instruction stringlengths 21 6.8k | input stringclasses 1
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Given any positive integer \( N \), show that there is just one solution to \( m + \frac{1}{2}(m + n - 1)(m + n - 2) = N \) in positive integers. | <think>
Okay, so I need to show that for any positive integer N, there's exactly one pair of positive integers (m, n) that satisfies the equation m + ½(m + n - 1)(m + n - 2) = N. Hmm, let's start by trying to understand the structure of this equation.
First, let's rewrite the equation to make it a bit clearer. The mai... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1305597,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Given a triangle $ABC$ with inradius $r$, circumradius $R$, and semiperimeter $s$, let $r_a$, $r_b$, and $r_c$ denote the exradii. Define $f(R, r, s) = \frac{R}{r} + 1 - \left(\frac{r_a + r_b + r_c}{s}\right)^2$. Calculate the minimum value of $f(R, r, s)$ over all possible triangles $ABC$. | <think>
Okay, so I have this problem here about a triangle ABC with inradius r, circumradius R, and semiperimeter s. The exradii are r_a, r_b, r_c. The function to minimize is f(R, r, s) = R/r + 1 - ( (r_a + r_b + r_c)/s )². Hmm, interesting. I need to find the minimum value of this function over all possible triangles... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1195915,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find integers \( n \) and \( m \) such that \( 3n^2 + 3n + 7 = m^3 \). | <think>
Okay, so I need to find integers n and m such that 3n² + 3n + 7 equals m cubed. Hmm, let's see. Let me start by understanding the problem. The equation is 3n² + 3n + 7 = m³. Both n and m are integers, so I need to find integer solutions.
First, maybe I should try plugging in some small integer values for n an... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3267285,
"category": "math",
"source": "OpenMathReasoning"
} | |
A point moves along the parabola \( y = x^2 \). Its \( x \)-coordinate is increasing uniformly at 2 units per minute. At what rate is its distance from the point \( (1, -1) \) changing when it passes through \( (2, 4) \)? | <think>
Okay, so I have this problem here where a point is moving along the parabola y equals x squared. The x-coordinate is increasing at a constant rate of 2 units per minute. I need to find the rate at which the distance from the point (1, -1) is changing when the point passes through (2, 4). Hmm, let me think throu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2099870,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(Q\) be a point on \(BC\) such that \(BQ = 2QC\), and let \(T\) be a point on \(AQ\) such that \(\angle BTQ = \angle BAC\). If \(\angle BAC = \theta\), find the value of \(\angle CTQ\) in terms of \(\theta\). | <think>
Okay, so I have this geometry problem here. Let me read it again and try to visualize it. The problem says:
"Let ABC be an isosceles triangle with AB = AC. Let Q be a point on BC such that BQ = 2QC, and let T be a point on AQ such that angle BTQ = angle BAC. If angle BAC = θ, find the value of angle CTQ in ter... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3900537,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a sequence of integers of length n and an integer m. Your task is to find the smallest integer x such that the sum of the floor division of each element in the sequence by x is at least m.
Note that floor division means dividing and taking the floor of the result (e.g., 7 // 3 = 2).
If no such x exists... | <think>
Okay, so I need to find the smallest integer x such that the sum of the floor divisions of each element in the sequence by x is at least m. Hmm. Let me think about how to approach this.
First, the problem is asking for the minimal x. Since x can range from 1 up to the maximum element in the array (because if x... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2171521,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Determine the value of \( n \) such that \(\frac{1! \cdot 2! \cdot 3! \cdots 100!}{n!}\) is a perfect square. | <think>
Okay, so I need to find the value of n such that when you divide the product of all factorials from 1! up to 100! by n!, the result is a perfect square. Hmm, let me think about how to approach this.
First, I know that a perfect square has all exponents even in its prime factorization. So, the problem boils dow... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1725764,
"category": "math",
"source": "OpenMathReasoning"
} | |
How many homomorphisms are there from \( \mathbb{Z}_n \) to \( \mathbb{Z}_n \), and what are they? | <think>
Okay, so I need to figure out how many homomorphisms there are from ℤ_n to ℤ_n and describe what they are. Hmm, homomorphisms between cyclic groups... I remember that cyclic groups have the property that their homomorphisms are determined by where the generator is sent. Let me think.
First, ℤ_n is a cyclic gro... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3926347,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve the following system in $\mathbb{R}$:
\[
\sin(x+y) = \frac{1}{2}
\]
\[
\cos(x-y) = \frac{\sqrt{2}}{2}
\] | <think>
Alright, let's see. I need to solve the system of equations involving sine and cosine. The equations are:
sin(x + y) = 1/2
and
cos(x - y) = √2/2
Okay, so first, I remember that sine and cosine have specific angles where their values are known. For sin(θ) = 1/2, the general solutions are θ = π/6 + 2πn or θ =... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3915145,
"category": "math",
"source": "OpenMathReasoning"
} | |
In triangle \(ABC\) with \(ABC\) not isosceles, the incircle touches the sides \(BC\), \(CA\), and \(AB\) at \(K\), \(L\), and \(M\) respectively. The line parallel to \(LM\) through \(B\) meets \(KL\) at \(D\), and the line parallel to \(LM\) through \(C\) meets \(KM\) at \(E\). Let \(F\) be the midpoint of \(\overlin... | <think>
Okay, so I need to find the ratio FG/GL in triangle ABC where the incircle touches the sides BC, CA, and AB at K, L, and M respectively. There are some lines drawn: a line parallel to LM through B meets KL at D, and another line parallel to LM through C meets KM at E. F is the midpoint of LM, and DE intersects ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1020534,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a function such that for any \( x, y \in \mathbf{R} \), we have \( f[x + f(y)] \geq f[f(x) + y] \) and \( f(0) = 0 \). Determine all such functions \( f \). | <think>
Okay, so I have this problem here where I need to find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for any real numbers \( x \) and \( y \), the inequality \( f[x + f(y)] \geq f[f(x) + y] \) holds, and also \( f(0) = 0 \). Hmm, functional inequalities can be tricky, but let's try to break... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3024664,
"category": "math",
"source": "NuminaMath-CoT"
} | |
The sum of several not necessarily distinct positive numbers did not exceed 100. Each of these numbers was replaced by a new number in the following way: first, they were logarithmed to base 10, then rounded to the nearest integer, and finally, 10 was raised to the power of the resulting integer. Could it happen that t... | <think>
Okay, let's try to figure out this problem. So, the question is about whether replacing numbers with 10 raised to the nearest integer of their base-10 logarithm can result in the sum exceeding 300, given that the original sum didn't exceed 100. Hmm, interesting.
First, let me parse the problem again. We have s... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 380535,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find the minimum value of the function \( f(x, y, z) = \sin x \sin y \sin z \) subject to the constraint \( x + y + z = \frac{\pi}{2} \) with \( x, y, z > 0 \). | <think>
Okay, so I need to find the minimum value of the function f(x, y, z) = sin x sin y sin z, given that x + y + z = π/2 and all variables are positive. Hmm, constraints optimization. I remember that Lagrange multipliers are a method used for finding the extrema of a function subject to equality constraints. Let me... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2530273,
"category": "math",
"source": "OpenMathReasoning"
} | |
Solve the equation $p^{2} \pm 2^{q} = q^{p}$ for prime numbers $p$ and $q$. | <think>
Okay, let me try to figure out how to solve the equation \( p^{2} \pm 2^{q} = q^{p} \) where \( p \) and \( q \) are prime numbers. Hmm, primes... So possible primes are 2, 3, 5, 7, 11, etc. The equation has two cases: one with a plus and one with a minus. I should check both.
First, maybe start with small pri... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 213323,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the minimum value of the function \( f(x) = (x+1)(x+2)(x+3)(x+4) \). | <think>
Okay, so I need to find the minimum value of the function f(x) = (x+1)(x+2)(x+3)(x+4). Hmm, let's see. I remember that when dealing with polynomials, especially quadratics, completing the square or finding the vertex helps. But this is a fourth-degree polynomial, which is more complicated. Maybe I can simplify ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1227206,
"category": "math",
"source": "OpenMathReasoning"
} | |
The chef is trying to solve some pattern problems, Chef wants your help to code it. Chef has one number K to form a new pattern. Help the chef to code this pattern problem.
-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow.
- Each test case contains a single line of ... | <think>
Okay, let's see. The problem is to generate a certain pattern based on the input K. Let's look at the sample inputs and outputs to understand the pattern.
Sample Input 3 has K=4, and the output is four lines. For K=2, the output is two lines. Wait, no, looking again: the first sample input is 2, output is 1121... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1056555,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Determine all functions \( f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) such that for all pairs of positive integers \( a \) and \( b \),
\[ f^{b}(a) + f^{a}(b) = c(a+b) \]
for a fixed positive integer \( c \). | <think>
Okay, so I need to find all functions f from the positive integers to the positive integers such that for any positive integers a and b, applying f b times to a and applying f a times to b and adding them gives c times (a + b), where c is a fixed positive integer. Hmm, this seems a bit abstract. Let me try to b... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3797104,
"category": "math",
"source": "OpenMathReasoning"
} | |
A company is making bulbs for flashlights. The probability of a bulb being defective is $\frac{1}{100}$. The company makes a profit of $0.80 for every non-defective bulb it sells, but each defective bulb costs the company $8 in replacement costs. How much money does the company make per bulb on average? | <think>
Okay, let's see. So the problem is about a company that makes flashlight bulbs, and we need to find out the average profit per bulb. Hmm. The probability of a bulb being defective is 1/100. For each non-defective bulb, they make a profit of $0.80, but each defective one costs them $8 in replacement costs. So, I... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 928720,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( A = \{x \mid 0 \leqslant x \leqslant 1\} \), and \( f \) be a mapping from \( A \) to the set of real numbers \( \mathbf{R} \), satisfying:
(1) \( f(1) = 1 \);
(2) \( f(x) \geqslant 0 \) for all \( x \in A \);
(3) If \( x, y, x+y \in A \), then \( f(x+y) \geqslant f(x) + f(y) \).
Prove that for any \( x \in A \... | <think>
Okay, so I need to prove that for any x in the interval [0,1], the function f(x) is less than or equal to 2x. The function f has some given properties: f(1) = 1, it's non-negative everywhere on [0,1], and it satisfies this superadditivity condition where f(x + y) is at least f(x) + f(y) whenever x, y, and x + y... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 415414,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \(ABC\) be a triangle with circumcenter \(O\), and let \(A'\) be the orthogonal projection of \(A\) on the line \(BC\). Let \(X\) be a point on the open ray \(AA'\) emanating from \(A\). The internal bisector of angle \(BAC\) meets the circumcircle of \(ABC\) again at \(D\). Let \(M\) be the midpoint of segment \(D... | <think>
Okay, so I need to find the value of \(\tan(2\theta)\) in terms of the side lengths \(a\), \(b\), and \(c\) of triangle \(ABC\), given some geometric conditions. Let me try to parse the problem step by step.
First, the problem states: Let \(ABC\) be a triangle with circumcenter \(O\). The orthogonal projection... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3619186,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the most general holomorphic function of the form \( f(x+iy) = u(x) + iv(y) \), where \( u \) and \( v \) are real-valued functions. | <think>
Alright, so I need to find the most general holomorphic function f(x + iy) = u(x) + iv(y), where u and v are real-valued functions. Hmm, okay, let's start by recalling what it means for a function to be holomorphic. A function is holomorphic if it's complex differentiable at every point in its domain. For funct... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2869816,
"category": "math",
"source": "OpenMathReasoning"
} | |
Define a zesty number as one that can be written as the sum or difference of two positive cubes in an odd number of ways. How many zesty numbers are there between $1$ and $10,000$, inclusive? | <think>
Okay, let's tackle this problem about zesty numbers. The question is asking how many numbers between 1 and 10,000 inclusive can be written as the sum or difference of two positive cubes in an odd number of ways. These are called zesty numbers. Hmm, okay, so first I need to understand what exactly constitutes a ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2448586,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \( E/K \) be a finite field extension where \( E \) is a perfect field. Define \( p \) as the characteristic of \( K \). Suppose \( a \in K \) is an element such that \( a^{p^n} \in K \) for some positive integer \( n \). Find the smallest positive integer \( m \) such that \( a^{p^m} \in K \) for all \( a \in K \)... | <think>
Okay, let's try to tackle this problem step by step. The question is about finding the smallest positive integer \( m \) such that \( a^{p^m} \in K \) for all \( a \in K \), given that \( E/K \) is a finite extension with \( E \) being a perfect field, and the characteristic of \( K \) is \( p \). Also, we know... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2834963,
"category": "math",
"source": "OpenMathReasoning"
} | |
In triangle \( \triangle ABC \), \( AB = 2 \) is a fixed length. From \( A \), draw the median \( AD \) of \( BC \), \( AD = 1 \frac{1}{2} \). If \( D \) is a moving point, what is the trajectory of vertex \( C \)?
(A) Line \( AB \), from \( A \) of length \( 1 \frac{1}{2} \).
(B) A circle with center \( A \) and rad... | <think>
Okay, so I need to figure out the trajectory of vertex C in triangle ABC where AB is fixed at 2 units, and the median AD from A to BC is always 1.5 units long. D is moving, so as D moves, point C moves along some path. The options given are different types of curves: lines, circles, an ellipse. Let me think ste... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1708564,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(xf(y)) = yf(x) \]
and
\[ \lim_{x \to \infty} f(x) = 0. \] | <think>
Okay, so I need to find all functions f from the positive real numbers to the positive real numbers that satisfy the functional equation f(xf(y)) = yf(x) and also have the limit as x approaches infinity of f(x) being zero. Hmm, functional equations can be tricky, but maybe I can start by plugging in some specif... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 37696,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given the `root` of a Binary Search Tree (BST), convert it to a Greater Tree such that every key of the original BST is changed to the original key plus sum of all keys greater than the original key in BST.
As a reminder, a binary search tree is a tree that satisfies these constraints:
The left subtree of a node conta... | <think>
Okay, so I need to convert a binary search tree into a Greater Tree. Hmm, what's a Greater Tree? Oh right, every key is replaced with the original key plus the sum of all keys greater than it. So like, if there are nodes with values higher than the current node, their sum is added to the current node's value.
... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 907042,
"category": "code",
"source": "leetcode"
} | |
What is the phase shift of the function $f(x) = \sin(4x + \pi)$? | <think>
Okay, so I need to find the phase shift of the function f(x) = sin(4x + π). Hmm, phase shift... right, that's the horizontal shift of the sine wave. I remember that the general form of a sine function is f(x) = A sin(Bx + C) + D. The phase shift formula I learned is -C/B. Let me confirm that.
Wait, phase shif... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3385683,
"category": "math",
"source": "OpenMathReasoning"
} | |
Consider \( n \) arbitrary points in the plane, where \( D \) is the maximum distance between any two points and \( d \) is the minimum distance between any two points. If \( n = 100 \), find the smallest possible value of \( \frac{D}{d} \). | <think>
Okay, let me try to figure out this problem. So, the question is: Given 100 points in the plane, where D is the maximum distance between any two points and d is the minimum distance between any two points, what's the smallest possible ratio D/d we can get? Hmm, interesting. I remember that arranging points in a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 558094,
"category": "math",
"source": "OpenMathReasoning"
} | |
Chef works as a cook in a restaurant. Each morning, he has to drive down a straight road with length K$K$ to reach the restaurant from his home. Let's describe this road using a coordinate X$X$; the position of Chef's home is X=0$X = 0$ and the position of the restaurant is X=K$X = K$.
The road has exactly two lanes (n... | <think>
Okay, let's see. I need to solve this programming problem. Hmm. Let's read the problem carefully.
So Chef is driving from home (X=0) to the restaurant (X=K) on a road with two lanes. There are obstacles in either lane. Each obstacle is at some X coordinate. The problem is to find the maximum distance Chef can ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1807871,
"category": "code",
"source": "OpenCodeReasoning"
} | |
You are given two very large numbers made up of 1 or 0 only. You have to find the digit by digit XOR of the two numbers, i.e., the i-th digit of the answer is 1 if and only if the i-th digit of the two given numbers differ otherwise the i-th digit of the answer is 0. The number of digits in both the numbers is same.
In... | <think>
Okay, I need to solve this problem where I have to compute the XOR of two binary numbers digit by digit. Let's see. The XOR operation between two bits is 1 if the bits are different, and 0 otherwise. So for each corresponding pair of digits in the two input numbers, I check if they are different. If they are, t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1858590,
"category": "code",
"source": "OpenCodeReasoning"
} | |
As behooves any intelligent schoolboy, Kevin Sun is studying psycowlogy, cowculus, and cryptcowgraphy at the Bovinia State University (BGU) under Farmer Ivan. During his Mathematics of Olympiads (MoO) class, Kevin was confronted with a weird functional equation and needs your help. For two fixed integers k and p, where... | <think>
Okay, let's try to solve this problem. So, the problem is to find the number of distinct functions f that satisfy the given functional equation for certain parameters p (a prime) and k. The equation is f(kx mod p) = k f(x) mod p. And this must hold for all x in 0 to p-1.
Hmm. Let me start by understanding what... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3753304,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Evaluate the definite integral $\int_0^1 \sqrt{t^2 + 1}\ dt$ by substitution $t = \frac{e^s - e^{-s}}{2}$. | <think>
Okay, so I need to evaluate the definite integral from 0 to 1 of sqrt(t² + 1) dt using the substitution t = (e^s - e^(-s))/2. Hmm, let's start by understanding the substitution they've given. The substitution t = (e^s - e^(-s))/2 looks familiar. Wait, isn't that the definition of the hyperbolic sine function? B... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 966137,
"category": "math",
"source": "OpenMathReasoning"
} | |
Create a function that accepts dimensions, of Rows x Columns, as parameters in order to create a multiplication table sized according to the given dimensions. **The return value of the function must be an array, and the numbers must be Fixnums, NOT strings.
Example:
multiplication_table(3,3)
1 2 3 ... | <think>
Okay, I need to create a Python function called multiplication_table that takes two parameters, the number of rows and columns, and returns a multiplication table as an array of arrays (lists in Python). The example given is for 3 rows and 3 columns, which results in [[1,2,3],[2,4,6],[3,6,9]]. Each element in t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2438187,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given a right trapezoid with bases \( a \) and \( b \) (\(a < b\)), it is known that a certain line, parallel to the bases, divides it into two trapezoids, each of which can have a circle inscribed in it. Find the radii of these circles. | <think>
Okay, let me try to figure out this problem. So, we have a right trapezoid with bases a and b, where a is shorter than b. A line is drawn parallel to the bases, dividing the original trapezoid into two smaller trapezoids, each of which can have a circle inscribed in them. We need to find the radii of these circ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3293602,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Sasha cut a paper triangle into two triangles. Then, every minute he cut one of the previously obtained triangles into two triangles. After some time, no less than an hour, all the triangles obtained by Sasha turned out to be equal. List all the original triangles for which such a situation is possible. | <think>
Okay, let's try to figure out this problem. So, Sasha starts with a triangle, cuts it into two triangles. Every minute after that, he takes one of the existing triangles and cuts it into two. After at least an hour, all the triangles are equal. We need to find which original triangles make this possible.
First... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 163510,
"category": "math",
"source": "NuminaMath-CoT"
} | |
One day Vasya heard a story: "In the city of High Bertown a bus number 62 left from the bus station. It had n grown-ups and m kids..."
The latter events happen to be of no importance to us. Vasya is an accountant and he loves counting money. So he wondered what maximum and minimum sum of money these passengers could h... | <think>
Okay, let's tackle this problem. Hmm, the task is to find the minimum and maximum possible total fare given n adults and m children. Oh right, the rules are a bit tricky.
First, I need to understand the conditions. So, each adult can take up to one child for free. Wait, no. Wait, the problem says "no more tha... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3810367,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Let \(ABCD\) be a convex quadrilateral and \(P\) be a point such that \(\angle APC = \angle BPD = 30^\circ\). If \(AB = 5\), \(AC = 6\), \(AD = 7\), \(BC = 8\), \(BD = 9\), and \(CD = 10\), calculate the minimum possible value of \(PA + PB + PC + PD\). | <think>
Okay, let's see. I need to find the minimum value of PA + PB + PC + PD in this convex quadrilateral ABCD with those given angles and side lengths. Hmm. The problem states that angles APC and BPD are both 30 degrees. Alright, let's break this down step by step.
First, let's visualize the quadrilateral ABCD. It'... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2178879,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to check if the given sequence of brackets is balanced or not. A sequence of brackets is called balanced if for every opening bracket there is a corresponding closing bracket and they are correctly nested.
Solve the problem starting with the provided function header.
Function header:
```
class Soluti... | <think>
Okay, I need to write a Python function called check_brackets in the Solution class that determines if a given string of brackets is balanced. Hmm. Let's think about what balanced means here. Oh right, every opening bracket must have a corresponding closing bracket in the correct order. Like, if there's a [ fol... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1327313,
"category": "code",
"source": "opc-sft-stage2"
} | |
King Tle4Ever of Time Limit Exceeded is really fascinated about Tic Tac Toe. He organizes a national level contest for Tic Tac Toe every year in Time Limit Exceeded. (Though I agree you need to be really stupid to loose a game of Tic Tac Toe but for the sake of question assume playing Tic Tac Toe for them is same as pl... | <think>
Okay, so I need to find the minimum and maximum possible sum of wi squared for n participants in a Tic Tac Toe contest, where each wi is the number of wins a participant has. And n is always an odd integer. Hmm. Let's think.
First, the problem says that for each test case with given n (odd), I have to compute ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 415536,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Many people are aware of DMCA – Digital Millennium Copyright Act. But another recently proposed DMCA – Digital Millennium Calculation Act – is much less known.
In this problem you need to find a root of a number according to this new DMCA law.
Input
The input contains a single integer a (1 ≤ a ≤ 1000000).
Output
Ou... | <think>
Okay, let's see. I need to find the root of a number according to this DMCA law. Wait, the examples given are 1 gives 1 and 81 gives 9. So for 81, the square root is 9. But wait, 81 could also be 3^4, but the fourth root would be 3, but the output is 9. Hmm, so maybe the problem is to find the largest possible ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 915257,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Consider a cyclic quadrilateral $ABCD$. Let $f(n)$ be the minimum number of cuts required to dissect $ABCD$ into $n$ cyclic quadrilaterals for $n \geq 4$. Determine the value of $f(5)$. | <think>
Okay, let's see. The problem is about a cyclic quadrilateral ABCD, and we need to find the minimum number of cuts required to dissect it into 5 cyclic quadrilaterals. The function f(n) gives this minimum number for n quadrilaterals, and we need f(5). Hmm, cyclic quadrilaterals are four-sided figures where all t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3059108,
"category": "math",
"source": "OpenMathReasoning"
} | |
The diagonals $BD$ of a quadrilateral $ABCD$ that are symmetric with respect to the angle bisectors of angles $B$ and $D$ pass through the midpoint of diagonal $AC$.
Prove that the diagonals $AC$ that are symmetric with respect to the angle bisectors of angles $A$ and $C$ pass through the midpoint of diagonal $BD$. | <think>
Okay, so I have this geometry problem here. Let me read it again and try to understand what's being asked.
The problem states: "The diagonals BD of a quadrilateral ABCD that are symmetric with respect to the angle bisectors of angles B and D pass through the midpoint of diagonal AC. Prove that the diagonals AC... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2984276,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Solve the equation \( e^{-2x} - 3e^{-x} + 2 = 0 \). | <think>
Okay, so I need to solve the equation \( e^{-2x} - 3e^{-x} + 2 = 0 \). Hmm, let me think about how to approach this. I remember that when dealing with exponential equations, substitution can sometimes simplify things. Let me see if that works here.
First, notice that both terms \( e^{-2x} \) and \( e^{-x} \) a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 24763,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to find the difference matrix for a given binary matrix. The difference matrix is defined as: for each cell in the matrix, the value is calculated as the number of 1s in the same row and column minus the number of 0s in the same row and column.
Solve the problem starting with the provided function hea... | <think>
Okay, I need to write a function called ones_minus_zeros that takes a binary matrix (grid) and returns the difference matrix. The difference matrix's each cell is calculated as the number of 1s in the same row and column minus the number of 0s in the same row and column. Hmm, let's think about how to approach t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1455237,
"category": "code",
"source": "opc-sft-stage2"
} | |
Let \(a, b, c,\) and \(k\) be positive real numbers such that \(a + b + c = \sqrt{k}\). For \(n = 4\), find the minimum value of the expression
\[ \frac{a}{\sqrt{1 + 4bc}} + \frac{b}{\sqrt{1 + 4ac}} + \frac{c}{\sqrt{1 + 4ab}}. \] | <think>
Okay, so I have this problem here where I need to find the minimum value of the expression \(\frac{a}{\sqrt{1 + 4bc}} + \frac{b}{\sqrt{1 + 4ac}} + \frac{c}{\sqrt{1 + 4ab}}\) given that \(a + b + c = \sqrt{k}\) and \(n = 4\). Wait, actually, the problem statement says \(n = 4\), but in the expression, there's no... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1305957,
"category": "math",
"source": "OpenMathReasoning"
} | |
Nastya likes reading and even spends whole days in a library sometimes. Today she found a chronicle of Byteland in the library, and it stated that there lived shamans long time ago. It is known that at every moment there was exactly one shaman in Byteland, and there were n shamans in total enumerated with integers from... | <think>
Okay, let's tackle this problem. Hmm, the problem is about determining if there's a king-shaman after each update. A king-shaman is one where their power is exactly equal to the sum of all previous shamans. So, for each query, after updating a position, we need to check if any shaman j satisfies a[j] = sum of a... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 431124,
"category": "code",
"source": "OpenCodeReasoning"
} | |
In an acute-angled triangle \(ABC\), points \(D\), \(E\), and \(F\) are the feet of the perpendiculars from \(A\), \(B\), and \(C\) onto \(BC\), \(AC\), and \(AB\), respectively. \(H\) is the intersection of \(AD\) and \(BE\). Given \(\sin A = \frac{3}{5}\) and \(BC = 39\), find the length of \(AH\). | <think>
Okay, let's see. I need to find the length of AH in an acute-angled triangle ABC where D, E, F are the feet of the perpendiculars from A, B, and C respectively. H is the intersection of AD and BE. They gave me sin A = 3/5 and BC = 39. Alright, where do I start?
First, maybe drawing a diagram would help, but si... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1816919,
"category": "math",
"source": "OpenMathReasoning"
} | |
An octagon with side lengths 3, 3, 11, 11, 15, 15, 15, 15 cm is inscribed in a circle. What is the area of the octagon in cm²? | <think>
Okay, so I have this problem here: an octagon with side lengths 3, 3, 11, 11, 15, 15, 15, 15 cm is inscribed in a circle. I need to find the area of the octagon. Hmm, let's start by visualizing the problem. An octagon inscribed in a circle means all its vertices lie on the circumference of the circle. That usua... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3381301,
"category": "math",
"source": "OpenMathReasoning"
} | |
Certainly, everyone is familiar with tic-tac-toe game. The rules are very simple indeed. Two players take turns marking the cells in a 3 × 3 grid (one player always draws crosses, the other — noughts). The player who succeeds first in placing three of his marks in a horizontal, vertical or diagonal line wins, and the g... | <think>
Okay, I need to solve this problem. Let's read the problem statement carefully.
The problem is about determining the state of a Tic-Tac-Toe game based on a given 3x3 grid. The possible outputs are whether it's the first or second player's turn next, or one of the verdicts like illegal, someone has already won,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3684668,
"category": "code",
"source": "OpenCodeReasoning"
} | |
You are given a [directed graph](https://en.wikipedia.org/wiki/Directed_graph) consisting of n vertices and m edges (each edge is directed, so it can be traversed in only one direction). You are allowed to remove at most one edge from it.
Can you make this graph [acyclic](https://en.wikipedia.org/wiki/Directed_acyclic... | <think>
Okay, I need to solve this problem where I have to determine if removing at most one edge from a directed graph can make it acyclic. Let's think about the approach here.
First, the problem is to check if the graph has a cycle, and if so, whether there's an edge in that cycle that, when removed, breaks all poss... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1117779,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a program, which takes an integer N and if the number is less than 10 then display "What an obedient servant you are!" otherwise print "-1".
Input
The first line contains an integer T, total number of testcases. Then follow T lines, each line contains an integer N.
Output
Output the given string or -1 dependin... | <think>
Okay, I need to solve this programming problem. Let's read the question carefully. The task is to write a Python program that takes an integer N for multiple test cases and outputs "What an obedient servant you are!" if N is less than 10, otherwise output -1. Oh right, the input starts with T, the number of tes... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2372614,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a python function to find the minimum number of moves required to make a string a palindrome. A move is defined as removing any character from the string.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def min_moves_to_palindrome(self, s):
```
Please place t... | <think>
Okay, I need to find the minimum number of moves (deletions) required to turn a given string into a palindrome. Hmm, how to approach this. Let's think.
Wait, a palindrome reads the same forwards and backwards. So the problem reduces to finding the longest palindromic subsequence (LPS) because the minimum delet... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 945617,
"category": "code",
"source": "opc-sft-stage2"
} | |
Vasya and Petya wrote down all integers from 1 to n to play the "powers" game (n can be quite large; however, Vasya and Petya are not confused by this fact).
Players choose numbers in turn (Vasya chooses first). If some number x is chosen at the current turn, it is forbidden to choose x or all of its other positive in... | <think>
Okay, let's see. The problem is about a game where two players take turns choosing numbers from 1 to n. The rule is that once a number x is chosen, you can't choose any of its higher powers (like x², x³, etc.) in future moves. The player who can't make a move loses. We need to determine who wins if both play op... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1033322,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Write a python function to find the unique elements in a list while preserving the order of their first occurrence.
Solve the problem starting with the provided function header.
Function header:
```
class Solution:
def unique_ordered_elements(self, lst):
```
Please place the solution code in the following format:
`... | <think>
Okay, I need to write a Python function called unique_ordered_elements in the Solution class. The goal is to find the unique elements in a list while preserving the order of their first occurrence. Hmm, how do I approach this?
Wait, so the problem is similar to removing duplicates but keeping the first occurre... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2850804,
"category": "code",
"source": "opc-sft-stage2"
} | |
Let \(a, b, c > 0\). Find the minimum value of:
\[ \frac{a}{a+2b} + \frac{2b}{3a+2c} + \frac{3c}{4a+b} \] | <think>
Okay, let's see. The problem is to find the minimum value of the expression \(\frac{a}{a+2b} + \frac{2b}{3a+2c} + \frac{3c}{4a+b}\) where \(a\), \(b\), and \(c\) are positive real numbers. Hmm, so we need to minimize this sum. I remember that for such optimization problems, inequalities like Cauchy-Schwarz or A... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1521812,
"category": "math",
"source": "OpenMathReasoning"
} | |
Ophelia is paid $51 for the first week of her part-time summer job. For each week after the first, she is paid $100. How many weeks in total does she have to work for her average weekly pay to be $93? | <think>
Okay, let's see. So Ophelia gets paid $51 the first week, and then $100 each week after that. We need to find out how many total weeks she has to work so that her average weekly pay is $93. Hmm. Alright, let's break this down step by step.
First, the average pay is calculated by taking the total amount she's e... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2042358,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that
\[ f(f(n)) + f(n+1) = n+2. \] | <think>
Okay, let's try to solve this functional equation: find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that \( f(f(n)) + f(n+1) = n + 2 \). Hmm, functional equations can be tricky, but maybe I can figure this out step by step.
First, let me understand what's being asked. We need to find a function that,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1998136,
"category": "math",
"source": "OpenMathReasoning"
} | |
For a given sequence of integers $A = \\{a_0, a_1, ..., a_{n-1}\\}$, perform the following operations.
* count($b, e, k$): print the number of the specific values $k$ in $a_b, a_{b+1}, ..., a_{e-1}$.
Constraints
* $1 \leq n \leq 1,000$
* $-1,000,000,000 \leq a_i, k_i \leq 1,000,000,000$
* $1 \leq q \leq 1,000$
* $0 ... | <think>
Okay, let's see. I need to solve this programming problem. Let's read the problem statement carefully.
The task is, given a sequence of integers and multiple queries, each asking for the count of a specific value k in a subarray from index b to e-1. So for each query, I have to find how many times k appears be... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3281224,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) which satisfy the functional equation
\[ xf(x) - yf(y) = (x-y)(f(x+y) - xy) \]
for all real numbers \( x \) and \( y \). | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy the equation xf(x) - yf(y) = (x - y)(f(x + y) - xy) for all real x and y. Hmm, functional equations can be tricky, but maybe I can try plugging in specific values for x and y to simplify things and figure out a patter... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2438718,
"category": "math",
"source": "OpenMathReasoning"
} | |
In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AXB = 50^\circ$, $\angle CAD = 51^\circ$, $\angle AED = 80^\circ$. Find the value of angle $\angle AXC$. | <think>
Okay, let's try to tackle this geometry problem. Hmm, it's a convex quadrilateral ABCD with some angles and equal diagonals. Let me jot down the given information first.
We have AC = BD, which might come into play later. Point X is on side AD. The diagonals intersect at point E. The angles ∠ABX and ∠AXB are bo... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3605962,
"category": "math",
"source": "OpenMathReasoning"
} | |
Evaluate the limit as \( x \) approaches 0 of \( \frac{\ln(1 - x) \sin(x)}{1 - \cos^2(x)} \). | <think>
Okay, so I need to find the limit as x approaches 0 of [ln(1 - x) * sin(x)] divided by [1 - cos²(x)]. Hmm, let me start by recalling some limit properties and maybe Taylor series expansions. Let's see...
First, I know that 1 - cos²(x) is the same as sin²(x) because of the Pythagorean identity. So the denominat... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 488583,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the n-th integral of $x \ln(x)$. | <think>
Okay, so I need to find the n-th integral of x ln(x). Hmm, integrating multiple times. Let me start by recalling how to integrate by parts because the integral of x ln(x) is probably a standard integration by parts problem. But since it's the n-th integral, I have to do this repeatedly, maybe find a pattern.
F... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 767008,
"category": "math",
"source": "OpenMathReasoning"
} | |
Can Vieta's Formulas be applied to find the roots of the polynomial \( x^3 + 21x + 7.88 \)? Specifically, solve the system of equations derived from Vieta's Formulas:
\[
\begin{cases}
r_1 + r_2 + r_3 = 0 \\
r_1r_2 + r_1r_3 + r_2r_3 = 21 \\
r_1r_2r_3 = -7.88
\end{cases}
\] | <think>
Okay, so I need to figure out if Vieta's formulas can be applied to find the roots of the polynomial \( x^3 + 21x + 7.88 \). The problem also asks to solve the system of equations from Vieta's formulas. Let me start by recalling what Vieta's formulas are. For a cubic equation \( x^3 + ax^2 + bx + c = 0 \), the ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1248816,
"category": "math",
"source": "OpenMathReasoning"
} | |
Write a function to find the largest values in each row of a binary tree. The binary tree is represented as a list of lists where the index of any element e at index i in the list represents that e is the parent of the elements at indices 2*i+1 and 2*i+2.
You should include the predefined classes here:
class TreeNode:... | <think>
Okay, I need to write a function called largest_values_in_each_row that takes a tree represented as a list of lists and returns a list of the largest values in each row. Hmm. Wait, the binary tree is represented as a list where the index of any element e at index i in the list represents that e is the parent of... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 80930,
"category": "code",
"source": "opc-sft-stage2"
} | |
Find all increasing arithmetic sequences of natural numbers, in which for any natural $n$, the product of the first $n$ terms divides the product of the next $n$ terms (from $n+1$ to $2n$). | <think>
Okay, so I need to find all increasing arithmetic sequences of natural numbers where, for any natural number n, the product of the first n terms divides the product of the next n terms (from term n+1 to 2n). Hmm. Let me start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence wher... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2466187,
"category": "math",
"source": "NuminaMath-CoT"
} | |
In how many ways can the 7 spokes of a wheel be colored using 3 distinct colors, considering that two colorings which are rotations of each other are the same? | <think>
Okay, so I have this problem: I need to find the number of ways to color the 7 spokes of a wheel using 3 distinct colors. But there's a catch—two colorings that are rotations of each other are considered the same. Hmm, okay. Let me try to work through this step by step.
First, if I ignore the rotation part, ho... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2912879,
"category": "math",
"source": "OpenMathReasoning"
} | |
Determine all values of \( \lambda \) such that all roots of the polynomial \( P(x) = x^n + \lambda \sum_{r=1}^n (-1)^r x^{n-r} \) are real numbers, where \( n > 2 \). | <think>
Okay, so I need to figure out for which values of λ all the roots of the polynomial P(x) = x^n + λ Σ_{r=1}^n (-1)^r x^{n-r} are real. The problem states that n is greater than 2. Hmm, let me start by understanding the polynomial better. Let me write it out more explicitly.
First, let me expand the sum part. T... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2463812,
"category": "math",
"source": "OpenMathReasoning"
} | |
Three students, Xena, Yolanda, and Zachary, generate random numbers in the interval [0, 2]. Find the probability that the sum of their numbers is at least 1 but no more than 2. | <think>
Okay, so I need to find the probability that the sum of three random numbers generated by Xena, Yolanda, and Zachary, each between 0 and 2, is at least 1 and at most 2. Hmm, let's start by understanding the problem.
First, each student picks a number uniformly from [0, 2]. So each of the three variables, say X... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3209383,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the number of ways to select 5 people out of 15 people sitting on a round table such that exactly two of the selected people are consecutive and no other two selected people are consecutive. | <think>
Okay, let's try to tackle this problem. So, we need to find the number of ways to select 5 people out of 15 sitting around a round table, with exactly two of them being consecutive and the other three not being consecutive to anyone else. Hmm, that's a bit tricky, but let me break it down step by step.
First, ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3217163,
"category": "math",
"source": "OpenMathReasoning"
} | |
A sequence of natural numbers \( c_1, c_2, \dots \) is called PERFECT if every natural number \( m \) with \( 1 \le m \le c_1 + \dots + c_n \) can be represented as \( m = \frac{c_1}{a_1} + \frac{c_2}{a_2} + \dots + \frac{c_n}{a_n} \). Given \( n \), find the maximum possible value of \( c_n \) in a perfect sequence \(... | <think>
Okay, so I need to figure out the maximum possible value of \( c_n \) in a perfect sequence \( (c_1, c_2, \ldots, c_n) \). The definition says that a sequence is PERFECT if every natural number \( m \) from 1 up to the sum \( S = c_1 + c_2 + \dots + c_n \) can be expressed as a sum of fractions \( \frac{c_i}{a_... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 769722,
"category": "math",
"source": "OpenMathReasoning"
} | |
What is the largest number of elements that can be contained in a set of vectors such that the set contains the sum of any two different vectors in the set? | <think>
Okay, so I need to figure out the largest number of elements in a set of vectors where the set contains the sum of any two different vectors. Hmm, let me start by understanding the problem properly.
First, the set has to be such that if I pick any two distinct vectors from the set and add them together, the r... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 13087,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(ABCD\) be a convex cyclic quadrilateral with circumcenter \(U\), in which the diagonals are perpendicular to each other. Let \(g\) be the line obtained by reflecting the diagonal \(AC\) across the angle bisector of \(\angle BAD\).
Show that the point \(U\) lies on the line \(g\). | <think>
Okay, let's try to tackle this geometry problem. So we have a convex cyclic quadrilateral ABCD with circumcenter U, and the diagonals AC and BD are perpendicular. We need to show that the circumcenter U lies on the line g, which is the reflection of diagonal AC across the angle bisector of angle BAD.
First, le... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3257877,
"category": "math",
"source": "NuminaMath-CoT"
} | |
For positive integer \( n \), the sequence \( a_1, a_2, \ldots, a_n, \ldots \) is defined by \( a_1 = 1 \) and \( a_n = \left(\frac{n+1}{n-1}\right)(a_1 + a_2 + \cdots + a_{n-1}) \) for \( n > 1 \). Determine the value of \( a_{1997} \). | <think>
Okay, so I have this problem here where I need to find the value of a_{1997} in a sequence defined by a_1 = 1 and for n > 1, a_n = ( (n+1)/(n-1) ) times the sum of all previous terms. Hmm, let's try to break this down step by step.
First, let's write down what's given. The first term a_1 is 1. For each subsequ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1887667,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a geometric progression of five terms, each a positive integer less than $100$, with the sum of the five terms being $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then what is $S$? | <think>
Okay, so I need to solve this problem about a geometric progression with five terms. Each term is a positive integer less than 100, and their total sum is 211. Then, S is the sum of those terms in the progression that are perfect squares. I need to find S.
First, let me recall what a geometric progression (GP)... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2384302,
"category": "math",
"source": "OpenMathReasoning"
} | |
In a park, there are \( n \geq 3 \) birds, each with a unique nest. The distances between any two nests are all different. Each bird then searches for a new nest such that every nest is occupied again. For all \( i, j, r, s \) with \( 1 \leq i \leq j \leq r \leq s \leq n \) and \( i \neq j \), \( r \neq s \), if the di... | <think>
Okay, let's see. The problem is about birds in a park with unique nests. Each bird moves to a new nest, and after they all move, every nest is occupied again. The key condition is about the distances between nests before and after the switch. Specifically, if before the switch, the distance between nests K_i an... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1742813,
"category": "math",
"source": "OpenMathReasoning"
} | |
In a large semicircle of radius 10, there are two smaller semicircles of radii 6 and 4 such that their diameters are lined up along the larger semicircle's diameter, and the smaller semicircles are tangent to each other and the larger semicircle. There is a tangent line \( LM \) that is tangent to both smaller semicirc... | <think>
Okay, let me try to figure out this geometry problem. So, we have a large semicircle with radius 10. Inside it, there are two smaller semicircles with radii 6 and 4. Their diameters are lined up along the larger semicircle's diameter. The smaller semicircles are tangent to each other and also tangent to the lar... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3696394,
"category": "math",
"source": "OpenMathReasoning"
} | |
Five points are on the surface of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points? | <think>
Okay, so I need to find the maximum value of the smallest distance between any two points among five points on the surface of a unit sphere. This is like arranging five points as far apart as possible from each other on a sphere, and then figuring out the smallest distance between any pair in that arrangement. ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1440417,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given a triangle \( \triangle ABC \) with centroid \( G \), denote \( d_a \) as the distance from \( G \) to the internal bisector of \( \angle A \). Similarly, define \( d_b \) and \( d_c \) for \( \angle B \) and \( \angle C \) respectively. Calculate the value of \( \frac{d_a}{\sin \frac{A}{2}} + \frac{d_b}{\sin \fr... | <think>
Okay, let's tackle this geometry problem. So, we have triangle ABC with centroid G. The centroid is the intersection point of the medians, right? It divides each median into a ratio of 2:1, with the longer part being closer to the vertex. Now, the problem defines d_a as the distance from G to the internal bisec... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1897830,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find the smallest positive integer $n$ for which the expression $|x-1| + |x-2| + |x-3| + \ldots + |x-n|$ is greater than or equal to $2019$. | <think>
Okay, so I need to find the smallest positive integer n such that the sum |x - 1| + |x - 2| + ... + |x - n| is greater than or equal to 2019 for some x. Hmm, let's see. I remember that the sum of absolute values like this is minimized when x is the median of the numbers from 1 to n. So maybe the minimal sum occ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3905679,
"category": "math",
"source": "OpenMathReasoning"
} | |
The line segment \( PA \) intersects the circle \( \odot O \) at point \( A \), and \( AB \) is a chord of \( \odot O \). Given that the radius of \( \odot O \) is 1, \( PA = 1 \), and \( AB = \sqrt{2} \), find the length of \( PB \).
(A) 1
(B) \( \sqrt{5} \)
(C) 1 or \( \sqrt{5} \)
(D) Unable to determine | <think>
Okay, so I need to solve this geometry problem. Let me try to visualize it first. There's a circle with center O and radius 1. Point P is outside the circle, and the line segment PA intersects the circle at point A. So PA is a secant line, right? AB is a chord of the circle, and they give me that AB is √2. The ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1039095,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Solve over positive real numbers the functional equation
$$
f(x) = f(f(f(x)) + y) + f(x f(y)) f(x + y).
$$ | <think>
Okay, so I need to solve this functional equation over positive real numbers:
f(x) = f(f(f(x)) + y) + f(x f(y)) f(x + y).
Hmm, functional equations can be tricky. Let me start by recalling some common strategies. Maybe trying specific values for x or y could help. Let's see.
First, let's consider substituti... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1126718,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Let \(a\), \(b\), and \(c\) be the sides of a triangle. The area of the triangle is given by \((a - b + c)(a + b - c)\). Find the length of the angle bisector opposite to side \(a\) in terms of \(b\) and \(c\). | <think>
Okay, let's see. I need to find the length of the angle bisector opposite to side a in terms of b and c. The problem states that the area of the triangle is given by (a - b + c)(a + b - c). Hmm, interesting. First, maybe I should recall the formula for the area of a triangle. Normally, the area can be calculate... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2287227,
"category": "math",
"source": "OpenMathReasoning"
} | |
Now our heroes - Maga and Alex are working for Oil Company as developers. Recently they have faced a great problem.
They couldn’t find the correct solution. Could you help them?
We are given r – radius of lower base and s – slant height. The figure can be cylinder or truncated cone. You have to find as largest volum... | <think>
Okay, let's try to figure out how to solve this problem. So the task is to find the maximum possible volume between a cylinder and a truncated cone (which is a frustum) given the radius r of the lower base and the slant height s. Hmm, right.
First, I need to understand the parameters. The slant height s is the... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2356060,
"category": "code",
"source": "OpenCodeReasoning"
} | |
How many solutions are there to the equation $w + x + y + z = 20$ where $w, x, y,$ and $z$ are all positive integers? | <think>
Okay, so I have this problem here: How many solutions are there to the equation w + x + y + z = 20 where w, x, y, and z are all positive integers. Hmm, I need to figure out the number of solutions. Let me think about how to approach this.
First, I remember that when dealing with equations like this, it's a com... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3344708,
"category": "math",
"source": "OpenMathReasoning"
} | |
For any positive integer \( q_0 \), consider the sequence defined by
$$
q_i = \left(q_{i-1} - 1\right)^3 + 3 \quad (i = 1, 2, \cdots, n)
$$
If every \( q_i (i = 1, 2, \cdots, n) \) is a power of a prime number, find the largest possible value of \( n \). | <think>
Okay, so I need to solve this problem where we start with a positive integer q₀, and then each subsequent term in the sequence is defined by qᵢ = (qᵢ₋₁ - 1)³ + 3. The condition is that every term in the sequence up to qₙ must be a power of a prime number. The question is asking for the largest possible value of... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2306865,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Given 666 points on a plane that cannot be covered by 10 lines, determine the maximum number of points that can be chosen such that any subset of 66 points from these chosen points also cannot be covered by 10 lines. | <think>
Okay, let's try to figure out this problem. So, we have 666 points on a plane, and it's given that these 666 points can't be covered by 10 lines. The question is asking for the maximum number of points we can choose from these 666 such that any subset of 66 points from the chosen ones also can't be covered by 1... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2262830,
"category": "math",
"source": "OpenMathReasoning"
} | |
Determine for which real values of \( x \) the inequality \( |x^2 - 10x + 21| \le |3x - 15| \) holds. | <think>
Okay, let's try to solve this inequality: |x² -10x +21| ≤ |3x -15|. Hmm, absolute value inequalities can be a bit tricky, but I remember that they usually involve considering different cases based on the expressions inside the absolute values being positive or negative. Let me think step by step.
First, maybe ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1864829,
"category": "math",
"source": "OpenMathReasoning"
} | |
A dead body was found in a room with a constant temperature of 70°F. At the time of discovery, the core temperature of the body was 85°F. One hour later, the core temperature was 80°F. Assuming the time of death corresponds to \( t = 0 \) and the core temperature at that time was 98.6°F, how many hours elapsed before t... | <think>
Okay, let me try to figure out this problem step by step. So, the problem is about a dead body found in a room, and we need to determine how many hours passed between the time of death and the discovery of the body. The given information includes the core temperatures at the time of discovery and one hour later... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 993092,
"category": "math",
"source": "OpenMathReasoning"
} | |
Arthur and Alexander are number busters. Today they've got a competition.
Arthur took a group of four integers a, b, w, x (0 ≤ b < w, 0 < x < w) and Alexander took integer с. Arthur and Alexander use distinct approaches to number bustings. Alexander is just a regular guy. Each second, he subtracts one from his number... | <think>
Okay, let's see. The problem is about Arthur and Alexander decreasing their numbers over time. We need to find the minimum number of seconds until Alexander's number (c) becomes less than or equal to Arthur's (a).
So, Alexander's part is straightforward: each second, he subtracts 1 from c. So after t seconds,... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3818625,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given an array of integers `nums` and an integer `k`, return _the number of contiguous subarrays where the product of all the elements in the subarray is strictly less than_ `k`.
**Example 1:**
**Input:** nums = \[10,5,2,6\], k = 100
**Output:** 8
**Explanation:** The 8 subarrays that have product less than 100 are:
... | <think>
Okay, I need to solve this problem where I have to find the number of contiguous subarrays whose product is strictly less than k. Let's see. So given the examples, like the first one with nums = [10,5,2,6] and k=100, the output is 8. Let me think about how to approach this.
Hmm, the brute force way would be to... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 669635,
"category": "code",
"source": "leetcode"
} | |
Given two integer arrays `A` and `B`, return the maximum length of an subarray that appears in both arrays.
Example 1:
Input:
A: [1,2,3,2,1]
B: [3,2,1,4,7]
Output: 3
Explanation:
The repeated subarray with maximum length is [3, 2, 1].
Note:
1 <= len(A), len(B) <= 1000
0 <= A[i], B[i] < 100
Write Python code to sol... | <think>
Okay, so I need to find the maximum length of a subarray that appears in both arrays A and B. Hmm, subarray here means a contiguous sequence, right? So unlike a subsequence where elements can be non-contiguous, here they have to be next to each other in both arrays.
Let me think about possible approaches. The... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1617840,
"category": "code",
"source": "leetcode"
} | |
Write a function that checks if a given directed graph is a valid course schedule. A course schedule is valid if for every course, all its prerequisites have been taken. The graph is represented as a list of courses and a list of prerequisites, where each prerequisite is represented as a list with two elements: the cou... | <think>
Okay, I need to write a function called can_finish_courses that checks if a given directed graph is a valid course schedule. The problem says that the schedule is valid if all prerequisites are properly taken. So, this sounds like detecting if there's a cycle in the directed graph. Because if there's a cycle, t... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3409028,
"category": "code",
"source": "opc-sft-stage2"
} | |
There are 2008 balls in a bag, numbered from 1 to 2008. Three balls are randomly picked from the bag, and let \( T \) denote the largest number among these three balls. What is the expected value of \( T \)? | <think>
Okay, so I need to find the expected value of the largest number T when three balls are randomly picked from a bag containing balls numbered 1 to 2008. Hmm, let's see. I remember that expected value problems often involve probabilities and sums. Since we're dealing with the largest of three numbers, maybe I can... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1804596,
"category": "math",
"source": "OpenMathReasoning"
} | |
Anatoly lives in the university dorm as many other students do. As you know, cockroaches are also living there together with students. Cockroaches might be of two colors: black and red. There are n cockroaches living in Anatoly's room.
Anatoly just made all his cockroaches to form a single line. As he is a perfectioni... | <think>
Okay, let's see. I need to find the minimum number of moves to make the cockroach colors alternate. The moves can be swapping any two cockroaches or changing a cockroach's color. Hmm, so the problem is to find the minimum steps between the current arrangement and a valid alternating pattern.
First, what's a va... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 939268,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Given that $A$ is one of the intersection points of two circles $O_{1}$ and $O_{2}$ with unequal radii in a plane. The points of tangency of an external common tangent $P_{1} P_{2}$ are $P_{1}$ and $P_{2}$, and the points of tangency of another external common tangent are $Q_{1}$ and $Q_{2}$. Let $M_{1}$ and $M_{2}$ be... | <think>
Okay, so I have this geometry problem here. Let me try to visualize it first. There are two circles, O₁ and O₂, with different radii. They intersect at point A, and there's another intersection point which probably isn't mentioned here. The problem talks about external common tangents. There are two external co... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 636461,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Equilateral triangle $RST$ is inscribed in equilateral triangle $ABC$ such that $RS$ is perpendicular to $AB$. Find the area of triangle $RST$ with respect to the area of triangle $ABC$. | <think>
Alright, so I need to find the area of equilateral triangle RST that's inscribed in another equilateral triangle ABC, where RS is perpendicular to AB. Hmm, okay. Let me start by visualizing this. Both triangles are equilateral, so all sides are equal, and all angles are 60 degrees.
First, let me sketch this i... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3054023,
"category": "math",
"source": "OpenMathReasoning"
} | |
What are the factors of $4$? | <think>
Okay, let's see... I need to find the factors of 4. Hmm, factors are numbers that multiply together to give the original number, right? So, like, which numbers can multiply in pairs to equal 4. Let me start from 1 and go up.
First, 1 times 4 is 4, so 1 and 4 are factors. Then, 2 times 2 is 4, so 2 is also a fa... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 319083,
"category": "math",
"source": "OpenMathReasoning"
} | |
Develop an equation that gives the distance \( d \) from a tower as a function of time \( t \), given that the tower is 100 feet east of the road, you are traveling at 60 feet per second, and at time \( t = 0 \), you are 1000 feet south of the point on the road that is due west of the tower. | <think>
Okay, so I need to come up with an equation that gives the distance d from the tower as a function of time t. Let's see. The problem says the tower is 100 feet east of the road, and I'm traveling on the road at 60 feet per second. At time t=0, I'm 1000 feet south of the point on the road that's due west of the ... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 323132,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that \( |y - f(f(x))| \geq |f(x)^2 + x f(y)| \) for any \( x, y \in \mathbb{R} \). | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers that satisfy the inequality |y - f(f(x))| ≥ |f(x)² + x f(y)| for all real numbers x and y. Hmm, functional equations can be tricky, but let's take it step by step.
First, let's parse the problem. We need a function f such that fo... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1940757,
"category": "math",
"source": "OpenMathReasoning"
} | |
Given triangle \(ABC\) and point \(P\), let \(H_a\), \(H_b\), and \(H_c\) be the orthocenters of triangles \(\triangle PBC\), \(\triangle PCA\), and \(\triangle PAB\), respectively. Calculate the ratio of the area of triangle \([H_aH_bH_c]\) to the area of triangle \([ABC]\). | <think>
Okay, let me try to figure out this problem. So, we have triangle ABC and a point P somewhere. Then, for each side of triangle ABC, they take point P and the opposite vertex to form three new triangles: PBC, PCA, and PAB. For each of these three triangles, we find their orthocenters, which are H_a, H_b, and H_c... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2242910,
"category": "math",
"source": "OpenMathReasoning"
} |
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