instruction stringlengths 21 6.8k | input stringclasses 1
value | output stringlengths 815 71.5k | system stringclasses 1
value | extra_info dict |
|---|---|---|---|---|
In an isosceles trapezoid with perpendicular diagonals, find the distance from the center of the circumscribed circle to the point of intersection of the diagonals, given that the lengths of the bases are $a$ and $b$. | <think>
Okay, so I need to solve this problem about an isosceles trapezoid with perpendicular diagonals. The question is asking for the distance from the center of the circumscribed circle to the point where the diagonals intersect. The bases have lengths a and b. Hmm, let me try to visualize this first.
An isosceles ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 204457,
"category": "math",
"source": "OpenMathReasoning"
} | |
Levko loves permutations very much. A permutation of length n is a sequence of distinct positive integers, each is at most n.
Let’s assume that value gcd(a, b) shows the greatest common divisor of numbers a and b. Levko assumes that element pi of permutation p1, p2, ... , pn is good if gcd(i, pi) > 1. Levko considers ... | <think>
Okay, let's see. We need to find a permutation of numbers from 1 to n such that exactly k elements are "good". A good element is one where the gcd of its index (i) and the value (p_i) is greater than 1. If there's no such permutation, we return -1.
Hmm. So first, the constraints. For each position i in the per... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1844637,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Consider the set \( X = (\mathbb{Q} \times \{0\}) \cup ((\mathbb{R} \setminus \mathbb{Q}) \times \{1\}) \) with the metric \( d((x,y), (u,v)) = |x - u| \). Calculate the sum of all possible values of \( d((\sqrt{2}, 1), (q, 0)) \) for \( q \in \mathbb{Q} \) such that \( d((\sqrt{2}, 1), (q, 0)) < 1 \). | <think>
Okay, let's try to figure out this problem. So we have this set X, which is a union of two parts: all the rational numbers paired with 0, and all the irrational numbers paired with 1. The metric d between two points (x,y) and (u,v) is just the absolute difference of their first components, right? So the distanc... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 343007,
"category": "math",
"source": "OpenMathReasoning"
} | |
What is the geometric mean of all the real numbers between 1 and 2? | <think>
Okay, so I need to find the geometric mean of all the real numbers between 1 and 2. Hmm, geometric mean usually is for a set of numbers, right? But here, it's all the real numbers between 1 and 2. That's an infinite set, so the regular method of multiplying all numbers and taking the nth root won't work because... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3405139,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the derivative of the function.
\[ y = \frac{1}{18 \sqrt{2}} \ln \frac{1 + \sqrt{2} \operatorname{cth} x}{1 - \sqrt{2} \operatorname{cth} x} \] | <think>
Okay, let's see. I need to find the derivative of this function y. The function is given as y equals 1 divided by (18 times the square root of 2) multiplied by the natural logarithm of (1 plus sqrt(2) coth x) over (1 minus sqrt(2) coth x). Hmm, that looks a bit complicated, but maybe I can break it down step by... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1444812,
"category": "math",
"source": "NuminaMath-CoT"
} | |
In the city of Saint Petersburg, a day lasts for 2^{100} minutes. From the main station of Saint Petersburg, a train departs after 1 minute, 4 minutes, 16 minutes, and so on; in other words, the train departs at time 4^k for each integer k ≥ 0. Team BowWow has arrived at the station at the time s and it is trying to co... | <think>
Okay, let's see. The problem is about counting how many trains have departed before a given time s. The trains leave at times that are powers of 4. Wait, right, because 4^k. So 1 (4^0), 4 (4^1), 16 (4^2), 64 (4^3), and so on.
The input is given as a binary string. So first, I need to process this binary number... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1931800,
"category": "code",
"source": "OpenCodeReasoning"
} | |
In this kata you will be given a sequence of the dimensions of rectangles ( sequence with width and length ) and circles ( radius - just a number ).
Your task is to return a new sequence of dimensions, sorted ascending by area.
For example,
```python
seq = [ (4.23, 6.43), 1.23, 3.444, (1.342, 3.212) ] # [ rectangle... | <think>
Okay, I need to solve this problem where I have a sequence of rectangle dimensions and circles, and I have to sort them by their area in ascending order. Hmm. Let me think about how to approach this.
First, I need to determine for each element in the sequence whether it's a rectangle or a circle. The rectangle... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3609291,
"category": "code",
"source": "TACO"
} | |
Write a Python function to find the maximum length of a strictly increasing subsequence in a given list of integers. A strictly increasing subsequence is a sequence of numbers where each number is greater than the previous number.
Solve the problem starting with the provided function header.
Function header:
```
clas... | <think>
Okay, I need to write a Python function to find the maximum length of a strictly increasing subsequence in a given list of integers. The function is part of a Solution class. Hmm, how do I approach this?
Wait, I remember that there's a standard problem called the Longest Increasing Subsequence (LIS), which thi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1968034,
"category": "code",
"source": "opc-sft-stage2"
} | |
A square pyramid of balls consists of square layers of balls stacked on top of each other. The i th (1-based indexing )layer from the top consists of exactly i^2 balls. Image
You have received one such beautiful square pyramid on your birthday, with each layer having a unique color. However, being the clumsy doofus yo... | <think>
Okay, I need to solve this programming problem. Let's read the problem carefully.
The problem is about a square pyramid made of balls. Each layer from the top (1-based index) has i² balls. So the first layer has 1²=1, second 2²=4, third 3²=9, and so on. But when the pyramid was knocked over, some balls from ea... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3791248,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \( f : \mathbb{R} \to \mathbb{R} \) be a function satisfying the conditions:
\[ f(0) = f(1) = 0 \]
\[ \min_{x \in [0,1]} f(x) = -1 \]
Find \( \min(\max_{x \in [0,1]} f''(x)) \). | <think>
Okay, so I need to find the minimum possible value of the maximum of the second derivative of a function f that goes from R to R. The function has to satisfy f(0) = f(1) = 0, and its minimum on the interval [0,1] is -1. Hmm, right. Let me break this down.
First, the function is defined on all real numbers, but... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3711428,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider an infinite sequence \( x_1, x_2, \dots \) of positive integers such that, for every integer \( n \geq 1 \):
- If \( x_n \) is even, \( x_{n+1} = \frac{x_n}{2} \);
- If \( x_n \) is odd, \( x_{n+1} = \frac{x_n - 1}{2} + 2^{k-1} \), where \( 2^{k-1} \leq x_n < 2^k \).
Determine the smallest possible value of \... | <think>
Okay, so I need to find the smallest starting number x₁ such that the sequence defined by the given rules includes the number 2020. Hmm, let's try to understand the problem step by step.
First, the sequence is defined by these two rules:
1. If xₙ is even, then xₙ₊₁ = xₙ / 2.
2. If xₙ is odd, then xₙ₊₁ = (xₙ -... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1846125,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given that \( f(x) = x^2 + 4 \) and \( f(y) = x^2 + 23 \) for positive integers \( x \) and \( y \), find \( f(x+y) \). | <think>
Okay, let's see. I need to solve this problem where we have two functions f(x) and f(y) defined as x² + 4 and x² + 23, respectively. Wait, hold on. Let me read the problem again carefully.
The problem states: Given that f(x) = x² + 4 and f(y) = x² + 23 for positive integers x and y, find f(x+y).
Hmm, okay. So... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 959065,
"category": "math",
"source": "OpenMathReasoning"
} | |
Develop a Python function that accepts an array as an input parameter and calculates the total quantity of elements contained within it.
Write Python code to solve the problem. Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to create a Python function that takes an array as input and returns the total number of elements in it. Let me think about how to approach this.
First, I remember that in Python, there's a built-in function called len() which gives the length of a list. So maybe the function is as simple as retur... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2338031,
"category": "code",
"source": "MagicoderEvolInstruct"
} | |
There was a big bank robbery in Tablecity. In order to catch the thief, the President called none other than Albert – Tablecity’s Chief of Police. Albert does not know where the thief is located, but he does know how he moves.
Tablecity can be represented as 1000 × 2 grid, where every cell represents one district. Eac... | <think>
Okay, let's see. So the problem is to find a strategy for the police to check two districts every hour such that no matter where the thief starts and how he moves each hour, he will be caught within 2015 hours. The thief can move to one of six adjacent districts each hour, as long as they exist in the 1000x2 gr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3773198,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Sereja has an array a, consisting of n integers a1, a2, ..., an. The boy cannot sit and do nothing, he decided to study an array. Sereja took a piece of paper and wrote out m integers l1, l2, ..., lm (1 ≤ li ≤ n). For each number li he wants to know how many distinct numbers are staying on the positions li, li + 1, ...... | <think>
Okay, so I need to solve this programming problem. Let's see what the problem says. The task is, given an array and multiple queries, each asking for the number of distinct elements from a position li to the end of the array. For each query li, I have to compute how many distinct elements are there from index l... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1893566,
"category": "code",
"source": "OpenCodeReasoning"
} | |
John has \( x \) red dogs and 5 blue dogs. If the probability that he picks 2 dogs of the same color is \( \frac{5}{11} \), what is \( x \)? | <think>
Okay, let me try to figure out this problem. So John has x red dogs and 5 blue dogs. We need to find x such that the probability of picking two dogs of the same color is 5/11. Hmm, probability problems can sometimes be tricky, but let's break it down step by step.
First, let's understand the problem. There are... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 350057,
"category": "math",
"source": "OpenMathReasoning"
} | |
Mole is hungry again. He found one ant colony, consisting of n ants, ordered in a row. Each ant i (1 ≤ i ≤ n) has a strength si.
In order to make his dinner more interesting, Mole organizes a version of «Hunger Games» for the ants. He chooses two numbers l and r (1 ≤ l ≤ r ≤ n) and each pair of ants with indices betwe... | <think>
Okay, let's see. The problem is about ants fighting in a segment, and we need to find out how many ants Mole eats after each query. The key is that an ant is freed if its battle points equal the number of other ants in the segment. Because each fight between two ants in the segment counts. Wait, the battle poin... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2749890,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given the function $f(x) = n^2 + 3n + 2$, find the value of the product $\left(1- \frac{2}{f(1)}\right) \left(1- \frac{2}{f(2)}\right)\left(1- \frac{2}{f(3)}\right) \cdots \left(1- \frac{2}{f(n)}\right) = k$. | <think>
Okay, let's see. I need to find the value of the product from 1 to n of (1 - 2/f(k)), where f(k) is k² + 3k + 2. Hmm, first maybe I should figure out what f(k) simplifies to. Let me compute f(k) first.
So f(k) = k² + 3k + 2. Wait, that quadratic can be factored, right? Let me check: k² + 3k + 2. The factors of... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3625281,
"category": "math",
"source": "OpenMathReasoning"
} | |
If \(xyz = x^3 + y^3 + z^3 = 4\) and \(xy^2 + x^2y + yz^2 + y^2z + zx^2 + z^2x = 12\), find the real value of \(xy + yz + zx\). | <think>
Okay, so I have this algebra problem here. Let me try to understand it step by step. The problem says:
If \(xyz = 4\) and \(x^3 + y^3 + z^3 = 4\), and also \(xy^2 + x^2y + yz^2 + y^2z + zx^2 + z^2x = 12\), find the real value of \(xy + yz + zx\).
Hmm, I need to find \(xy + yz + zx\). Let me recall some algebr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1700876,
"category": "math",
"source": "OpenMathReasoning"
} | |
There are 5 blue chips and 3 yellow chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction. | <think>
Okay, so I've got this problem here about probability with chips in a bag. Let me try to break it down step by step. Let me read it again to make sure I get all the details.
Alright, there are 5 blue chips and 3 yellow chips in a bag. So total chips would be 5 + 3, which is 8. Then, one chip is drawn, put back... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1395415,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( x \) and \( y \) be distinct natural numbers and \( p \) be a prime. Given that \(\frac{\sqrt{x^2 + py}}{2} + \frac{\sqrt{y^2 + px}}{2}\) is a prime, find the value of \( p \) if \( x = 10 \) and \( y = 6 \). | <think>
Okay, let's see. I need to find the value of the prime number p given that x is 10 and y is 6. The expression given is the average of two square roots: [sqrt(x² + p*y) + sqrt(y² + p*x)] divided by 2, and this average is a prime number. Hmm, let's break this down step by step.
First, substitute x and y with 10 ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1340402,
"category": "math",
"source": "OpenMathReasoning"
} | |
Buses run between the cities A and B, the first one is at 05:00 AM and the last one departs not later than at 11:59 PM. A bus from the city A departs every a minutes and arrives to the city B in a ta minutes, and a bus from the city B departs every b minutes and arrives to the city A in a tb minutes.
The driver Simion... | <think>
Okay, let's see. So the problem is about calculating how many buses Simion will meet while driving from city A to city B. The key here is to figure out all the buses from B that are on the road during the time Simion is traveling, but not counting those that he meets exactly at the start or end points.
First, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2948638,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Evaluate the expression $1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \cdots - 1998^2 + 1999^2$. | <think>
Okay, so I need to evaluate this alternating sum of squares: 1² minus 2² plus 3² minus 4² and so on, up to 1999². The pattern is that the odd-numbered terms are positive and the even-numbered terms are negative. Let me see how to approach this.
First, let me write out the first few terms to see if I can spot a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 810859,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to find the alien dictionary order given a list of words. The alien dictionary order is defined by the order of characters in the alien language. The words in the list are sorted according to this alien dictionary order. Your task is to determine this order.
Solve the problem starting with the provide... | <think>
Okay, I need to solve this problem where I have to find the alien dictionary order based on a list of words that are sorted according to this order. The function is part of the Solution class, so I have to implement the alien_dictionary method.
First, I remember that this problem is similar to topological sort... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3933860,
"category": "code",
"source": "opc-sft-stage2"
} | |
The terminal side of an angle in standard position intersects a circle centered at the origin at the point \((-6\sqrt{5}, 2\sqrt{15})\). What is the radian measure of the angle? | <think>
Okay, let's see. I need to find the radian measure of an angle whose terminal side intersects a circle at the point (-6√5, 2√15). The angle is in standard position, so it starts from the positive x-axis and rotates around the origin. The circle is centered at the origin, so the radius should be the distance fro... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 161229,
"category": "math",
"source": "OpenMathReasoning"
} | |
Suppose \( a, b, c, d \) are real numbers such that
\[ |a-b| + |c-d| = 99 \]
\[ |a-c| + |b-d| = 1 \]
Determine all possible values of \( |a-d| + |b-c| \). | <think>
Okay, let's try to solve this problem. So, we have four real numbers a, b, c, d. The given conditions are:
1. |a - b| + |c - d| = 99
2. |a - c| + |b - d| = 1
And we need to determine all possible values of |a - d| + |b - c|.
Hmm, absolute values often involve distances on the number line, right? Maybe if I c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2288842,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find the smallest number of natural numbers whose sum of squares is $2021.$ | <think>
Okay, so I need to find the smallest number of natural numbers whose sum of squares equals 2021. Hmm, let me start by understanding the problem. Natural numbers here mean positive integers, right? So, I need to express 2021 as a sum of squares of the least possible number of such numbers.
First, maybe I shoul... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3751414,
"category": "math",
"source": "OpenMathReasoning"
} | |
A 2010*2010 rectangular table is tiled completely with 4040100 1*1 tiles. A circular coin with diameter 2 is randomly tossed onto the table. Given that the coin lands completely on the table, what is the probability that it completely covers a tile? | <think>
Okay, let's try to solve this problem. So we have a 2010x2010 rectangular table that's completely tiled with 4040100 1x1 tiles. Then there's this circular coin with a diameter of 2, so its radius is 1. We need to find the probability that when we randomly toss the coin onto the table, it completely covers a til... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2473986,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the coefficient of \(x^{16}\) in the expansion of \(\left(5x^2 - x^4\right) \left(1 - x\right)^{-3}\). | <think>
Okay, so I need to find the coefficient of x^16 in the expansion of (5x² - x⁴)(1 - x)^{-3}. Hmm, let's see. I remember that the expansion of (1 - x)^-n is related to the binomial theorem for negative exponents, which uses the formula for a binomial series. The general term in the expansion of (1 - x)^{-3} would... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 764539,
"category": "math",
"source": "OpenMathReasoning"
} | |
Owl Sonya gave a huge lake puzzle of size n × m to hedgehog Filya as a birthday present. Friends immediately started to assemble the puzzle, but some parts of it turned out to be empty — there was no picture on them. Parts with picture on it are denoted by 1, while empty parts are denoted by 0. Rows of the puzzle are n... | <think>
Okay, let's see. I need to solve this problem where for each query, I have to find the maximum square of 1's within a given rectangle in a grid. The constraints are pretty tight, especially with t up to a million. So, a brute force approach for each query isn't going to work. I need a more efficient way.
Hmm. ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 995995,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that:
\[ f(xf(x+y)) = f(yf(x)) + x^2 \] | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy the functional equation f(xf(x+y)) = f(yf(x)) + x². Hmm, functional equations can be tricky, but maybe I can start by plugging in some specific values for x and y to see if I can find a pattern or figure out what f mi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1836054,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to convert a given decimal number into its equivalent English words.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def numberToWords(self, num):
```
Please place the solution code in the following format:
```python
# Your solution code her... | <think>
Okay, I need to write a Python function called numberToWords that converts a given decimal number into its equivalent English words. The function is part of a Solution class. Let's think about how to approach this.
First, I remember that breaking down the number into chunks is a common strategy. Like, numbers ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 989416,
"category": "code",
"source": "opc-sft-stage2"
} | |
Let \( a = \underbrace{19191919 \ldots 1919}_{19 \text{ is repeated } 3838 \text{ times}} \). What is the remainder when \( a \) is divided by 13? | <think>
Okay, let's tackle this problem. So, we have this number 'a' which is just the sequence "19" repeated 3838 times. We need to find the remainder when this huge number is divided by 13. Hmm, modulus operation on a very large number... I remember that there are some tricks with modular arithmetic for repeating num... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 421067,
"category": "math",
"source": "OpenMathReasoning"
} | |
Alook was composing magical spells for his mage master. The mage was good at spellcasting but since spell design requires intricate mathematics, this task was given to Alook who had the gift of numbers.
The power contained in a spell is a function of its lexicographical structure, which is why Alook wants to extensive... | <think>
Okay, let's see. I need to solve this problem where I have to find the number of phonetically different spells a mage casts, given the maximum phoneme length n. The answer has to be computed for each test case, and since n can be up to 1e9, I need an efficient solution, probably using matrix exponentiation or s... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2343329,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Solve the equation \(2\sin^3 x - \cos 2x + \cos x = 0\). | <think>
Okay, so I need to solve the equation 2 sin³x - cos2x + cosx = 0. Hmm, let's see. I remember that trigonometric equations can sometimes be simplified using identities. Let me think about which ones might help here. The cos2x term stands out. I recall that there are a few different forms for the double angle ide... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 876049,
"category": "math",
"source": "OpenMathReasoning"
} | |
Point P lies on a circle with radius 2. A point A is chosen at random inside the circle. Find the probability that the distance between A and P is more than \(2\sqrt{3}\). | <think>
Okay, so I need to find the probability that a randomly chosen point A inside a circle of radius 2 is more than 2√3 units away from a fixed point P on the circle. Hmm, let's break this down step by step.
First, the setup: there's a circle with radius 2, and point P is somewhere on the circumference. Point A is... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3617725,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the integral $\int_{\frac{\sqrt{3}}{2}}^{1} \frac{1}{1-\sqrt{(1-x^{2})}} \, dx$. | <think>
Okay, let's see. I need to evaluate the integral from sqrt(3)/2 to 1 of 1 over (1 minus the square root of (1 - x²)) dx. Hmm, that looks a bit complicated. Maybe I can simplify the denominator first. Let me write it down again: ∫ from √3/2 to 1 of 1/(1 - √(1 - x²)) dx.
The denominator has 1 minus the square r... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1191529,
"category": "math",
"source": "OpenMathReasoning"
} | |
In a triangle \( \triangle ABC \), point \( D \) is on \( BC \). A circle is drawn tangent to \( BC \), \( AD \), and the circumcircle of \( \triangle ABC \) at points \( F \), \( G \), and \( H \) respectively. If an inversion is performed with respect to point \( A \), what are the images of the tangency points \( F ... | <think>
Alright, let's tackle this geometry problem step by step. So, we have triangle ABC with point D on BC. There's a circle that's tangent to BC at F, tangent to AD at G, and tangent to the circumcircle of ABC at H. We need to find the images of points F, G, and H after an inversion with respect to point A.
First,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3754144,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate $\lim_{x \to e}\frac{e^{x^e} - x^{e^x}}{x^x \tan\left(\frac{\pi e}{x}\right)}$ | <think>
Okay, let's see. I need to find the limit as x approaches e of [e^{x^e} - x^{e^x}] divided by [x^x times tan(πe/x)]. Hmm. That looks a bit complicated, but maybe I can break it down step by step.
First, let me write down the expression again to make sure I have it right:
Limit as x → e of [e^{x^e} - x^{e^x}] ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1704208,
"category": "math",
"source": "OpenMathReasoning"
} | |
Geek lives in a special city where houses are arranged in a hierarchial manner. Starting from house number 1, each house leads to two more houses.
1 leads to 2 and 3.
2 leads to 4 and 5.
3 leads to 6 and 7. and so on.
Given the house numbers on two houses x and y, find the length of the shortest path between them.... | <think>
Okay, let's see. I need to find the shortest path between two house numbers in this hierarchical tree structure. Hmm. So the houses are arranged in a binary tree where each node has two children. Like, house 1 has 2 and 3 as children, 2 has 4 and 5, and so on. So the structure is a perfect binary tree.
The pro... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3284469,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Cammeron, Aidan, Edward, and Avi are sitting at a round table. If First comes, they will pull up 4 more chairs to ensure nobody sits next to First. If First doesn't come, they will pull up 3 more chairs and sit at random among the 7 chairs. What is the probability that there will be exactly 2 people eating without any ... | <think>
Okay, let me try to figure out this probability problem. Hmm, so there are Cammeron, Aidan, Edward, and Avi sitting around a round table. Depending on whether First comes or not, they adjust the number of chairs. If First comes, they add 4 more chairs so that nobody sits next to First. If First doesn't come, th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3235987,
"category": "math",
"source": "OpenMathReasoning"
} | |
In triangle ABC, AB = 13, AC = 14, and BC = 15. Let M denote the midpoint of AC. Point P is placed on line segment BM such that AP ⊥ PC. Suppose that the area of triangle APC can be written in the form \(\frac{p\sqrt{q}}{r}\), where p, q, and r are positive integers with p and r relatively prime and q squarefree. What ... | <think>
Okay, so I need to find the area of triangle APC where AP is perpendicular to PC. The triangle ABC has sides AB=13, AC=14, and BC=15. M is the midpoint of AC, and P is a point on BM such that AP is perpendicular to PC. The answer should be in the form p√q/r with p and r coprime and q square-free, then find p+q+... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 7847,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to check if a given string matches a given wildcard pattern. The pattern can contain '?' and '*' wildcard characters. '?' matches any single character, while '*' matches any sequence of characters (including the empty sequence).
Solve the problem starting with the provided function header.
Function h... | <think>
Okay, I need to write a function called isMatch that takes two strings, s and p, and returns a boolean indicating whether s matches the wildcard pattern p. The pattern can have '?' and '*' where '?' matches any single character and '*' matches any sequence, including empty. Hmm.
So, the problem is similar to r... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3629861,
"category": "code",
"source": "opc-sft-stage2"
} | |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[ f(f(x) + y) = f(f(x) - y) + 4f(x)y \]
for all \( x, y \in \mathbb{R} \). | <think>
Okay, so I have this functional equation to solve: f(f(x) + y) = f(f(x) - y) + 4f(x)y for all real numbers x and y. Hmm, functional equations can be tricky, but let me see what I can do step by step.
First, I should recall some standard techniques for solving functional equations. Common approaches include sub... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2437537,
"category": "math",
"source": "OpenMathReasoning"
} | |
«Polygon» is a system which allows to create programming tasks in a simple and professional way. When you add a test to the problem, the corresponding form asks you for the test index. As in most cases it is clear which index the next test will have, the system suggests the default value of the index. It is calculated ... | <think>
Okay, I need to solve this programming problem. Let's read the problem carefully.
The task is to find the smallest positive integer that's not present in a given list of integers. These integers represent the indexes of previously added tests. So the next test's default index should be the smallest positive in... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 246930,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Find all positive integers \( n \) satisfying \( P(n) = 15n \), where \( P(n) \) is the product of the positive divisors of \( n \). | <think>
Okay, let's try to solve this problem where I need to find all positive integers n such that P(n) = 15n. Here, P(n) is the product of the positive divisors of n. Hmm, I remember that there's a formula related to the product of divisors. Let me recall... Oh right, for a number n with prime factorization n = p₁^a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 284221,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( G \) be a group whose order does not exceed the continuum. Construct a metric space \( M \) whose group of isometries is precisely \( G \). | <think>
Okay, so I have this problem: Given a group G whose order doesn't exceed the continuum, I need to construct a metric space M such that the group of isometries (bijective distance-preserving maps) of M is exactly G. Hmm, interesting. Let me think through how to approach this.
First, I remember that for finite g... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2319028,
"category": "math",
"source": "OpenMathReasoning"
} | |
In the convex quadrilateral \(ABCD\), point \(K\) is the midpoint of \(AB\), point \(L\) is the midpoint of \(BC\), point \(M\) is the midpoint of \(CD\), and point \(N\) is the midpoint of \(DA\). For some point \(S\) lying inside quadrilateral \(ABCD\), it is given that \(KS = LS\) and \(NS = MS\). Prove that \(\angl... | <think>
Okay, so I need to prove that angle KSN is equal to angle MSL in this convex quadrilateral ABCD with midpoints K, L, M, N. Hmm, let me try to visualize the problem first. Let me sketch a rough diagram in my mind. ABCD is a convex quadrilateral, so all its interior angles are less than 180 degrees. The midpoints... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 100883,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Some terrorist attacks on Indian border. Now Indian Army have to send his soldiers to fight against terrorist. There are total N soldiers in the camp. Every soldier has a skill denoted by a single character lies between A-Z (all skills are in CAPITAL LETTERS). Now commander gave order to stand all the soldiers in a row... | <think>
Okay, let's see. I need to find the maximum number of soldiers that can be selected in a row such that all their skills are unique. Hmm, right. So the problem is similar to the longest substring without repeating characters problem. Oh right, that's a classic sliding window problem.
So the input is a string wh... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3747963,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \( f(x) = [n + p \sin x] \), where \( x \in (0, \pi) \), \( n \in \mathbb{Z} \), and \( p \) is a prime number. Here, \( [x] \) represents the greatest integer less than or equal to \( x \). Find the number of points at which \( f(x) \) is not differentiable. | <think>
Okay, so I need to find the number of points where the function f(x) = [n + p sin x] is not differentiable on the interval (0, π). Here, n is an integer, p is a prime number, and [x] is the floor function, which gives the greatest integer less than or equal to x. Alright, let's break this down step by step.
Fi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2942036,
"category": "math",
"source": "OpenMathReasoning"
} | |
Using a compass and a straightedge, construct a triangle given the point $H$ of the intersection of its altitudes, the center $O$ of its circumscribed circle, and the line $l$ on which one of its sides lies. | <think>
Okay, so I need to construct a triangle given the orthocenter H, the circumcenter O, and the line l where one of the sides lies. Hmm, let me start by recalling some properties of triangle centers.
First, I know that the circumcenter O is the intersection of the perpendicular bisectors of the triangle's sides ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2199067,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Given two sequences, one is increasing sequence a[] and another a normal sequence b[], find the K-th missing element in the increasing sequence which is not present in the given sequence. If no K-th missing element is there output -1
Example 1:
Input : Arr[] = {0, 2, 4, 6, 8, 10, 12, 14, 15}
Brr[] = {4, 10, 6, 8, 12}... | <think>
Okay, I need to solve this problem where I have to find the K-th missing element in the increasing sequence a[] that's not present in the normal sequence b[]. Let me think about how to approach this.
First, the problem says that a is an increasing sequence. So it's sorted in ascending order. The other array, b... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 541910,
"category": "code",
"source": "TACO"
} | |
Consider an \(m \times n\) board that is divided into unit squares. An L-triomino consists of three unit squares, a center square, and two leg squares. There is an L-triomino in the top left corner so that the center square is on the corner square. In one move, the L-triomino can be rotated by multiples of \(90^{\circ}... | <think>
Alright, let's tackle this problem step by step. So, we have an m x n board divided into unit squares. There's an L-triomino in the top left corner, with its center on the corner square. The L-triomino can be rotated around one of its leg squares' centers by 90-degree multiples in each move. We need to figure o... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3519879,
"category": "math",
"source": "NuminaMath-CoT"
} | |
You need to construct a string consists of parenthesis and integers from a binary tree with the preorder traversing way.
The null node needs to be represented by empty parenthesis pair "()". And you need to omit all the empty parenthesis pairs that don't affect the one-to-one mapping relationship between the string an... | <think>
Okay, so I need to construct a string from a binary tree using the preorder traversal. The problem says that null nodes should be represented by empty parenthesis "()", but we have to omit any unnecessary empty parentheses that don't affect the mapping. Hmm, right. So the key is to figure out when to include th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 315076,
"category": "code",
"source": "leetcode"
} | |
Let \( K \subseteq \mathbb{R} \) be a field with the operations from \( \mathbb{R} \) such that for any \( a \in K \setminus \mathbb{Q} \), there exists \( b \in K \setminus \mathbb{Q} \) so that \( a + b \in \mathbb{Q} \) and \( ab \in \mathbb{Q} \). Suppose \( K \) contains an element of the form \( x + y\sqrt{d} \) ... | <think>
Okay, so I need to solve this problem about a field K that's a subset of the real numbers. The conditions are that for any element a in K but not in Q (the rationals), there exists another element b in K but not in Q such that a + b and ab are both rational. Also, K contains an element of the form x + y√d where... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2422901,
"category": "math",
"source": "OpenMathReasoning"
} | |
Two cans containing coffee have the same shape and are made out of the same tin. One can weighs 2 kg and is 12 cm high; the other weighs 1 kg and is 9.5 cm high. Find the net weight of the coffee in both cans. | <think>
Okay, so I have this problem here about two coffee cans. They both have the same shape and are made from the same tin material. One can weighs 2 kg and is 12 cm high, the other is 1 kg and 9.5 cm high. I need to find the net weight of the coffee in both cans. Hmm, okay. Let me think.
First, since the cans are ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1337349,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to return the spiral order traversal of a matrix.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def spiral_traversal(self, matrix):
```
Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to write a Python function called spiral_traversal for a matrix. Let's think about how to approach this. The spiral order traversal means we go around the matrix in a spiral, right? Like, first the top row left to right, then right column top to bottom, then bottom row right to left, then left colu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2234849,
"category": "code",
"source": "opc-sft-stage2"
} | |
Given a convex 2000-gon, where no three diagonals intersect at a single point. Each of its diagonals is colored in one of 999 colors. Prove that there exists a triangle, all sides of which lie entirely on diagonals of the same color. (The vertices of the triangle do not necessarily have to be vertices of the original p... | <think>
Okay, so I have this problem about a convex 2000-gon where each diagonal is colored in one of 999 colors. I need to prove that there's a triangle with all sides of the same color. Hmm, let's see. The polygon is convex, so all the diagonals are inside it. Also, no three diagonals intersect at the same point, whi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1308545,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Determine a five-digit positive integer $n$ whose digit sum is the least and $n^3 - 1$ is divisible by $2556$. | <think>
Okay, so I need to find a five-digit number n such that n³ - 1 is divisible by 2556, and the digit sum of n is the smallest possible. Hmm, let me start by breaking down the problem.
First, the condition that n³ - 1 is divisible by 2556 means that n³ ≡ 1 mod 2556. So, n has to be a number where when you cube it... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3892524,
"category": "math",
"source": "OpenMathReasoning"
} | |
ABCD is a parallelogram. X is a point on the side BC such that triangles ACD, ACX, and ABX are all isosceles. Find the angles of the parallelogram. | <think>
Okay, so I need to find the angles of parallelogram ABCD where point X on BC makes triangles ACD, ACX, and ABX all isosceles. Hmm, let me start by recalling some properties of parallelograms. In a parallelogram, opposite sides are equal and parallel, and opposite angles are equal. Also, consecutive angles are s... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2065942,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \(ABC\) be a triangle and \(D, E\), and \(F\) be the midpoints of sides \(BC, CA\), and \(AB\) respectively. What is the maximum number of circles which pass through at least 3 of these 6 points? | <think>
Okay, so I need to figure out the maximum number of circles that can pass through at least three of the six points in a triangle with its midpoints. Let me try to visualize this. The triangle is ABC, and D, E, F are the midpoints of BC, CA, and AB respectively. So altogether, we have six points: the three verti... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 172808,
"category": "math",
"source": "NuminaMath-CoT"
} | |
The prime numbers are not regularly spaced. For example from `2` to `3` the step is `1`.
From `3` to `5` the step is `2`. From `7` to `11` it is `4`.
Between 2 and 50 we have the following pairs of 2-steps primes:
`3, 5 - 5, 7, - 11, 13, - 17, 19, - 29, 31, - 41, 43`
We will write a function `step` with parameters:
... | <think>
Okay, I need to solve this problem where I have to find the first pair of consecutive primes that are exactly a step 'g' apart. The primes must be between m and n inclusive, and I have to return the first such pair. If none exist, return None or an empty list or something like that. Let's see.
First, I need to... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2863110,
"category": "code",
"source": "TACO"
} | |
Write a function to detect if two rectangles defined by their top-left and bottom-right coordinates are overlapping.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def is_rectangle_overlap(self, rect1, rect2):
```
Please place the solution code in the following fo... | <think>
Okay, I need to write a Python function to determine if two rectangles overlap. The rectangles are given by their top-left and bottom-right coordinates. Hmm. Wait, wait, how are the coordinates structured here? Like, each rectangle is represented by two points: the top-left and bottom-right. So for rect1, the c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2328585,
"category": "code",
"source": "opc-sft-stage2"
} | |
Write a function to find the sum of all numbers that are present in a string.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def sum_in_string(self, s):
```
Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to write a Python function called sum_in_string that's part of the Solution class. The goal is to find the sum of all numbers present in a string. Hmm. Let's think about how to approach this.
So, the input is a string, and I have to extract all the numbers from it and sum them up. The numbers coul... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2365420,
"category": "code",
"source": "opc-sft-stage2"
} | |
Evaluate the limit \(\lim_{n \to \infty} (n^{1/n})^a\), where \(a = \sum_{k=3}^{n} \tan\left(\frac{\pi}{k}\right)\). | <think>
Okay, so I need to evaluate the limit as n approaches infinity of (n^(1/n))^a, where a is the sum from k=3 to n of tan(π/k). Hmm, let's start by understanding each part of this problem.
First, the base of the expression is n^(1/n). I remember that the limit of n^(1/n) as n approaches infinity is 1. Let me conf... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1495534,
"category": "math",
"source": "OpenMathReasoning"
} | |
< PREVIOUS KATA
NEXT KATA >
###Task:
You have to write a function `pattern` which returns the following Pattern(See Examples) upto (3n-2) rows, where n is parameter.
* Note:`Returning` the pattern is not the same as `Printing` the pattern.
####Rules/Note:
* The pattern should be created using only unit digits.
* I... | <think>
Okay, let's try to figure out how to generate this pattern. The example given for n=5 shows that there's a structure with upper and lower parts, and a middle section. Let's break it down.
First, the function is called pattern(n). The output for n=5 has 3n-2 = 13 rows. Wait, for n=5, 3*5-2 is 13. The example sh... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3135201,
"category": "code",
"source": "OpenCodeReasoning"
} | |
There are 5 boxes marked \( A \), \( B \), \( C \), \( D \), and \( E \). The combined weight of any two boxes can be 34, 41, 48, 52, 53, 60, 64, 67, 71, or 78 pounds. The weights are ordered such that \( B > D > E > A > C \). All of the boxes together weigh 142 pounds. Find the weights of \( A \), \( B \), \( C \), \(... | <think>
Okay, so I need to figure out the weights of boxes A, B, C, D, and E. The given information is that any two boxes combined can weigh 34, 41, 48, 52, 53, 60, 64, 67, 71, or 78 pounds. Also, the order of their weights is B > D > E > A > C, and the total weight of all boxes together is 142 pounds. Hmm, this sounds... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1589190,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to compute the total number of full staircase rows that can be formed given a certain number of bricks. A staircase row is defined as a row of bricks where the number of bricks in the row is equal to the row number. For instance, a staircase with 3 full rows would have 1 brick in the first row, 2 brick... | <think>
Okay, I need to solve this problem where I have to find out how many full staircase rows can be formed with a given number of bricks. Let me think about how the staircase works. Oh right, each row i has exactly i bricks. So the first row is 1, second 2, third 3, and so on. The total bricks used for k rows is th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 809100,
"category": "code",
"source": "opc-sft-stage2"
} | |
Solve the equation \( x = \sqrt{a - \sqrt{x + a}} \) where \( a \) is a strictly positive real number. | <think>
Okay, let's try to solve this equation: \( x = \sqrt{a - \sqrt{x + a}} \). Hmm, the variable x is on both sides and inside square roots. That might make things a bit tricky. Let me start by writing down the equation again to make sure I have it right.
So, the equation is x equals the square root of (a minus th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2440523,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all pairs of nonnegative integers $(x, y)$ such that $(14y)^x + y^{(x+y)} = 2007$. | <think>
Okay, let's tackle this problem: find all pairs of nonnegative integers (x, y) such that (14y)^x + y^(x+y) = 2007. Hmm, nonnegative integers, so x and y can be 0 or positive integers. Let me start by understanding the equation better.
First, the equation is (14y)^x + y^(x+y) = 2007. Let's break it down. The te... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2745289,
"category": "math",
"source": "OpenMathReasoning"
} | |
Jill was given a large jar of jam. She gave one sixth of the jam to Jan. Jill then gave one thirteenth of the remaining jam to Jas. Jill was left with 1 kg of jam. What was the weight, in kg, of the jam in Jill's jar at the start?
A. 1.2
B. 1.3
C. 1.4
D. 1.6
E. 1.9 | <think>
Okay, let's see. The problem is about Jill giving away parts of her jam to Jan and Jas, and then being left with 1 kg. I need to find the original amount. Hmm. Let me break it down step by step.
First, Jill starts with some amount of jam. Let me call that original amount "x" kilograms. That makes sense. Then s... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3424934,
"category": "math",
"source": "NuminaMath-CoT"
} | |
A wooden cube \( n \) units on a side is painted red on all six faces and then cut into \( n^3 \) unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is \( n \)? | <think>
Okay, let's try to solve this problem. So we have a wooden cube that's n units on each side. It's painted red on all six faces and then cut into n³ unit cubes. The key point here is that exactly one-fourth of the total number of faces of the unit cubes are red. We need to find n.
First, I need to figure out th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1993910,
"category": "math",
"source": "OpenMathReasoning"
} | |
You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? | <think>
Okay, so I need to figure out how many ways to make exactly eight folds on a square piece of paper, folding along any line of symmetry each time, and considering that the direction of the folds doesn't matter. Hmm, let's start by understanding the problem.
First, a square has several lines of symmetry. Let me ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1018093,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function that returns the length of the longest substring without repeating characters.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def compute_Longest_Substring(self, s):
```
Please place the solution code in the following format:
```python
# Y... | <think>
Okay, I need to write a Python function called compute_Longest_Substring that takes a string s and returns the length of the longest substring without repeating characters. Hmm, how do I approach this?
Let me think. The problem is a classic one. I remember that the sliding window technique is useful here. The ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3750453,
"category": "code",
"source": "opc-sft-stage2"
} | |
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point... | <think>
Okay, let's see. So, Freddy the frog starts at (0, 21) and wants to reach the river at y = 24. There's a fence at y = 0 that he can't cross. Each time he jumps, he picks a direction parallel to one of the coordinate axes, so either up, down, left, or right. But when he's at y = 0, he can't go down, so he has th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1691999,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a function \( f(x) \) such that \( f(0) = 0 \), \( \int_{0}^{\frac{\pi}{4}} [f'(x)]^2 \, dx = 2 \), and \( \int_{0}^{\frac{\pi}{4}} \sin(2x) f(x) \, dx = \frac{1}{2} \), find the value of \( \int_{0}^{\frac{\pi}{4}} f(x) \, dx \). | <think>
Okay, so I need to find the integral of f(x) from 0 to π/4, given some conditions. Let me see. The problem states that f(0) = 0, the integral of [f'(x)]² dx from 0 to π/4 is 2, and the integral of sin(2x) f(x) dx over the same interval is 1/2. Hmm, how do I connect these pieces?
First, maybe I should recall so... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3610122,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a grid with n rows and m columns. Each cell contains a number, either 0 or 1. In one operation, you can flip the value of any cell (change 0 to 1 or 1 to 0).
Determine the minimum number of operations needed to make some row or column consist entirely of 1s. If it is impossible, return -1.
Input:
The... | <think>
Okay, I need to solve this problem where I have to find the minimum number of flips to make any row or column all 1s. If it's impossible, return -1. Let's think about how to approach this.
First, let's understand the problem. The grid is made of 0s and 1s. Each flip changes a cell. The goal is to make at least... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 981478,
"category": "code",
"source": "OpenCodeReasoning"
} | |
How many solutions does the equation $[x] - 1977(x) = 1978$ have, where $[x]$ denotes the greatest integer less than or equal to $x$ and $(x)$ denotes the fractional part of $x$? | <think>
Okay, so I have this equation to solve: [x] - 1977(x) = 1978. Here, [x] is the greatest integer less than or equal to x, which I know is also called the floor function. And (x) denotes the fractional part of x, which should be x minus [x], right? So, fractional part is always between 0 (inclusive) and 1 (exclus... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2064218,
"category": "math",
"source": "OpenMathReasoning"
} | |
In $\triangle ABC$, prove that if $A \leqslant B \leqslant C < \frac{\pi}{2}$, then
$$
\sin 2A \geqslant \sin 2B \geqslant \sin 2C.
$$ | <think>
Alright, let's try to tackle this problem. So, we have triangle ABC where the angles satisfy A ≤ B ≤ C < π/2, and we need to prove that sin 2A ≥ sin 2B ≥ sin 2C. Hmm, okay. Let me think step by step.
First, I know that in any triangle, the sum of the angles is π radians. So, A + B + C = π. But the given condit... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3733055,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let's assume that we have a pair of numbers (a, b). We can get a new pair (a + b, b) or (a, a + b) from the given pair in a single step.
Let the initial pair of numbers be (1,1). Your task is to find number k, that is, the least number of steps needed to transform (1,1) into the pair where at least one number equals n... | <think>
Okay, let's see. The problem is to find the minimum number of steps needed to reach a pair where at least one number is n, starting from (1,1). Each step allows us to add the two numbers and replace one of them. Hmm.
So, for example, when the input is 5, the output is 3. The steps are (1,1) → (1,2) → (3,2) → (... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1917306,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Transformation
Write a program which performs a sequence of commands to a given string $str$. The command is one of:
* print a b: print from the a-th character to the b-th character of $str$
* reverse a b: reverse from the a-th character to the b-th character of $str$
* replace a b p: replace from the a-th charact... | <think>
Okay, I need to solve this programming problem. Let's read the problem carefully.
So, the task is to perform a series of commands on a given string. The commands are print, reverse, and replace. The input includes the initial string, the number of commands, and then each command. For each print command, I have... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2412255,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \( K \) be a convex cone in the \( n \)-dimensional real vector space \( \mathbb{R}^n \), and consider the sets \( A = K \cup (-K) \) and \( B = (\mathbb{R}^n \setminus A) \cup \{ 0 \} \). Define \( V_A \) as the subspace of \( \mathbb{R}^n \) that lies in \( A \) and \( V_B \) as the subspace of \( \mathbb{R}^n \)... | <think>
Okay, let's try to figure out this problem. So we have a convex cone K in ℝ⁵. A convex cone is a set that's closed under addition and non-negative scalar multiplication, right? So if you take any two points in K, their sum is also in K, and multiplying any point in K by a non-negative scalar keeps it in K. Also... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3415308,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve the equation \( \arcsin(x+1) - \arcsin(x) = \frac{\pi}{4} \) for \( x \in [-1, 0] \). | <think>
Okay, let's try to solve this equation: arcsin(x + 1) minus arcsin(x) equals π/4, where x is between -1 and 0. Hmm, I need to find the value of x that satisfies this equation. Let me start by recalling some properties of the arcsin function.
First, the domain of arcsin is from -1 to 1, so x + 1 must also be in... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2410191,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the largest member of the sequence \( a_n = (12 - |7 - n|) \cos(\pi n) \). | <think>
Okay, let's try to figure out the largest member of the sequence \( a_n = (12 - |7 - n|) \cos(\pi n) \). Hmm, sequences can be tricky, but let's break it down step by step.
First, let's understand the components of the sequence. The general term is given by \( a_n = (12 - |7 - n|) \cos(\pi n) \). So there are ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 187436,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a 2D grid of size m x n and an integer k. You need to shift the grid k times. In one shift operation: Return the 2D grid after applying shift operation k times.
Write Python code to solve the problem. Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, let's see. I need to solve this problem where I have a 2D grid of size m x n, and I have to shift it k times. Each shift operation moves all the elements in a specific way. The problem is to return the grid after applying the shift k times. Hmm.
First, I need to understand what exactly a "shift" operatio... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1891310,
"category": "code",
"source": "leetcode"
} | |
Write a function that takes a list of integers as an argument and returns a list of booleans. Each boolean indicates whether the sum of the absolute differences between each pair of adjacent numbers in the list is even. If the sum is even, the boolean is True; if it's odd, the boolean is False.
Solve the problem start... | <think>
Okay, I need to write a Python function called even_sum_diff inside the Solution class. The function takes a list of integers and returns a list of booleans. Each boolean in the output corresponds to whether the sum of absolute differences between each pair of adjacent numbers in the list is even. Wait, wait. W... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3443836,
"category": "code",
"source": "opc-sft-stage2"
} | |
Berlanders like to eat cones after a hard day. Misha Square and Sasha Circle are local authorities of Berland. Each of them controls its points of cone trade. Misha has n points, Sasha — m. Since their subordinates constantly had conflicts with each other, they decided to build a fence in the form of a circle, so that ... | <think>
Okay, let's see. The problem is to determine if we can draw a circle such that all of Misha's points are strictly inside the circle and all of Sasha's are strictly outside, or vice versa. So we need to check both possibilities: Misha inside and Sasha outside, or Sasha inside and Misha outside.
Hmm. How to appr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2335989,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Prateek and Chintu are working on different projects both with equal priority. They both need to run some batches of processes. A batch has processes which need some systems to run them irrespective of the number of process running on each dependent system. If a batch runs then the dependent systems are occupied by its... | <think>
Okay, I need to solve this problem. Let's read through the problem statement carefully.
So the problem is about Chintu wanting to replace some of Prateek's processes with his own batches in such a way that the total number of processes increases. The key points here are:
- Each system can only run processes f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3504359,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Find the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $$\text{LCM}[a,b,c]=\text{LCM}[a,b,d]=\text{LCM}[a,c,d]=\text{LCM}[b,c,d]= 2^r \times 3^s$$. | <think>
Okay, so I need to find the number of ordered quadruples (a, b, c, d) of positive integers where the least common multiple (LCM) of any three of them is equal to 2^r times 3^s for some non-negative integers r and s. That is, LCM[a, b, c] = LCM[a, b, d] = LCM[a, c, d] = LCM[b, c, d] = 2^r × 3^s. Hmm, interesting... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1846666,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider the sequence defined by \(a_1 = 0\) and \(|a_n| = |a_{n-1} + 1|\). Let \(S_n = \sum_{i=1}^{n}a_i\). Find the maximum value of \(n\) such that \(\frac{S_n}{n} = -\frac{1}{2}\). | <think>
Okay, let me try to figure out this problem. So, we have a sequence where a₁ is 0, and then each subsequent term is defined by the absolute value equation |aₙ| = |aₙ₋₁ + 1|. We need to find the maximum value of n such that the average Sₙ/n equals -1/2. Hmm, let's start by understanding the sequence first.
Sinc... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1256394,
"category": "math",
"source": "OpenMathReasoning"
} | |
Prove that there exists only one set of three different positive integers, which do not have a common divisor greater than 1, and have the property that the sum of any two of these integers is divisible by the third integer. | <think>
Okay, so I need to prove that there's only one set of three different positive integers with no common divisor greater than 1, such that the sum of any two is divisible by the third. Hmm, let me start by trying to understand the problem.
First, let's parse the requirements:
1. Three different positive integer... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2991527,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Compute the Laplacian $\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ for a smooth real-valued function $u(x,y) = u(r)$ where $r = \sqrt{x^2 + y^2}$. | <think>
Okay, so I need to compute the Laplacian of a function u(x, y) that depends only on r, which is the radial coordinate sqrt(x² + y²). Hmm, the Laplacian in Cartesian coordinates is the sum of the second partial derivatives with respect to x and y. But since u is a function of r, maybe it's easier to switch to po... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 450162,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to find the maximum profit from an array of stock prices, where you can make at most two transactions (buy and sell twice). Note that you cannot engage in multiple transactions at the same time (i.e., you must sell the stock before you buy again).
Solve the problem starting with the provided function ... | <think>
Okay, I need to solve this problem where I have to find the maximum profit from an array of stock prices with at most two transactions. The key here is that I can buy and sell twice, but I can't have overlapping transactions. So, each buy must be after the previous sell. Hmm, how do I approach this?
Let me thi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 874217,
"category": "code",
"source": "opc-sft-stage2"
} | |
Little Susie, thanks to her older brother, likes to play with cars. Today she decided to set up a tournament between them. The process of a tournament is described in the next paragraph.
There are n toy cars. Each pair collides. The result of a collision can be one of the following: no car turned over, one car turned ... | <think>
Okay, let's try to understand this problem. We need to find all the good cars based on the collision results matrix. A good car is one that didn't turn over in any collision.
Hmm, wait, so for a car to be good, in every collision it was involved in, it shouldn't have turned over. Because if in any collision i... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 240665,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \( f(x) = \left(\tan x\right)^{3/2} - 3\tan x + \sqrt{\tan x} \). Consider the three integrals:
\[ I_1 = \int_{0}^{1} f(x) \, dx \]
\[ I_2 = \int_{0.3}^{1.3} f(x) \, dx \]
\[ I_3 = \int_{0.5}^{1.5} f(x) \, dx \]
Determine the correct order of the integrals from greatest to least. The options are:
A) \( I_1 > I_2 > ... | <think>
Okay, so I need to figure out the order of the integrals I1, I2, and I3 from greatest to least. The function given is f(x) = (tan x)^(3/2) - 3 tan x + sqrt(tan x). The integrals are over different intervals: I1 from 0 to 1, I2 from 0.3 to 1.3, and I3 from 0.5 to 1.5. The options are A to D, each ordering these ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1800365,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( a, b, c > 0 \) such that \( abc = 1 \). Find the minimum value of the expression:
\[ 6 + 3 \left( \frac{a}{b} + \frac{a}{c} + \frac{b}{a} + \frac{b}{c} + \frac{c}{a} + \frac{c}{b} \right) - 2 \sum a^2 - 6 \sum ab \] | <think>
Okay, so I have this problem where I need to find the minimum value of a given expression with the condition that a, b, c are positive real numbers and their product is 1. The expression is:
6 + 3(a/b + a/c + b/a + b/c + c/a + c/b) - 2(a² + b² + c²) - 6(ab + bc + ca)
Hmm, let's start by understanding the comp... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 560213,
"category": "math",
"source": "OpenMathReasoning"
} | |
Is it possible to tile an $8 \times 8$ chessboard from which one corner has been removed, using $2 \times 1$ dominoes? And if two opposite corners have been removed? And if two non-opposite corners have been removed? | <think>
Okay, so I need to figure out if it's possible to tile different modified chessboards with dominoes. The dominoes are 2x1 in size, right? Let me start with the first question: can I tile an 8x8 chessboard that's missing one corner?
Hmm, a regular chessboard has 64 squares. If one corner is removed, that leaves... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 803101,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Write a Python function `replace_in_file` that takes three parameters: `filename`, `pattern`, and `replacement`. The function should search for the occurrence of `pattern` in the file specified by `filename`. If the pattern is found, it should replace all occurrences of the pattern with the `replacement` string in the ... | <think>
Okay, I need to write a Python function called replace_in_file. The parameters are filename, pattern, and replacement. The goal is to replace all occurrences of the pattern in the file with the replacement string. But there's a specific way to do this using re.search and os.rename. Let's think through the steps... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1124532,
"category": "code",
"source": "opc-sft-stage2"
} | |
Let $\sigma \in S_n$ be a derangement, i.e., it has no fixed points. Let $\Omega(n)$ be the maximal possible order of $\sigma$. Determine $\Omega(n)$ in terms of $n$ and describe how $\Omega(n)$ behaves asymptotically as $n$ gets large. | <think>
Okay, let's see. I need to figure out the maximal possible order of a derangement in the symmetric group S_n. A derangement is a permutation with no fixed points, so every element must move. The order of a permutation is the least common multiple (LCM) of the lengths of its cycles. Since derangements can't have... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1195734,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( n \) be any positive integer. Consider a sequence containing only the letters \( \mathrm{A} \) and \( \mathrm{B} \), where at most three consecutive \( \mathrm{A} \)s and at most three consecutive \( \mathrm{B} \)s are allowed. Let \( S(n) \) represent the number of such sequences of length \( n \). The remainde... | <think>
Okay, so I need to find the number of sequences of length 2015 made up of A's and B's where you can't have more than three of the same letter in a row. Then, take that number modulo 12 and choose the correct answer from the options given. Let me try to figure this out step by step.
First, the problem is about ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1306655,
"category": "math",
"source": "NuminaMath-CoT"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.