problem stringlengths 1 7.91k | generated_solution stringlengths 824 78.8k | expected_answer stringlengths 0 1.2k | problem_type stringclasses 1
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Let \(a, b, c\) be reals satisfying \(a^2 + b^2 + c^2 = 6\). Calculate the maximum value of the expression \((a-b)^3 + (b-c)(c-a)\). | <think>
Okay, so I have this problem where I need to find the maximum value of the expression (a - b)^3 + (b - c)(c - a) given that a² + b² + c² = 6. Hmm, let's see how to approach this.
First, I remember that optimization problems with constraints often use methods like Lagrange multipliers, but since this is a symm... | 24\sqrt{3} + 3 | nvidia/OpenMathReasoning|aops_c6_high_school_olympiads | math | null | hard | |
Solve the following problem. Make sure to put the answer (and only answer) inside \boxed{}.
Which of the following scheduling algorithms could potentially lead to starvation in a computer system?
A: First-Come, First-Served (FCFS)
B: Shortest Job First (SJF)
C: Round Robin (RR)
D: Priority Scheduling with aging
E: Pri... | <think>
I need to determine which scheduling algorithms could lead to starvation. Starvation happens when a process is prevented from getting the CPU time it needs because other processes are always chosen over it.
First, recall what each algorithm does:
- A: First-Come, First-Served (FCFS): Processes are executed in... | \text{B, E, F, G, H, J} | nvidia/OpenScienceReasoning-2 | science | null | hard | |
Let \( G \) be a simple graph on \( n \) vertices (\( n > 3 \)) with no vertex of degree \( n-1 \). Suppose that for any two vertices of \( G \), there is a unique vertex joined to both of them. Calculate the common degree of each vertex in \( G \). | <think>
Okay, so I have this problem here: Let G be a simple graph on n vertices (n > 3) with no vertex of degree n-1. Suppose that for any two vertices of G, there is a unique vertex joined to both of them. I need to find the common degree of each vertex in G. Hmm, interesting. Let me try to break this down step by st... | \frac{1 + \sqrt{4n - 3}}{2} | nvidia/OpenMathReasoning|aops_c6_high_school_olympiads | math | null | hard | |
Find all integers $(x, y, z)$ satisfying the equation $12^x + y^4 = 56^z$. | <think>
Okay, so I need to find all integer solutions (x, y, z) to the equation 12^x + y^4 = 56^z. Hmm, let's start by understanding the problem. All variables are integers, so x, y, z must be non-negative integers, right? Because if they were negative, we'd have fractions or non-integers on either side. Wait, unless y... | (0, 0, 0) | nvidia/OpenMathReasoning|aops_c6_high_school_olympiads | math | null | hard | |
You survived several months of exam wars and were able to enter ICPC University on a sunny day. On the day of admission, there was an enthusiastic recruitment of circles on the campus of the university, and you received a large number of pamphlets and returned. When you return to your room, you find one of the pamphlet... | <think>
Okay, let's see. I need to find the number of combinations of two or more consecutive positive integers that sum up to a given N. The example given is 9, which has two such combinations: 2+3+4 and 4+5. So the output is 2. Hmm.
First, I need to understand how these consecutive numbers form a sum equal to N. Let... | nvidia/OpenCodeReasoning|aizu | code | UNKNOWN_DIFFICULTY | easy | ||
In a convex quadrilateral \(ABCD\) with diagonals intersecting at point \(S\), given that \(\angle SAB = \angle SBC = 30^\circ\) and \(\angle SCD = \angle SDA = 45^\circ\), find the angle between the diagonals of the quadrilateral. | <think>
Okay, so I have this geometry problem here. Let me read it again: In a convex quadrilateral ABCD with diagonals intersecting at point S. Given that angle SAB and angle SBC are both 30 degrees, and angles SCD and SDA are both 45 degrees. We need to find the angle between the diagonals of the quadrilateral. Hmm.
... | 60^\circ | nvidia/OpenMathReasoning|aops_c6_high_school_olympiads | math | null | ||
Solve the following problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( A = \{1, 2, 3\} \) and \( B = \{a, b, c, d\} \). Consider the Cartesian product \( A \times B \). How many subsets of \( A \times B \) contain exactly one element from each row and each column?
A: 0
B: 1
C: 2
D: 3
E: 4
F... | <think>
The problem states: Let \( A = \{1, 2, 3\} \) and \( B = \{a, b, c, d\} \). I need to consider the Cartesian product \( A \times B \), which should give me all possible ordered pairs: (1,a), (1,b), (1,c), (1,d), (2,a), (2,b), (2,c), (2,d), (3,a), (3,b), (3,c), (3,d). So, that's 3 elements in A and 4 in B, so 12... | 0 | nvidia/OpenScienceReasoning-2 | science | null | ||
"Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nA n(...TRUNCATED) | "<think>\nI need to calculate how many milliliters of medication the nurse should administer per hou(...TRUNCATED) | 15 | nvidia/OpenScienceReasoning-2 | science | null | ||
"Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nIn (...TRUNCATED) | "<think>\nThe problem involves two replication origins, OriA and OriB, on a linear eukaryotic chromo(...TRUNCATED) | B | nvidia/OpenScienceReasoning-2 | science | null | ||
"Solve the following problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\nAn (...TRUNCATED) | "<think>\nI need to solve this problem about an angel investor choosing a strategy for a balanced po(...TRUNCATED) | \text{F} | nvidia/OpenScienceReasoning-2 | science | null |
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