instruction stringlengths 21 6.8k | input stringclasses 1
value | output stringlengths 815 71.5k | system stringclasses 1
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|---|---|---|---|---|
Given $\Delta ABC$ and a point $D$, let $X, Y, Z$ be the reflections of $D$ with respect to $BC, CA, AB$ respectively. Let $A', B', C'$ be the circumcenters of $\Delta XBC$, $\Delta YCA$, and $\Delta ZAB$ respectively, and let $D'$ be the circumcenter of $\Delta XYZ$. Let $T$ be the point such that $D'$ and $T$ are inv... | <think>
Okay, so I've got this geometry problem here. Let me try to parse it step by step. First, we have triangle ABC and a point D. Then, X, Y, Z are the reflections of D over the sides BC, CA, and AB respectively. So, reflecting D over each side gives us these three points. Then, A', B', C' are the circumcenters of ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3907658,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a string `s` containing just the characters `'('`, `')'`, `'{'`, `'}'`, `'['` and `']'`, determine if the input string is valid.
An input string is valid if:
1. Open brackets must be closed by the same type of brackets.
2. Open brackets must be closed in the correct order.
3. Every close bracket has a corres... | <think>
Okay, I need to solve this problem where I have to check if a string of parentheses is valid. Hmm, the rules are that the brackets must be closed in the correct order and same type. So like, if I have a '(', then the next closing should be a ')', and not something else. Also, the order matters. For example, "([... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3946896,
"category": "code",
"source": "leetcode"
} | |
Gena loves sequences of numbers. Recently, he has discovered a new type of sequences which he called an almost arithmetical progression. A sequence is an almost arithmetical progression, if its elements can be represented as:
- a1 = p, where p is some integer;
- ai = ai - 1 + ( - 1)i + 1·q (i > 1), where q is some int... | <think>
Okay, let's tackle this problem. Hmm, the task is to find the longest subsequence in a given sequence that forms an almost arithmetical progression. Let me first understand what an almost arithmetical progression (AAP) is.
From the problem statement, the definition is:
- The first element is p (any integer).
-... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2230601,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given a circular array of N distinct integers, find the maximum possible sum of K consecutive elements where K ≤ N. The sum should be maximized while ensuring that the difference between the maximum and minimum elements in the selected consecutive subarray is as small as possible.
INPUT:
First line contains N, the si... | <think>
Okay, I need to solve this problem where I have a circular array of N distinct integers. The task is to find the maximum possible sum of K consecutive elements. But there's a twist: among all possible subarrays of K elements that have the maximum sum, I need to choose the one where the difference between the ma... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2492528,
"category": "code",
"source": "OpenCodeReasoning"
} | |
For \(0^\circ \leq \theta < 180^\circ\), consider the function \(f(\theta) = 3\cos 2\theta + 4\sin \theta\). Find the maximum and minimum value of \(f(\theta)\). | <think>
Okay, so I have this problem here: For θ between 0 degrees and 180 degrees, the function is f(θ) = 3 cos 2θ + 4 sin θ. I need to find the maximum and minimum values of this function. Hmm, okay. Let's see. I remember that trigonometric functions can sometimes be tricky, but maybe I can simplify this using some i... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2600969,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given two integers N and K, write a program to compute the value of N^K (N raised to the power K) without using the '**' or '^' operators or the built-in pow() function. Instead, use only multiplication and bitwise shift operations to achieve the result.
Input:
First line contains T, the number of test cases.
Next T l... | <think>
Okay, I need to solve this problem where I have to compute N^K without using exponentiation functions or operators, but instead using multiplication and bitwise shifts. And also, the output has to be in a specific format that shows the step-by-step computation using shifts. Hmm, let's think about how to approac... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1379507,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a function to find the number that appears most frequently in the given list of numbers. If there are multiple numbers with the same highest frequency, return the number that appears first in the list.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def find_m... | <think>
Okay, I need to write a Python function called find_most_frequent_number in the Solution class. The goal is to find the number that appears the most in the given list. But if there are multiple numbers with the same highest frequency, I have to return the one that appeared first in the list.
Hmm, so first, I n... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 781090,
"category": "code",
"source": "opc-sft-stage2"
} | |
For \( n \) positive integers \( a_1, \ldots, a_n \), consider all their pairwise products \( a_i a_j \) where \( 1 \le i < j \le n \). Let \( N \) be the number of those products which are cubes of positive integers. Find the maximal possible value of \( N \) if it is known that none of the \( a_j \) is a cube of an i... | <think>
Okay, let's see. The problem is about finding the maximum number of pairwise products that are perfect cubes, given that none of the numbers themselves are cubes. Hmm, so we have n positive integers a1, a2, ..., an, none of which are cubes, and we want to maximize the number of pairs (ai, aj) such that ai*aj is... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 509249,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given the root of a binary tree and an integer limit, delete all insufficient nodes in the tree simultaneously, and return the root of the resulting binary tree. A node is insufficient if every root to leaf path intersecting this node has a sum strictly less than limit. A leaf is a node with no children.
Write Python ... | <think>
Okay, let's try to figure out how to solve this problem. So, the task is to delete all insufficient nodes in a binary tree. An insufficient node is one where every root-to-leaf path that goes through it has a sum strictly less than the given limit. Hmm, right. So, we need to check each node and see if all paths... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1818404,
"category": "code",
"source": "leetcode"
} | |
Let \( a, b, c \) be distinct real numbers. Calculate the minimum value of the expression
\[
\left| \frac{b-c}{c-a} \right| (|a-b| + |b-c|) + \left| \frac{c-a}{a-b} \right| (|b-c| + |c-a|) + \left| \frac{a-b}{b-c} \right| (|c-a| + |a-b|) + |a-b||b-c||c-a| + 1
\] | <think>
Okay, so I need to find the minimum value of this expression with three distinct real numbers a, b, c. The expression looks a bit complicated, but let me break it down step by step.
First, let me write out the expression again to make sure I have it right:
\[
\left| \frac{b-c}{c-a} \right| (|a-b| + |b-c|) + ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3088767,
"category": "math",
"source": "OpenMathReasoning"
} | |
Convex quadrilateral \(ABCD\) has right angles \(\angle A\) and \(\angle C\) and is such that \(AB = BC\) and \(AD = CD\). The diagonals \(AC\) and \(BD\) intersect at point \(M\). Points \(P\) and \(Q\) lie on the circumcircle of triangle \(AMB\) and segment \(CD\), respectively, such that points \(P\), \(M\), and \(Q... | <think>
Okay, so I have this geometry problem here. Let me try to parse it step by step. It's about a convex quadrilateral ABCD with right angles at A and C. Also, AB equals BC, and AD equals CD. The diagonals AC and BD intersect at point M. Then there are points P and Q: P is on the circumcircle of triangle AMB, and Q... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 877679,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train? | <think>
Okay, so there are two trains moving towards each other on parallel tracks. One is going at 60 km/h, and the other at 80 km/h. A passenger in the second train notices that the first train passes by him in 6 seconds. We need to find the length of the first train.
Hmm, let's see. Since they're moving towards ea... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3530090,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Jasmine had 3 paperclips on Monday, then she had 6 on Tuesday, and her number of paperclips proceeded to double on each subsequent day. On what day of the week did she first have more than 100 paperclips? | <think>
Okay, so Jasmine starts with 3 paperclips on Monday. Then on Tuesday, she has 6. The problem says her number of paperclips doubles each subsequent day. They want to know on which day of the week she first has more than 100 paperclips.
Let me start by listing out the days and the number of paperclips each day. ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2462276,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a **perfect binary tree** where all leaves are on the same level, and every parent has two children. The binary tree has the following definition:
struct Node {
int val;
Node \*left;
Node \*right;
Node \*next;
}
Populate each next pointer to point to its next right node. If there is no next righ... | <think>
Okay, let's see. I need to populate each next pointer in a perfect binary tree so that each node's next points to the next right node. If there's no next right node, it should be NULL. Hmm. Oh right, the tree is perfect, so all leaves are on the same level and every parent has two children. That probably helps ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2040872,
"category": "code",
"source": "leetcode"
} | |
Density is the ratio of an object's mass to its volume. The volume after compaction is $V_{2} = 0.8 V_{1}$. Given that the mass did not change due to compaction, the density after compaction is $\rho_{2} = \frac{1}{0.8}\rho_{1} = 1.25\rho_{1}$, which means it increased by $25\%$. | <think>
Okay, so I need to solve this density problem. Let me read through it again. The problem states that density is mass over volume. After compaction, the volume becomes V₂ = 0.8 V₁. The mass didn't change. They say the density after compaction is ρ₂ = (1/0.8)ρ₁, which is 1.25ρ₁, so a 25% increase. Hmm, they want ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2598963,
"category": "math",
"source": "NuminaMath-CoT"
} | |
A cone is inscribed in a sphere. The area of the lateral surface of the cone is $\frac{3}{8}$ the area of the sphere. Find the angle of inclination of the generatrix of the cone to the base plane. | <think>
Okay, so I have this problem where a cone is inscribed in a sphere. The lateral surface area of the cone is 3/8 the area of the sphere. I need to find the angle of inclination of the cone's generatrix to the base plane. Hmm, let's start by recalling some formulas and visualizing the problem.
First, the sphere ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1179440,
"category": "math",
"source": "OpenMathReasoning"
} | |
A triangle has integer sides, a perimeter of 9, and one side of length 3. Determine the possible lengths of the other two sides. | <think>
Okay, so I need to find the possible lengths of the other two sides of a triangle that has integer sides, a perimeter of 9, and one side that's length 3. Hmm, let me think step by step.
First, I remember that in a triangle, the sum of any two sides must be greater than the third side. That's the triangle inequ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1687280,
"category": "math",
"source": "OpenMathReasoning"
} | |
What is the least possible value for \( x^2 + 3x + 2 \) if \( x^2 - 3x - 2 \leq 0 \)? | <think>
Okay, so I need to find the least possible value of the quadratic expression \( x^2 + 3x + 2 \) given that \( x^2 - 3x - 2 \leq 0 \). Hmm, let me think about how to approach this.
First, maybe I should figure out the domain of x that satisfies the inequality \( x^2 - 3x - 2 \leq 0 \). To do that, I can start ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 332522,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the sum \(\sum_{i=0}^{n} \left[{n \choose i}^2\right]\). | <think>
Okay, let's see... I need to find the sum of the squares of the binomial coefficients from i=0 to n. So the problem is to evaluate ∑_{i=0}^n [C(n, i)]². Hmm, I remember that binomial coefficients have some identities, maybe there's a formula for this sum?
Wait, the binomial theorem says that (a + b)^n = ∑_{k=0... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 916410,
"category": "math",
"source": "OpenMathReasoning"
} | |
In triangle \(ABC\), let \(P\) be a point inside the triangle such that \(\angle PBA + \angle PCA = \angle PBC + \angle PCB\). If \(I\) is the incenter of triangle \(ABC\), and \(AP = x \cdot AI\), find the minimum possible value of \(x\). | <think>
Okay, so I'm trying to solve this geometry problem. Let me first make sure I understand it correctly.
We have triangle ABC, and a point P inside it such that the sum of the angles ∠PBA and ∠PCA is equal to the sum of the angles ∠PBC and ∠PCB. Then, I is the incenter of triangle ABC, and we need to find the mi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2856957,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the largest prime factor of $3^{18} + 2^9 \times 3^{13} + 2^{18} \times 3^7 + 2^{27}$. | <think>
Okay, let's see. I need to find the largest prime factor of this expression: 3^18 + 2^9 × 3^13 + 2^18 × 3^7 + 2^27. Hmm, that's a bit complex. Let me try to break it down step by step.
First, maybe I can factor out some common terms from each part of the expression. Let's look at each term individually:
1. Th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1202466,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a `pattern` and a string `s`, return `true` _if_ `s` _**matches** the_ `pattern`_._
A string `s` **matches** a `pattern` if there is some **bijective mapping** of single characters to strings such that if each character in `pattern` is replaced by the string it maps to, then the resulting string is `s`. A **bije... | <think>
Okay, I need to solve this problem where I have to check if a string s matches a given pattern based on a bijective mapping. Let me think about how to approach this.
So, the problem says that each character in the pattern must map to a string, and each character's mapping is unique. That means if the pattern i... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1889209,
"category": "code",
"source": "leetcode"
} | |
In an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle A = 20^\circ\), point \(D\) is on \(AB\) such that \(AD = BC\). Find \(\angle ACD\). | <think>
Okay, let's see. I have this geometry problem here. It says: In an isosceles triangle ABC with AB = AC and angle A is 20 degrees. There's a point D on AB such that AD equals BC. We need to find the measure of angle ACD. Hmm, okay. Let me try to visualize this.
First, since ABC is isosceles with AB = AC, that m... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3522156,
"category": "math",
"source": "OpenMathReasoning"
} | |
Problem
KND is a student programmer at the University of Aizu. He is known to be a sweet tooth. He will be staying in a city for a year and wants to visit all N sweets shops in the city during that time. So I think the best place to live for the past year is the best place to go around the sweets shop. As his neighbor... | <think>
Okay, let's tackle this problem. Hmm. So the problem is about finding the optimal point where KND should live so that the maximum time taken to reach any of the N sweet shops is minimized. The time to reach each shop is the Euclidean distance divided by the speed vi for that shop. So our goal is to find the poi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2662194,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given two strings s1 and s2, The task is to convert s1 into a palindrome such that s1 contain s2 as a substring in a minimum number of operation.
In a single operation, we can replace any word of s1 with any character.
Note: If it is not possible to convert s1 such that it is a palindrome as well as contains substring ... | <think>
Okay, I need to solve this problem where I have to find the minimum number of operations to convert s1 into a palindrome that includes s2 as a substring. The operations allowed are replacing any character in s1 with another. If it's not possible, return -1. Hmm.
First, let's understand the problem. The steps r... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 398989,
"category": "code",
"source": "TACO"
} | |
In the video game Fallout 4, the quest **"Road to Freedom "** requires players to reach a metal dial called the **"Freedom Trail Ring "** and use the dial to spell a specific keyword to open the door.
Given a string `ring` that represents the code engraved on the outer ring and another string `key` that represents the... | <think>
Okay, so I need to solve this problem where I have to find the minimum number of steps to spell a keyword using a ring, similar to the Fallout 4 game's mechanic. Let's see.
The problem says that the ring is circular, right? So rotating clockwise or anticlockwise moves the characters around. Each step is a sing... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2978594,
"category": "code",
"source": "leetcode"
} | |
In $\vartriangle ABC$, with $BC > CA > AB$, the inscribed circle is tangent to the sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$ respectively. Let $A'$, $B'$, and $C'$ be the symmetric points of $A$, $B$, and $C$ with respect to the lines $EF$, $DF$, and $DE$ respectively. If the area of $\vartriangle DEF$ is ... | <think>
Alright, let's try to tackle this geometry problem. Hmm, okay. So we have triangle ABC with sides BC > CA > AB. The incircle touches BC at D, CA at E, and AB at F. Then, A', B', and C' are the reflections of A, B, and C over the lines EF, DF, and DE respectively. The area of triangle DEF is 16, and we need to f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3857362,
"category": "math",
"source": "OpenMathReasoning"
} | |
Today, the bookstore owner has a store open for `customers.length` minutes. Every minute, some number of customers (`customers[i]`) enter the store, and all those customers leave after the end of that minute.
On some minutes, the bookstore owner is grumpy. If the bookstore owner is grumpy on the i-th minute, `grumpy... | <think>
Okay, I need to solve this problem where the bookstore owner can use a technique to not be grumpy for X consecutive minutes, and we need to maximize the number of satisfied customers. Let's think about how to approach this.
First, let's understand the problem. The customers array gives the number of customers ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2845075,
"category": "code",
"source": "leetcode"
} | |
One day Greg and his friends were walking in the forest. Overall there were n people walking, including Greg. Soon he found himself in front of a river. The guys immediately decided to get across the river. Luckily, there was a boat by the river bank, just where the guys were standing. We know that the boat can hold pe... | <think>
Okay, let's see. So the problem is about finding the minimum number of boat rides to get everyone across the river, and also the number of ways to do that in the minimum steps. The boat has a weight limit, and each person is either 50 or 100 kg. Hmm.
First, I need to figure out the constraints. The boat needs ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3413683,
"category": "code",
"source": "OpenCodeReasoning"
} | |
On the side \(BC\) of \(\triangle ABC\), take any point \(A'\). The perpendicular bisector of segment \(A'B\) intersects side \(AB\) at point \(M\), and the perpendicular bisector of segment \(A'C\) intersects side \(AC\) at point \(N\). Prove that the point symmetric to \(A'\) with respect to line \(MN\) lies on the c... | <think>
Okay, let's tackle this geometry problem. Hmm, so we have triangle ABC, and a point A' on BC. Then, the perpendicular bisector of A'B meets AB at M, and the perpendicular bisector of A'C meets AC at N. We need to show that the reflection of A' over line MN lies on the circumcircle of ABC.
First, let's recall ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1244244,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Write a python function to find the number of lucky triples in a list. A lucky triple is a tuple (lst[i], lst[j], lst[k]) where i < j < k and lst[i] divides lst[j] and lst[j] divides lst[k].
Solve the problem starting with the provided function header.
Function header:
```
def find_lucky_triples(lst):
```
Please pla... | <think>
Okay, I need to write a Python function called find_lucky_triples that counts the number of lucky triples in a given list. A lucky triple is a tuple (lst[i], lst[j], lst[k]) where the indices i < j < k, and lst[i] divides lst[j], and lst[j] divides lst[k].
Hmm, so first, what's a triple here? It's any three e... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 252613,
"category": "code",
"source": "opc-sft-stage2"
} | |
Solve the system of equations:
\[ \begin{cases} \cos x \sin y = \frac{1}{2} \\ \sin(2x) = -\sin(2y) \end{cases} \]
Problem 4a: Calculate:
\[ \cos \frac{4\pi}{9} \cos \frac{2\pi}{9} \cos \frac{\pi}{9} \]
Problem 4b: Calculate:
\[ \sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \]
Problem 4c: Calculate:
\[ \fr... | <think>
Alright, let me try to tackle these problems one by one. I'm a bit nervous because trigonometry can be tricky, but I'll take it step by step.
Starting with Problem 4a: Calculate the product cos(4π/9) cos(2π/9) cos(π/9).
Hmm, I remember there's a formula for the product of cosines at angles that are in a geome... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1084254,
"category": "math",
"source": "OpenMathReasoning"
} | |
Sum the series \( 1 + \dfrac{1}{3^3} + \dfrac{1}{5^3} - \dfrac{1}{7^3} - \dfrac{1}{9^3} - \dfrac{1}{11^3} + \dfrac{1}{13^3} + \dfrac{1}{15^3} + \dfrac{1}{17^3} \pm \ldots \) | <think>
Okay, so I need to find the sum of this series: 1 + 1/3³ + 1/5³ - 1/7³ - 1/9³ - 1/11³ + 1/13³ + 1/15³ + 1/17³ - ... and so on. The pattern seems to repeat every three terms with a sign change. Let me try to figure this out.
First, let's write down the series again to see the pattern clearly:
1 (positive)
+ ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1083001,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that for all real numbers \( x \) and \( y \):
$$
f\left(\frac{x+y}{2}\right)=\frac{f(x)}{2}+\frac{f(y)}{2}
$$ | <think>
Okay, so I have this problem where I need to find all functions f from the rationals to the rationals such that for all real numbers x and y, the function satisfies f((x + y)/2) = (f(x) + f(y))/2. Hmm, interesting. Let me try to unpack this step by step.
First, the function is defined on the rational numbers, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1379908,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \( f \) be a polynomial function satisfying \( f(2 + x) = f(1 - 3x) \) for all \( x \) in the set of all real numbers. If \( f \) has exactly 4 real zeroes, find the sum of these zeros. | <think>
Okay, let's see. I need to solve this problem where there's a polynomial function f that satisfies the equation f(2 + x) = f(1 - 3x) for all real numbers x. And it's given that f has exactly 4 real zeros. The question is to find the sum of these zeros. Hmm.
First, let me understand what this functional equatio... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 649462,
"category": "math",
"source": "OpenMathReasoning"
} | |
Can two ships $1 \times 4$, four ships $1 \times 3$, six ships $1 \times 2$, and eight ships $1 \times 1$ be arranged on a $10 \times 10$ square board such that no two ships share a common point? | <think>
Okay, let's try to figure out if we can arrange all these ships on a 10x10 board without any of them touching each other, not even at a corner. So the ships are: one 1x4, four 1x3s, six 1x2s, and eight 1x1s. Hmm. Let's start by calculating the total area needed. The 1x4 is 4 squares. Each 1x3 is 3, so four of t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1887823,
"category": "math",
"source": "OpenMathReasoning"
} | |
In how many ways can you make change for a dollar using pennies, nickels, dimes, quarters, and half dollars? | <think>
Okay, so the problem is to find out how many ways you can make change for a dollar using pennies, nickels, dimes, quarters, and half dollars. Let me think. Hmm, this is a classic combinatorics problem. I remember that these kinds of problems involve figuring out all the possible combinations of coins that add u... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2746805,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve in reals:
\[ 2x^3 + 3x^2 + 2x + 1 = x(2x + 3)\sqrt{x^2 + \frac{1}{x}} \] | <think>
Okay, so I have this equation to solve: 2x³ + 3x² + 2x + 1 = x(2x + 3)√(x² + 1/x). Hmm, looks a bit complicated. Let me start by understanding the equation and thinking about possible steps to solve it.
First, I should note that since there's a square root involved, √(x² + 1/x), the expression inside the squar... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3437324,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given an array of `n` integers `nums`, a 132 pattern is a subsequence of three integers `nums[i]`, `nums[j]` and `nums[k]` such that `i < j < k` and `nums[i] < nums[k] < nums[j]`.
Return `true` if there is a 132 pattern in `nums`, otherwise, return `false`.
Follow up: The `O(n^2)` is trivial, could you come up with t... | <think>
Okay, I need to solve this 132 pattern problem. Let me think. So the problem is to determine if there exists a subsequence of three elements where the first is less than the third, which is less than the second. And the indices are in order. Hmm.
The brute force approach would be O(n^3) time, checking all poss... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 551078,
"category": "code",
"source": "leetcode"
} | |
You are given a **0-indexed** integer array `nums`. For each index `i` (`1 <= i <= nums.length - 2`) the **beauty** of `nums[i]` equals:
* `2`, if `nums[j] < nums[i] < nums[k]`, for **all** `0 <= j < i` and for **all** `i < k <= nums.length - 1`.
* `1`, if `nums[i - 1] < nums[i] < nums[i + 1]`, and the previous co... | <think>
Okay, I need to solve this problem where I have to calculate the sum of beauty for each element in the array except the first and last. Let's see. The beauty is defined in three conditions.
First, the highest possible beauty is 2. That's when all elements to the left are smaller than nums[i], and all elements... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1989071,
"category": "code",
"source": "leetcode"
} | |
For a right square pyramid with volume \( V \) and surface area \( A \), it is given that \( \frac{V^2}{A^3} \le \frac{m}{n} \), where \( \gcd(m, n) = 1 \). Determine \( m + n \). | <think>
Okay, so I have this problem about a right square pyramid. The question says that for such a pyramid with volume V and surface area A, the ratio V²/A³ is less than or equal to m/n, where m and n are coprime integers. I need to find m + n. Hmm, let's break this down step by step.
First, let me recall the formul... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2280567,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( A_1, B_1, C_1 \) be the midpoints of the sides \( BC, AC, AB \) of an equilateral triangle \( ABC \). A circle \( \gamma \) is circumscribed around triangle \( A_1B_1C_1 \). Tangents \( B_2C_2 \), \( A_2C_2 \), and \( A_2B_2 \) are drawn to \( \gamma \), respectively, parallel to the sides \( BC \), \( AC \), an... | <think>
Okay, let me try to solve this geometry problem. So, we have an equilateral triangle ABC. The midpoints of the sides BC, AC, and AB are A1, B1, C1 respectively. Then, there's a circle γ circumscribed around triangle A1B1C1. Then, they draw tangents B2C2, A2C2, and A2B2 to γ, each parallel to the sides BC, AC, a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2636812,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to find the n-th number in the Pell number series.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def get_pell(self, 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 get_pell inside the Solution class that returns the nth Pell number. Hmm, what's a Pell number again? Oh right, Pell numbers are a sequence similar to Fibonacci numbers, but with a different recurrence relation. Let me think. The Pell sequence starts with P0 = 0, P... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 850538,
"category": "code",
"source": "opc-sft-stage2"
} | |
You are asked to write a function named `GiniIndex` that computes the Gini index for a given list of values. The Gini index is a measure of inequality where 0 represents perfect equality and 1 represents maximum inequality. The function uses the Lorenz curve and the perfect equality curve to compute the Gini index.
He... | <think>
Okay, I need to write a Python function called GiniIndex that calculates the Gini index for a given list of values. The steps are outlined, so I should follow them carefully.
First, step 1 says to sort the list. So I'll start by taking the input list X and sorting it in ascending order. Wait, but do I modify t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2917955,
"category": "code",
"source": "opc-sft-stage2"
} | |
Write a python function to find the number of equi-leaders in a list. An equi-leader is an index at which the list can be split into two sublists, and the most frequent number (mode) in the whole list is the mode in both sublists.
Solve the problem starting with the provided function header.
Function header:
```
def ... | <think>
Okay, I need to write a Python function called equi_leader that finds the number of equi-leaders in a list. Hmm, equi-leaders are indices where splitting the list into two parts (left and right of the index) results in both parts having the same leader as the entire list. The leader is the mode, the most freque... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1440671,
"category": "code",
"source": "opc-sft-stage2"
} | |
Is it possible to cover a \(5 \times 7\) rectangle with L-shaped tiles made of three cells (formed by removing one cell from a \(2 \times 2\) square) without extending beyond its borders, such that each cell of the rectangle is covered by the same number of cells belonging to the L-shaped tiles in multiple layers? | <think>
Okay, so I need to figure out if it's possible to cover a 5x7 rectangle using L-shaped tiles. Each tile is made of three cells, like a 2x2 square missing one cell. The catch is that the tiles can overlap, but every cell in the 5x7 grid must be covered by the same number of layers of tiles. Hmm, interesting.
Fi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2150985,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find the smallest natural number $k$ such that $k \cdot 2^n + 1$ is composite for every $n \in \{1, 2, 3, \ldots, 10\}$. | <think>
Okay, so I need to find the smallest natural number k such that for each n from 1 to 10, the expression k·2ⁿ +1 is composite. Hmm, let's break this down.
First, natural number k, so k has to be a positive integer. The condition is that for every n in {1,2,...,10}, when you calculate k multiplied by 2 to the p... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 463415,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve the equation \((x^{2}+y^{2})^{3}=(x^{3}-y^{3})^{2}\) for real numbers \(x\) and \(y\). | <think>
Alright, let's try to solve this equation: \((x^{2} + y^{2})^{3} = (x^{3} - y^{3})^{2}\) for real numbers x and y. Hmm, okay. So, I need to find all real pairs (x, y) that satisfy this equation. Let me start by expanding both sides to see if I can simplify things.
First, let's look at the left-hand side (LHS):... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2722298,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are working with a dataset sourced from multiple CSV files, and your task is to consolidate these files into a single DataFrame using the `pandas` library in Python. The dataset is distributed across 12 CSV files. The first file is explicitly provided, and you need to append the data from the remaining 11 files to ... | <think>
Okay, let's see. I need to create a function called csv_combiner that takes an initial DataFrame df and a verbose flag. The goal is to combine data from 11 more CSV files into this df. Hmm.
First, the initial DataFrame is from all_energy_statistics1.csv. The rest are 2 to 12. So there are 12 files in total. Th... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2465936,
"category": "code",
"source": "opc-sft-stage2"
} | |
What is the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle <ABC = 60 degrees? | <think>
Okay, let's try to figure out the area of the largest semicircle that can fit inside a rhombus with side length 1 and angle ABC of 60 degrees. Hmm, first, I need to visualize the problem. A rhombus has all sides equal, and opposite angles equal. Since one angle is 60 degrees, the adjacent angle must be 120 degr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 152667,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to count the number of ways to divide a number into a sum of positive integers. The order of the summands does not matter.
Solve the problem starting with the provided function header.
Function header:
```
def count_partitions(n, i = None):
```
Please place the solution code in the following format:... | <think>
Okay, I need to write a function called count_partitions that calculates the number of ways to partition a number n into a sum of positive integers, where the order doesn't matter. The function header is given as def count_partitions(n, i = None):. Hmm, the second parameter i is optional. I wonder what that rep... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 967928,
"category": "code",
"source": "opc-sft-stage2"
} | |
Write a function that takes a list of integers and returns the maximum sum that can be obtained by partitioning the list into sublists of size at most k and summing the maximum element in each sublist.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def max_partitio... | <think>
Okay, let's see. I need to write a Python function called max_partition_sum that takes a list of integers and an integer k, and returns the maximum sum possible by partitioning the list into sublists of size at most k. Each sublist contributes the maximum element of that sublist to the total sum. So I need to s... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1040710,
"category": "code",
"source": "opc-sft-stage2"
} | |
In a plane with 137 seats, the first passenger randomly takes any seat. Subsequent passengers take their assigned seats if available, or a random available seat if their seat is occupied. What is the probability that the last passenger finds her assigned seat unoccupied? | <think>
Okay, let's try to figure out this probability problem. So, the setup is a plane with 137 seats, right? The first passenger boards and randomly picks a seat. Then each subsequent passenger takes their assigned seat if it's available, otherwise they pick a random available seat. The question is, what's the proba... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3860247,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to find the maximum number of flowers that can be planted in a garden without violating the given conditions. The garden is represented by a list where 0 indicates an empty spot and 1 indicates a spot with a flower. You can plant a flower in an empty spot if it is not adjacent to another flower. Return... | <think>
Okay, I need to solve this problem where I have to find the maximum number of flowers that can be planted in a garden without violating the adjacent rule. The garden is represented by a list of 0s and 1s. 0 means empty, 1 means already has a flower. The rule is that you can't plant a flower next to another one.... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2308934,
"category": "code",
"source": "opc-sft-stage2"
} | |
In the X-ray lab at KTH some experiments require evacuated tubes between source and sample and between sample and detector so that the X-rays are not absorbed by the air. Since the positions of object and detector vary between different experiments, several tubes of different lengths are available. The tubes should be ... | <think>
Okay, let's see. The problem is about selecting four tubes from a list such that the sum of the first two is <= L1 and the sum of the next two is <= L2. And we need to maximize the total sum of all four tubes. If it's not possible to find such four tubes, output Impossible.
Hmm, right. So the approach here is ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1656747,
"category": "code",
"source": "TACO"
} | |
What is the relationship between the sides of two triangles inscribed in the same circle? | <think>
Okay, so I need to figure out the relationship between the sides of two triangles that are inscribed in the same circle. Hmm, let's start by recalling some basic geometry concepts. First, when a triangle is inscribed in a circle, it's called a cyclic triangle, and the circle is the circumcircle of the triangle.... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3758894,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \( a_n = \frac{(2+\sqrt{3})^{2^n} + (2-\sqrt{3})^{2^n}}{2} \) for \( n \geq 1 \). If \( p \) is a prime such that \( p \mid a_n \) and \( 12 \mid (p-1) \), find the smallest possible value of \( k \) such that \( 2^k \mid (p-1) \) for \( n = 4 \). | <think>
Okay, let me try to work through this problem step by step. So, the problem is about finding the smallest possible value of k such that 2^k divides (p-1), where p is a prime that divides a_4, and 12 divides (p-1). The sequence a_n is given by that formula involving (2+√3) and (2-√3) raised to 2^n, all divided b... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3324009,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a python function to rotate an array to the left by one position.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def rotate_left_one(self, arr):
```
Please place the solution code in the following format:
```python
# Your solution code here
``` | <think>
Okay, I need to write a Python function called rotate_left_one that takes an array and rotates it to the left by one position. Let me think about how to do that.
So, rotating left by one means that each element moves one position to the left, and the first element moves to the end. For example, if the array is... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2164016,
"category": "code",
"source": "opc-sft-stage2"
} | |
Read problem statements in [Hindi], [Bengali], [Mandarin Chinese], [Russian], and [Vietnamese] as well.
Chef likes assigning scores to pairs of numbers in an unusual way:
Consider a pair $(X, Y)$ of integers in decimal representation. If they don't have the same number of digits, add leading zeros to the smaller of th... | <think>
Okay, let's see. The problem is to compute the sum of the scores for all pairs (i,j) where i and j are in [L, R]. The score is the minimum possible sum of absolute differences of digits when Y is permuted optimally. Oh right, and the numbers can have leading zeros to make their lengths equal. So for each pair (... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2966637,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Solve the inequality
\[\frac{(x - 2)(x - 3)(x - 4)}{(x - 1)(x - 5)(x - 6)} > 0.\] | <think>
Okay, so I have this inequality to solve: \[\frac{(x - 2)(x - 3)(x - 4)}{(x - 1)(x - 5)(x - 6)} > 0.\] Hmm, alright, let me start by recalling how to solve rational inequalities. The idea is to find where the expression is positive, right? So first, I need to identify the critical points where the numerator or ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 536379,
"category": "math",
"source": "OpenMathReasoning"
} | |
A circle is inscribed in quadrilateral $ABCD$, tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle. | <think>
Okay, let me try to figure out this problem. So we have a quadrilateral ABCD with an inscribed circle. That means the circle is tangent to all four sides. The circle touches AB at point P and CD at point Q. We’re given the lengths: AP is 19, PB is 26, CQ is 37, and QD is 23. We need to find the square of the ra... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2202926,
"category": "math",
"source": "OpenMathReasoning"
} | |
Memory and his friend Lexa are competing to get higher score in one popular computer game. Memory starts with score a and Lexa starts with score b. In a single turn, both Memory and Lexa get some integer in the range [ - k;k] (i.e. one integer among - k, - k + 1, - k + 2, ..., - 2, - 1, 0, 1, 2, ..., k - 1, k) and add... | <think>
Okay, let's try to solve this problem. So, the task is to find the number of possible games where Memory ends up with a strictly higher score than Lexa after t turns. Both players add a random integer from [-k, k] each turn for t times. The answer needs to be computed modulo 1e9+7.
Hmm. Let's think about the v... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 204521,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given a garland with $n$ lights, where some lights are on and some are off, in one move we can turn off a turned-on light and also change the state of its neighboring lights. For what values of $n$ is it always possible to turn off all the lights after some moves? | <think>
Okay, so I have this problem about a garland of n lights. Some are on, some are off. The move allowed is turning off a lit light and flipping the state of its neighbors. The question is for which values of n it's always possible to turn all the lights off, no matter the starting configuration. Hmm, interesting.... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3439825,
"category": "math",
"source": "OpenMathReasoning"
} | |
Misha made a homemade darts game during the summer at the dacha. A circular board is divided into several sectors by concentric circles, and darts can be thrown at it. Points are awarded based on the number written in the sector, as shown in the diagram.
Misha threw 8 darts three times. In the second attempt, he score... | <think>
Okay, let's try to figure out this problem step by step. So, the problem is about Misha playing darts. He threw 8 darts three times. The second attempt's score was twice the first, and the third was 1.5 times the second. We need to find the score of the second attempt.
Hmm, wait, the problem says he threw 8 da... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1751829,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Evaluate the sum $\sum_{n\geq 1} \frac{1}{2^{2^{2^n}}}$. | <think>
Okay, so I need to evaluate the sum that's given: the sum from n ≥ 1 of 1 over 2 raised to the power of 2^(2^n). Hmm, that seems pretty complicated. Let me write it out to visualize better. The general term is 1/(2^{2^{2^n}}). So when n starts at 1, the exponents would be 2^(2^1) = 2^2 = 4, then for n=2, it's 2... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3376479,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the integral
\[ \int_0^{\infty} x^{-x} \, dx. \] | <think>
Okay, so I have this integral to solve: the integral from 0 to infinity of x^(-x) dx. Hmm, that's written as ∫₀^∞ x^{-x} dx. I need to evaluate this. Alright, let me think about how to approach this.
First, x^{-x} is the same as 1/(x^x). So the integrand is 1 over x raised to the x power. That seems a bit tric... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3622387,
"category": "math",
"source": "OpenMathReasoning"
} | |
Determine $ k$ so that the equation $ x^4 - 14x^2 + 24x - k = 0$ has 4 distinct roots. | <think>
Okay, so I need to find the value of k such that the equation x⁴ -14x² +24x -k =0 has four distinct roots. Hmm, let's think about this step by step.
First, I remember that a quartic equation can have up to four real roots, but the problem specifies four distinct roots. So, the equation must cross the x-axis f... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3616735,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate $\int x \sqrt{2x + 1} \, dx$. | <think>
Okay, so I need to evaluate the integral of x times the square root of (2x + 1) dx. Hmm, let's see. I remember that when you have an integral with a function inside another function, like a square root here, substitution might be a good approach. Let me try that.
First, let me set u equal to the inside of the ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3348291,
"category": "math",
"source": "OpenMathReasoning"
} | |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Missy, Lauren, and Liz. In how many ways can we choose 6 starters if the only restriction is that not all 3 triplets can be in the starting lineup? | <think>
Okay, so the problem is about the volleyball team with 14 players, including three triplets: Missy, Lauren, and Liz. We need to figure out how many ways we can choose 6 starters with the restriction that not all three triplets can be in the starting lineup. Hmm, let me think through this step by step.
First, I... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3051060,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a set $T$, which is a subset of $S$. The set $S$ consists of $0, 1, ... n-1$. Print all subsets of $T$. Note that we represent $0, 1, ... n-1$ as 00...0001, 00...0010, 00...0100, ..., 10...0000 in binary respectively and the integer representation of a subset is calculated by bitwise OR of existing elemen... | <think>
Okay, let's tackle this problem. So, the task is to generate all subsets of the given set T and output them sorted by their integer representation. The integer for a subset is the bitwise OR of all elements in it. Each element in the original set S (from 0 to n-1) is represented as a bit in binary. For example,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2756752,
"category": "code",
"source": "OpenCodeReasoning"
} | |
A fair coin is flipped 10 times. Let \( \frac{i}{j} \), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \( i+j \). | <think>
Okay, so I need to find the probability that when flipping a fair coin 10 times, there are no two heads in a row. Then express that probability as a reduced fraction i/j and find i + j. Hmm, let's start breaking this down.
First, when dealing with probabilities involving sequences and restrictions like no cons... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2065212,
"category": "math",
"source": "OpenMathReasoning"
} | |
What is the remainder when $6^{2006} - 1$ is divided by 43? | <think>
Okay, so I need to find the remainder when 6^2006 minus 1 is divided by 43. Hmm, let's see. I remember that when dealing with exponents modulo some number, Euler's theorem might be helpful. Let me recall: Euler's theorem states that if a and n are coprime, then a^φ(n) ≡ 1 mod n, where φ is Euler's totient funct... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3554954,
"category": "math",
"source": "OpenMathReasoning"
} | |
The image shows how we can obtain the Harmonic Conjugated Point of three aligned points A, B, C.
- We choose any point L, that is not in the line with A, B and C. We form the triangle ABL
- Then we draw a line from point C that intersects the sides of this triangle at points M and N respectively.
- We draw the diago... | <think>
Okay, let's tackle this problem step by step. The goal is to find the harmonic conjugate point D given three collinear points A, B, C in order xA < xB < xC, such that xD is between xA and xB.
First, I need to recall the formula for the harmonic conjugate. From the problem statement, there's a mention of using... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3060823,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Restore $\triangle ABC$ given the lengths of sides $b$ and $c$, and the length of the angle bisector between them. | <think>
Okay, so I need to restore triangle ABC where I know the lengths of sides b and c, and the length of the angle bisector between them. Let me think about how to approach this.
First, let me recall some properties of angle bisectors in triangles. The angle bisector theorem states that the angle bisector divides ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2529016,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given two triangles \( A_1 A_2 A_3 \) and \( B_1 B_2 B_3 \), "describe" around triangle \( A_1 A_2 A_3 \) a triangle \( M_1 M_2 M_3 \) with the largest area, similar to triangle \( B_1 B_2 B_3 \) (vertex \( A_1 \) must lie on line \( M_2 M_3 \), vertex \( A_2 \) must lie on line \( A_1 A_3 \), vertex \( A_3 \) must lie... | <think>
Okay, so I have this problem here where I need to describe a triangle M1M2M3 around triangle A1A2A3 such that M1M2M3 is similar to another triangle B1B2B3 and has the largest possible area. Also, each vertex of triangle A1A2A3 has to lie on a specific side of triangle M1M2M3. Let me try to break this down step ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1579075,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Write a function to find the shortest path from a source to multiple destinations in a directed graph. The graph is represented as a list of roads connecting nodes.
Solve the problem starting with the provided function header.
Function header:
```
def find_shortest_paths(n, roads, source, destinations):
```
Please p... | <think>
Okay, I need to write a Python function called find_shortest_paths that finds the shortest paths from a source node to multiple destinations in a directed graph. The graph is represented by a list of roads, which I guess are the edges. Let's think about the parameters. The function takes n, which is the number ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 839732,
"category": "code",
"source": "opc-sft-stage2"
} | |
Find all real solutions to the system
$$
x^{4} - 17 = y^{4} - 7 = z^{4} + 19 = u^{4} + 5 = xyzu
$$
Solve the equation
$$
5^{x} + 12^{x} = 13^{x}
$$ | <think>
Alright, let's try to tackle these two problems. First, the system of equations with x, y, z, u and then the exponential equation. Let me start with the first one.
The system is given as:
x⁴ - 17 = y⁴ - 7 = z⁴ + 19 = u⁴ + 5 = xyzu
Hmm, so all these expressions are equal to the same value, which is also equal... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3570429,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya stood exactly 9 meters behind Vasya at the starting point. Who finished first in the second race and by how many meters did he lead his opponent? (Assume that each boy ran both times at his own con... | <think>
Okay, so Petya and Vasya had a 60-meter race. When Petya finished, Vasya was 9 meters behind. Now, in the second race, Petya starts 9 meters behind Vasya. We need to figure out who wins the second race and by how much. Let me break this down step by step.
First, let's understand their speeds. In the first race... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1630843,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Determine if the limit
\[ \lim_{\varepsilon \downarrow 0} \int_{|x| > \varepsilon} \frac{f(x)}{x} \, dx \]
exists and is finite when \( f \) is merely continuous on \([-1,1]\). | <think>
Okay, so I need to figure out whether the limit as epsilon approaches zero from above of the integral from |x| > epsilon of f(x)/x dx exists and is finite when f is just continuous on [-1,1]. Hmm. Let me start by recalling what this kind of integral is called. Oh right, this looks like a principal value integra... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2002301,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the number of distinguishable non-square rectangles, with corners at points with integer coordinates, that can be counted on an 8 by 8 checkerboard placed with its corners at (0,0), (8,0), (0,8), and (8,8). | <think>
Okay, so I need to find the number of distinguishable non-square rectangles on an 8x8 checkerboard. The corners have to be at integer coordinates, and it's an 8x8 grid, which I think means the points go from (0,0) to (8,8). Wait, but the checkerboard usually has 8x8 squares, so each square is from (0,0) to (1,1... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3432518,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider the product
\[
P = (2x_1^2 + 3y_1^2)(2x_2^2 + 3y_2^2) \ldots (2x_{2011}^2 + 3y_{2011}^2)
\]
where \( x_1, x_2, \ldots, x_{2011}, y_1, y_2, \ldots, y_{2011} \) are positive integers. Let \( S \) be the sum of all possible values of \( x_i \) and \( y_i \) such that \( P \) is closest to but not a perfect square... | <think>
Okay, let's tackle this problem step by step. The question is about finding the sum \( S \) of all possible values of \( x_i \) and \( y_i \) such that the product \( P \) is closest to but not a perfect square. Then we need to find \( S \mod 1000 \).
First, let me make sure I understand the problem correctly... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3656401,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to check if a given list of integers can be sorted by swapping two elements. If the list can be sorted by a single swap, return the indices of the elements to be swapped. If the list is already sorted or cannot be sorted by a single swap, return an empty list.
Solve the problem starting with the provi... | <think>
Okay, I need to solve this problem where I have to check if a list of integers can be sorted by swapping two elements. If it can, return their indices. Otherwise, return an empty list. Hmm, let's think about how to approach this.
First, the problem says that if the list is already sorted, return empty. So I ne... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1136856,
"category": "code",
"source": "opc-sft-stage2"
} | |
I am faced with a challenge that requires me to identify the K Closest Elements. The task is as follows: You are provided with a sorted integer array, denoted as `arr`, along with two integers, `k` and `x`. Your mission is to return the `k` integers in the array that are closest to `x`, with the output also being sorte... | <think>
Okay, let's see. I need to find the k closest elements to x in a sorted array. The result should be sorted in ascending order. Hmm. So the array is already sorted, which should help. Let's think about the examples given.
In the first example, when x is 3, the closest elements are 1,2,3,4. Since 3 is in the arr... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3313538,
"category": "code",
"source": "MagicoderEvolInstruct"
} | |
Let \(\xi_{1}, \ldots, \xi_{m}\) and \(\eta_{1}, \ldots, \eta_{n}\) be random variables. Form the random vectors \(X=\left(\xi_{1}, \ldots, \xi_{m}\right)\) and \(Y=\left(\eta_{1}, \ldots, \eta_{n}\right)\). Suppose the following conditions are satisfied:
1) The random variables \(\xi_{1}, \ldots, \xi_{m}\) are indepen... | <think>
Okay, so I need to prove that all the random variables ξ₁, ..., ξₘ, η₁, ..., ηₙ are independent given that the vectors X = (ξ₁, ..., ξₘ) and Y = (η₁, ..., ηₙ) are independent, and within each vector, the variables are independent. Let me think about how to approach this.
First, I remember that independence of ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 489300,
"category": "math",
"source": "NuminaMath-CoT"
} | |
In triangle ABC, side AC and the perpendicular bisector of BC meet at point D, and BD bisects ∠ABC. If AD = 9 and DC = 7, what is the area of triangle ABD? | <think>
Okay, let's see. I need to find the area of triangle ABD given some conditions. Hmm, let me start by drawing a rough sketch to visualize the problem. So there's triangle ABC. The perpendicular bisector of BC meets AC at point D. Also, BD bisects angle ABC. AD is 9 units, and DC is 7 units. That makes AC a total... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1574928,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to check if a given string is a rotation of a palindrome. A palindrome is a word, phrase, number, or other sequence of characters that reads the same backward or forward. A rotation of a string is a rearrangement of its characters by moving some of them to the beginning or end of the string.
For examp... | <think>
Okay, I need to write a function called is_rotation_of_palindrome that checks if a given string is a rotation of a palindrome. Hmm. Let me think about what this means.
So a palindrome is a string that reads the same forwards and backwards. Like "racecar" or "madam". A rotation of a string means that you shift ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 927855,
"category": "code",
"source": "opc-sft-stage2"
} | |
Dr. Sato, a botanist, invented a number of special fertilizers for seedlings. When you give the fertilizer to the seedlings, the size of the seedlings changes in a blink of an eye. However, it was found that fertilizer has the following side effects.
* The size of the seedlings does not change with the fertilizer give... | <think>
Okay, I need to solve this programming problem. Let me read the problem statement carefully.
So the problem is about finding the maximum possible size of a seedling after applying fertilizer m times. The fertilizers are applied in a sequence, and the growth rate depends on the previous fertilizer used. The fir... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3370953,
"category": "code",
"source": "OpenCodeReasoning"
} | |
How many positive integers \( n \) satisfy the inequality \( 1 + \sqrt{n^2 - 9n + 20} > \sqrt{n^2 - 7n + 12} \)? | <think>
Okay, so I have this inequality to solve: 1 + sqrt(n² - 9n + 20) > sqrt(n² - 7n + 12). I need to find how many positive integers n satisfy this. Hmm, let me think step by step.
First, I should check the domains of the square roots because the expressions inside them must be non-negative. So, let me start by fi... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3759597,
"category": "math",
"source": "OpenMathReasoning"
} | |
This is an interactive problem. In the output section below you will see the information about flushing the output.
On Sunday Leha the hacker took Nura from the house where she lives and went with her to one of the most luxurious restaurants in Vičkopolis. Upon arrival, they left the car in a huge parking lot near the... | <think>
Okay, let's try to figure out how to solve this problem. So, the problem is that Leha needs to guess two dishes that Noora ordered by asking up to 60 queries. Each query is two numbers x and y, and Noora responds based on some minimum distance calculations. The challenge is to find two of the ordered dishes eff... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1499883,
"category": "code",
"source": "OpenCodeReasoning"
} | |
When 5655 is divided by a two-digit positive integer \( N \), the remainder is 11. When 5879 is divided by the same positive integer \( N \), the remainder is 14. What is the sum of the digits of \( N \)? | <think>
Okay, let me try to figure out this problem. So, the problem says that when 5655 is divided by a two-digit number N, the remainder is 11. And when 5879 is divided by the same N, the remainder is 14. We need to find the sum of the digits of N. Hmm, alright, let's break this down step by step.
First, I remember ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1117647,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Determine the probability that a real number \( r \) chosen from the interval \((0,2)\) satisfies \(\lfloor \log_{20} (24r) \rfloor = \lceil \log_{20} r \rceil\). | <think>
Okay, so I need to find the probability that a real number r chosen from the interval (0,2) satisfies the equation floor(log base 20 of (24r)) equals ceiling(log base 20 of r). Hmm, let me start by understanding what this equation means.
First, let's recall that floor(x) is the greatest integer less than or eq... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2231271,
"category": "math",
"source": "OpenMathReasoning"
} | |
Prove that if the plane is divided into parts by lines and circles, then the resulting map can be colored with two colors so that the parts sharing a segment or an arc will be of different colors. | <think>
Okay, so I need to prove that if the plane is divided into parts by lines and circles, the resulting map can be colored with just two colors such that any two adjacent regions (sharing a segment or an arc) have different colors. Hmm, this sounds like a graph coloring problem, right? But instead of a graph, it's... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2864764,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Seven pirates were dividing five identical treasure chests. They agreed that five of them would take one chest each, while the remaining two would receive fair compensation equal to the value of a chest. Each recipient of a chest paid 10,000 doubloons into a common fund, which was then distributed among the remaining p... | <think>
Okay, let's try to figure out this pirate treasure problem. So, there are seven pirates dividing five identical treasure chests. The agreement is that five pirates will each take one chest, and the remaining two pirates will get fair compensation equal to the value of a chest. Each of the five pirates who took ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2706406,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Consider the set of points in the \( x-y \) plane:
\[ A = \{(a, b) \mid a, b \in \mathbb{Z}, 0 \leq a \leq 9 \wedge 0 \leq b \leq 5\}. \]
Find the number of rectangles whose vertices are points in \( A \). | <think>
Okay, let's see. I need to find the number of rectangles that can be formed from the set of points A, where each point has integer coordinates (a, b) with a between 0 and 9 inclusive and b between 0 and 5 inclusive. Hmm, rectangles... right, so rectangles in a grid are usually defined by their opposite sides be... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 586864,
"category": "math",
"source": "OpenMathReasoning"
} | |
Read problems statements in Mandarin Chinese, Russian and Vietnamese as well.
Limak is a big grizzly bear who loves wrestling.
He is going to participate in a coming tournament.
There will be N = 2^{H} bears participating in the tournament.
Bears are numbered 1 through N and Limak has index 1.
Each bear has a unique... | <think>
Okay, I need to solve this problem. Let's read it carefully.
So the problem is about a tournament where bears fight in stages. Limak is bear 1, and he wants to win the tournament. The question is the minimum number of bribes he needs to do. Each bribe can flip the result of a match only if the strength differe... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2050728,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \( X \) be a vector space with two equivalent norms \( \|\cdot\|_1 \) and \( \|\cdot\|_2 \). Let \( f^*_1 \) and \( f^*_2 \) be the dual norms associated with \( \|\cdot\|_1 \) and \( \|\cdot\|_2 \) respectively. If \( \alpha \) and \( \beta \) are positive constants such that \( \alpha \|x\|_1 \leq \|x\|_2 \leq \b... | <think>
Okay, so I have this problem about dual norms in a vector space. Let me try to unpack it step by step.
First, the problem says that X is a vector space with two equivalent norms, ||·||₁ and ||·||₂. Then, there are dual norms f*₁ and f*₂ associated with these. The question is asking, given that α||x||₁ ≤ ||x||... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 928051,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the best constant \( k \) such that the inequality \( a^{3} + b^{3} + c^{3} - 3abc \ge 2\left(\frac{a+b}{k} - c\right)^{3} \) holds for all real numbers \( a, b, \) and \( c \). | <think>
Okay, so I need to find the best constant k such that the inequality a³ + b³ + c³ - 3abc ≥ 2((a+b)/k - c)³ holds for all real numbers a, b, c. Hmm, let me start by understanding the left-hand side and the right-hand side of this inequality.
First, I remember that the expression a³ + b³ + c³ - 3abc is a known i... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3800026,
"category": "math",
"source": "OpenMathReasoning"
} | |
Positive integers \( x \) and \( y \) satisfy \( 6x + 7y = 2012 \). Let the minimum value of \( x + y \) be \( p \) and the maximum value be \( q \). Find \( p + q \). | <think>
Okay, so I need to solve this problem where positive integers x and y satisfy the equation 6x + 7y = 2012. I have to find the minimum and maximum values of x + y, called p and q respectively, and then add them together for the final answer. Let me think through how to approach this step by step.
First, since b... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 580854,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Watchmen are in a danger and Doctor Manhattan together with his friend Daniel Dreiberg should warn them as soon as possible. There are n watchmen on a plane, the i-th watchman is located at point (x_{i}, y_{i}).
They need to arrange a plan, but there are some difficulties on their way. As you know, Doctor Manhattan co... | <think>
Okay, let's see. The problem is to find the number of pairs of points where the Manhattan distance equals the Euclidean distance. Wait, so Manhattan distance is |x_i - x_j| + |y_i - y_j|, and Euclidean is sqrt((x_i -x_j)^2 + (y_i - y_j)^2). We need these two to be equal.
Hmm, let's think about when these two d... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 336828,
"category": "code",
"source": "OpenCodeReasoning"
} | |
A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 15 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes. | <think>
Okay, so I need to solve this geometry problem about a crosswalk on a street. Let me try to visualize it first. The street has two parallel curbs that are 40 feet apart. There's a crosswalk with two parallel stripes crossing the street at an angle. The problem says that the length of the curb between the stripe... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2565521,
"category": "math",
"source": "OpenMathReasoning"
} |
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