Hmm, let's think about how to approach this.
First, the expression has two parts: the sum of reciprocals and a fraction involving products of variables. Sin... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2738279,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that:
\[ f(x^2) = (f(x))^2 \]
\[ f(x+1) = f(x) + 1 \] | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy two conditions: first, that f of x squared equals f of x squared, which sounds a bit circular, but actually, wait, let me parse that again. The problem says f(x²) = (f(x))². So for any real number x, when you square x... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3467583,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given \( n \) points in the plane, prove that the shortest polygonal chain with vertices at these points does not have self-intersections. | <think>
Okay, so I need to prove that the shortest polygonal chain connecting n points in the plane doesn't have any self-intersections. Hmm, let me think about how to approach this.
First, let me recall what a polygonal chain is. It's a connected series of line segments, right? And a self-intersection would be where... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2451464,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Calculate the integral \( J = \int \frac{dx}{1 + \tan x} \). | <think>
Okay, so I need to solve the integral J = ∫ dx/(1 + tan x). Hmm, let's think. I remember that integrating functions with trigonometric expressions can sometimes be tricky, but maybe there's a substitution or a trick to simplify it.
First, let me recall that tan x is sin x over cos x. So, maybe I can rewrite t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1663823,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the limit \(\lim_{n\to\infty}{n\left(\left(1+\frac{1}{n}\right)^{n+1}-\mathrm{e}\right)}\). | <think>
Okay, so I need to find the limit as n approaches infinity of n times the quantity (1 + 1/n)^(n+1) minus e. Hmm, let me think. I remember that (1 + 1/n)^n approaches e as n goes to infinity. But here we have (1 + 1/n)^(n+1), which is like multiplying that by another (1 + 1/n) term. So maybe (1 + 1/n)^(n+1) is c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1120930,
"category": "math",
"source": "OpenMathReasoning"
} | |
In this task, you are required to implement the Gauss-Seidel method to solve a system of linear equations of the form `A * x = b`. The Gauss-Seidel method is an iterative technique for solving a square system of \( n \) linear equations with \( n \) unknowns.
Given:
- `A` is an \( n \times n \) matrix of coefficients... | <think>
Okay, I need to implement the Gauss-Seidel method to solve the system of linear equations A*x = b. The stopping condition is when the Euclidean distance between consecutive iterations is less than or equal to eps. Hmm, let's think about how the Gauss-Seidel method works.
First, I remember that Gauss-Seidel is ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2945270,
"category": "code",
"source": "opc-sft-stage2"
} | |
Alice and Bob play a game where they take turns drawing line segments of length 1 in the Euclidean plane. Alice starts and draws the first segment. Each subsequent segment must start from the endpoint of the previously drawn segment. It is not allowed to draw a segment that overlaps with the previously drawn segment. I... | <think>
Okay, let's try to figure out this problem. So Alice and Bob are taking turns drawing line segments of length 1 in the plane. Alice starts, and each new segment has to start where the previous one ended. Also, you can't draw a segment that overlaps with the previous one. If a new segment intersects any of the p... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1232716,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Bear Limak likes watching sports on TV. He is going to watch a game today. The game lasts 90 minutes and there are no breaks.
Each minute can be either interesting or boring. If 15 consecutive minutes are boring then Limak immediately turns TV off.
You know that there will be n interesting minutes t1, t2, ..., tn. Yo... | <think>
Okay, let's see. I need to solve this problem where Limak watches a game that's 90 minutes long. The problem says that if there are 15 consecutive boring minutes, he turns off the TV immediately. But the interesting minutes are given, so the rest are boring. So I need to find out how many minutes he watches bef... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2052332,
"category": "code",
"source": "OpenCodeReasoning"
} | |
An infinite sequence \( x_1, x_2, \ldots \) of positive integers satisfies \[ x_{n+2} = \gcd(x_{n+1}, x_n) + 2006 \] for each positive integer \( n \). Does there exist such a sequence which contains exactly \( 10^{2006} \) distinct numbers? | <think>
Okay, let's try to figure out this problem. So we have an infinite sequence of positive integers where each term after the second is the greatest common divisor (gcd) of the previous two terms plus 2006. The question is whether such a sequence can exist that contains exactly 10^{2006} distinct numbers. Hmm, tha... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2677198,
"category": "math",
"source": "OpenMathReasoning"
} | |
Yukiya, the owner of Aizuyama Ski Resort, has prepared a course for advanced skiers with obstacles and jumping hills. There are various ways to slide on the course, and users who can slide in all patterns during the season will be given a gift.
Let's create a program for the oil tree shop that outputs the number of pa... | <think>
Okay, I need to solve this programming problem. Let's read the problem statement carefully.
So the task is to calculate the number of possible sliding paths through a grid that represents a ski course. The movement rules involve possible jumps due to jumping platforms and obstacles that block certain cells.
L... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3493396,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a function to find the number of triplets in a list that form a geometric progression with a given common ratio.
Solve the problem starting with the provided function header.
Function header:
```
def count_geometric_triplets(arr, r):
```
Please place the solution code in the following format:
```python
# Your ... | <think>
Okay, I need to write a function called count_geometric_triplets that takes an array and a common ratio r, and returns the number of triplets in the array that form a geometric progression. Hmm, a geometric progression triplet would be three elements a, b, c such that b = a*r and c = b*r. And the order matters?... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1712680,
"category": "code",
"source": "opc-sft-stage2"
} | |
Frantisek Nudilek was writing consecutive natural numbers in the sequence: 1234567891011... At some point, he finished the number he was writing and noticed that there are five consecutive ones in the sequence of digits he wrote.
1. What is the minimum number of consecutive natural numbers Frantisek had to write?
2. W... | <think>
Okay, so Frantisek is writing consecutive natural numbers like 1, 2, 3, ..., and so on, right? He's concatenating them into a long string: 123456789101112... and so forth. At some point, he notices that there are five consecutive ones ('11111') in the digits he's written. The problem has two parts: first, find ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1980871,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? Express your answer as a common fraction. | <think>
Okay, so the problem is: Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? Hmm, let's see. Alright, probability questions can sometimes be tricky, but let's break it down step by step.
First, I need to recall that each coin flip is an independent ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3858235,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to find the maximum XOR value of any subarray within the given array.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def maxSubarrayXOR(self, arr, n):
```
Please place the solution code in the following format:
```python
# Your solution cod... | <think>
Okay, I need to find the maximum XOR of any subarray in the given array. Hmm, how do I approach this? Let's think about subarrays. A subarray is a contiguous part of the array. For each possible subarray, I can compute the XOR and keep track of the maximum. But wait, if the array is large, say n is 10^5, then c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 192509,
"category": "code",
"source": "opc-sft-stage2"
} | |
Find the $n$th term of a non-recursive sequence whose first terms are: 1, 5, 14, 30, 55, 91, ... Assume that $a_1 = 1$, $a_2 = 5$, ... | <think>
Okay, so I need to find the nth term of this sequence: 1, 5, 14, 30, 55, 91,... Let's see. The user mentioned it's a non-recursive sequence, so I can't just look for a recurrence relation. I need an explicit formula for the nth term. Let me start by writing down the terms and their positions to see if I can spo... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1795608,
"category": "math",
"source": "OpenMathReasoning"
} | |
Determine all the 3-regular graphs $G$ with the minimum number of vertices such that $g(G)=5$, where $g(G)$ is the girth of the graph $G$ (i.e., the length of the shortest cycle in $G$). | <think>
Okay, so I need to find all the 3-regular graphs with the minimum number of vertices where the girth is 5. Hmm, let me start by recalling some definitions. A 3-regular graph, also known as a cubic graph, is a graph where every vertex has degree 3. The girth is the length of the shortest cycle in the graph. So, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 787854,
"category": "math",
"source": "OpenMathReasoning"
} | |
Richard is planning to build a bowling alley for his gaming zone.Help Richard build it by coding a scoreboard for the same.
Details of this scoreboard:
A player will be given three chances to hit the bottles ,after every hit 9 new bottles will be replaced.
Player scores 10 points with a 9/9 hit out of the given three... | <think>
Okay, I need to solve this problem where I have to calculate the final score of a bowling alley game based on the given rules. Let's see the problem statement again to make sure I understand all the conditions.
The problem says that a player has three chances (A, B, C) to hit the bottles. After each hit, 9 new... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3008866,
"category": "code",
"source": "OpenCodeReasoning"
} | |
If the equation $2x + 2y + z = n$ has 28 solutions in positive integers $x, y, z$, then $n$ must be either:
A) 14 or 15
B) 15 or 16
C) 16 or 17
D) 17 or 18
E) 18 or 19 | <think>
Okay, so I need to solve this problem where the equation 2x + 2y + z = n has 28 solutions in positive integers x, y, z, and figure out what n must be. The options are given as A) 14 or 15, B) 15 or 16, etc. Let me think step by step.
First, the equation is 2x + 2y + z = n. All variables x, y, z are positive in... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 739122,
"category": "math",
"source": "OpenMathReasoning"
} | |
In an acute triangle, \( h \) is the length of the altitude drawn to the side of length \( c \), and \( a \) and \( b \) are the lengths of the other two sides. Given that \( c < h\sqrt{2} \), calculate the maximum possible value of \( \frac{a^2 + b^2 - c^2}{h^2} \). | <think>
Okay, so I need to find the maximum possible value of the expression (a² + b² - c²)/h² given that in an acute triangle, c is less than h√2. Hmm, let me start by recalling some triangle properties.
First, in any triangle, the area can be expressed as (base * height)/2. Here, since h is the altitude to side c, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1698011,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), the equality holds:
\[ f(f(x)f(y) + f(xy)) + f(x + y + 1) = f(2xy) \] | <think>
Okay, so I need to find all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation:
\[ f(f(x)f(y) + f(xy)) + f(x + y + 1) = f(2xy) \]
for all real numbers \( x \) and \( y \). Hmm, functional equations can be tricky, but maybe I can start by plugging in specific values for \( x \) and \( y \) ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2303856,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the power series representation of the function \( f(x) = \frac{x^5}{(x-1)^2} \). | <think>
Okay, so I need to find the power series representation for the function f(x) = x⁵/(x - 1)². Hmm, power series... right. Let me recall how to approach this. I think power series expansions often start with a known series and then manipulate it. The function here is x⁵ divided by (x - 1) squared.
First, maybe ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3751970,
"category": "math",
"source": "OpenMathReasoning"
} | |
On an $8 \times 8$ chessboard, we can repaint the squares such that in one move the color of all squares in one row and one column (a total of 15 squares) is changed. If all squares were initially white, is it possible to achieve the usual chessboard coloring pattern? | <think>
Okay, so I have this problem about an 8x8 chessboard. The question is whether it's possible to achieve the usual chessboard coloring pattern starting from an all-white board by repainting squares. The catch is that each move allows me to change the color of all squares in one row and one column, totaling 15 squ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 991753,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction. | <think>
Okay, let's see. The problem is about finding the probability that when two different numbers are selected from the set {1,2,3,4,5,6,7}, their positive difference is 2 or greater. Hmm, right. So first, I need to figure out the total number of possible pairs, and then the number of pairs that meet the condition,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2152914,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given an integer N print total number of divisors of N. It is guaranteed that N is a multiple of prime numbers less than 10^6.
Input
First line contains T (number of test cases).
Next T lines contains integer N.
Constraints
1 ≤ T ≤ 100
1 ≤ N ≤ 10^18
SAMPLE INPUT
6
1
2
3
4
5
20
SAMPLE OUTPUT
1
2
2
3
2
6
Write ... | <think>
Okay, I need to solve this problem where, given a number N, I have to find the total number of its divisors. The constraints are that N can be up to 1e18, but it's guaranteed that N is a multiple of primes less than 1e6. Hmm. Let's think.
So, the standard approach for finding the number of divisors is to facto... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 647040,
"category": "code",
"source": "OpenCodeReasoning"
} | |
What is the value of $$\lim_{n\to\infty}\frac{e_p(n)}{e_q(n)}$$ for primes $p$ and $q$, where $e_p(n)$ is the exponent of the prime $p$ in the prime factorization of $n!$? | <think>
Okay, so I need to find the limit as n approaches infinity of e_p(n) divided by e_q(n), where e_p(n) and e_q(n) are the exponents of the primes p and q in the prime factorization of n! Hmm. Alright, let me start by recalling how exponents of primes in n! are calculated.
I remember that there's a formula for f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3930982,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( p \) and \( q \) be positive prime numbers such that \( p + q \) and \( p + 7q \) are perfect squares. Find all possible values of \( pq \). | <think>
Okay, let's tackle this problem where we need to find all possible values of \( pq \) given that \( p \) and \( q \) are positive primes, and both \( p + q \) and \( p + 7q \) are perfect squares. Hmm, interesting. Let me start by writing down what's given.
So, we have two equations:
1. \( p + q = a^2 \) for ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1887898,
"category": "math",
"source": "OpenMathReasoning"
} |
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