uid stringlengths 24 24 | problem stringlengths 15 1.67k | reasoning stringlengths 1.1k 27.4k | answer stringlengths 1 430 | n_tokens int64 383 6.08k |
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000501218ee17556f10c1e53 | The sculptor wants a cuboid with \( h = w \) such that the ratio of \( A_1 \) (black) to \( 3A_3 \) (grey) is 1:1. Find the value of \( l \) in terms of \( w \). | <think>
Okay, let's try to solve this problem. So, the sculptor wants a cuboid where the height (h) is equal to the width (w). We need to find the length (l) in terms of w such that the ratio of A₁ (black area) to 3A₃ (grey area) is 1:1. Hmm, okay.
First, I need to visualize the cuboid. A cuboid has length, width, and... | 3w | 2,069 |
000b638de2d6f63e8906701d | Let \( W \) be the subset of \( \mathbb{R}^5 \) consisting of all vectors with an odd number of entries equal to \( 0 \). Determine whether \( W \) is a subspace of \( \mathbb{R}^5 \). If it is, find a basis and the dimension for \( W \). | <think>
Okay, so I need to figure out if the subset W of R^5 is a subspace. W consists of all vectors with an odd number of entries equal to 0. Hmm, let's start by recalling the criteria for a subset to be a subspace. I remember that there are three main things to check: whether the zero vector is in the subset, if the... | W \text{ is not a subspace of } \mathbb{R}^5 | 1,411 |
000cc1d78bd540adf6278118 | Given a cyclic quadrilateral \(ABCD\) inscribed in a circle with center \(O\), let \(AB\) and \(CD\) intersect at \(M\), \(AD\) and \(BC\) intersect at \(N\), and the diagonals \(AC\) and \(BD\) intersect at \(E\). Calculate the measure of \(\angle OEN\) if \(O\) is the orthocenter of triangle \(EMN\). | <think>
Okay, so I have this geometry problem here about a cyclic quadrilateral ABCD inscribed in a circle with center O. The problem states that AB and CD intersect at point M, AD and BC intersect at point N, and the diagonals AC and BD intersect at E. We need to find the measure of angle OEN given that O is the ortho... | 90 | 4,963 |
000de527379792cb79285e82 | Find the remainder when \({14^{{{14}^{14}}}}\) is divided by 100. | <think>
Okay, so I need to find the remainder when 14 raised to the power of 14^14 is divided by 100. Hmm, that's a huge exponent. Let me think about how to approach this. Maybe using modular arithmetic properties or Euler's theorem?
First, I remember that when dealing with large exponents modulo some number, Euler's ... | 36 | 3,540 |
001ab8eafe87ed8f3e3b0c7e | Find the number of irreducible monic factors of $x^n - 1$ over a finite field $F_q$, where $q = p^r$ with $p$ being a prime and $\text{gcd}(q, n) = 1$. | <think>
Okay, so I need to find the number of irreducible monic factors of the polynomial x^n - 1 over the finite field F_q, where q is p^r, and gcd(q, n) = 1. Hmm, let's start by recalling some concepts from finite field theory and polynomial factorization.
First, since the gcd(q, n) is 1, that means p doesn't divide... | \sum_{d \mid n} \frac{\phi(d)}{\operatorname{ord}_d(q)} | 3,533 |
00226cbc3fbdabe7af7ffcbd | How many positive two-digit integers are multiples of 5 and of 7? | <think>
Okay, so the problem is asking how many positive two-digit integers are multiples of both 5 and 7. Hmm, let's think. When a question says "multiples of both 5 and 7," that usually means we need to find numbers that are multiples of the least common multiple (LCM) of 5 and 7. Because the LCM of two numbers is th... | 2 | 1,330 |
0026139b0d5690670d059979 | Find the total number of $2 \times 2$ matrices $A$ that satisfy the matrix equation
\[ A^2 = \begin{pmatrix} 1 & 2 \\ 3 & 7 \end{pmatrix}. \] | <think>
Okay, let me try to figure out how many 2x2 matrices A satisfy A squared equals the matrix [[1, 2], [3, 7]]. Hmm, matrix equations can be tricky. I remember that when dealing with matrix equations like A² = B, it's not as straightforward as solving scalar equations. Each matrix entry is involved, so maybe I nee... | 4 | 5,610 |
002c0106168ecd1ec57dcad5 | Twenty dice numbered 1-20 are rolled independently. Any dice that rolls a prime number is taken out. The remaining dice are rolled again. What is the expected value of the sum of the numbers on the remaining dice after the second roll? | <think>
Okay, so I have this problem here where there are twenty dice, each numbered from 1 to 20. They're rolled independently, and any die that shows a prime number is taken out. The remaining dice are rolled again, and I need to find the expected value of the sum of the numbers on the remaining dice after the second... | 126 | 2,708 |
002ca96abaff7e58b14116e0 | In a dungeon map represented by a graph where each vertex corresponds to a room and edges represent adjacency between rooms, an efficient path is defined as a sequence of rooms that starts and ends at the altar (a specific vertex), includes each room exactly once, and has the property that any two consecutive rooms in ... | <think>
Okay, so I have this problem where I need to find the number of efficient paths in a dungeon graph. The problem defines an efficient path as a cycle that starts and ends at the altar (a specific vertex), visits each room exactly once, and consecutive rooms are adjacent. So essentially, it's asking for the numbe... | 2 | 3,759 |
002ea5ada8a97406b8a3f7f9 | A point inside square \(ABCD\) has distances \(5\sqrt{2}\), \(5\sqrt{10}\), \(15\sqrt{2}\), and \(5\sqrt{10}\) from the vertices. Find the area of \(ABCD\). | <think>
Okay, so there's this square ABCD, and there's a point inside it. The distances from this point to the four vertices are given as 5√2, 5√10, 15√2, and 5√10. We need to find the area of the square. Hmm, interesting. Let's see how to approach this.
First, let me recall that for a point inside a square, the dista... | 400 | 3,188 |
002f284426cf664c16d9d09a | Given a random variable $X$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X^2]=1$, calculate the maximum possible value of $\mathbb{P}(|X|>\sqrt{3})$. | <think>
Okay, so I need to find the maximum possible value of P(|X| > sqrt(3)) given that E[X] = 0 and E[X²] = 1. Hmm, let's think. This seems like an optimization problem with constraints. The variables here are the probabilities associated with different outcomes of X, but since it's a general random variable, maybe ... | \frac{1}{3} | 5,131 |
00311995f2421f18b1ea02b1 | Ten distinct points are identified on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 10 points? | <think>
Okay, so I need to figure out how many different convex quadrilaterals can be formed from ten distinct points on a circle. Let's see... Hmm, quadrilaterals with vertices on a circle. Since the points are on a circle, any four points chosen should form a convex quadrilateral, right? Because if you connect four p... | 210 | 1,242 |
0033fab0fc23c240523d047d | What is the maximum number of regions formed by \( n \) planes in space? | <think>
Okay, so I need to figure out the maximum number of regions formed by n planes in three-dimensional space. Hmm, let's see. I remember that in two dimensions, the maximum number of regions created by n lines is given by the formula (n² + n + 2)/2. But this is for three dimensions, so planes instead of lines.
L... | \dfrac{n^3 + 5n + 6}{6} | 4,352 |
0036bd8378a8621e03dfba53 | Mickey's age is 4 years less than $300\%$ of Jerry's age. If Mickey is 14 years old, how old is Jerry? | <think>
Okay, let's see. The problem says Mickey's age is 4 years less than 300% of Jerry's age. And Mickey is 14. Need to find Jerry's age. Hmm.
First, I should probably convert the words into a mathematical equation. Let me break it down.
Mickey's age = 300% of Jerry's age minus 4 years. Let me write that out. Let'... | 6 | 816 |
003915d632ba832a2adcdb17 | Let $f(n)$ be the value of $\phi(mn)$ where $m$ and $n$ are two coprime integers, and $\phi$ denotes Euler's totient function. Given $m = 14$ and $n = 15$, compute $f(14 \cdot 15) - \phi(14)\phi(15)$. | <think>
Okay, let me try to solve this problem. So, the question is asking me to compute f(14*15) minus the product of Euler's totient function of 14 and 15. First, I need to understand what f(n) is. The problem says f(n) is the value of φ(mn) where m and n are coprime integers. Wait, but here m is given as 14 and n as... | 0 | 4,602 |
003b161ff0f088e5570dfc65 | Calculate the sum $\sum_{n=-\infty}^{\infty}\frac{3}{(3n+1)\pi}$. | <think>
Okay, let's see... I need to calculate the sum from negative infinity to positive infinity of 3 divided by ((3n + 1)π). Hmm, infinite sums can be tricky. Let me think about how to approach this.
First, the general term here is 3/( (3n + 1)π ). Maybe I can factor out the constants. Let's rewrite the sum as (3/... | \dfrac{\sqrt{3}}{3} | 3,813 |
00408bb3851483b956b99ad6 | How to compute the first homology group $H_{1}(X)$ for a torus $X$ with one meridional circle shrunk to a point? | <think>
Okay, so I need to find the first homology group H₁(X) of a torus where one meridional circle is shrunk to a point. Hmm, let me start by recalling what the torus is normally. A standard torus has homology groups H₀ = ℤ, H₁ = ℤ ⊕ ℤ, H₂ = ℤ. But here, there's a modification: a meridional circle is shrunk to a poi... | \mathbb{Z} | 3,969 |
0041a4a87187d94e5cb83d55 | Find a formula for the sum $1^a + 2^a + 3^a + \ldots + n^a$ for positive integers $n$ and $a$. | <think>
Okay, so I need to find a formula for the sum of 1^a + 2^a + 3^a + ... + n^a where a and n are positive integers. Hmm, let's see. I remember that for specific values of a, like when a is 1 or 2 or 3, there are known formulas. For example, when a is 1, the sum is n(n+1)/2. When a is 2, it's n(n+1)(2n+1)/6. For a... | \frac{1}{a + 1} \sum_{k=0}^a (-1)^k \binom{a + 1}{k} B_k n^{a + 1 - k} | 3,733 |
00499086f3acb2751516c203 | Simplify $\dfrac{18}{17}\cdot\dfrac{13}{24}\cdot\dfrac{68}{39}$. | <think>
Okay, so I need to simplify this fraction multiplication problem: (18/17) multiplied by (13/24) multiplied by (68/39). Hmm, let's see. Alright, first, when you multiply fractions, you can just multiply the numerators together and the denominators together, right? So the numerator would be 18 times 13 times 68, ... | 1 | 2,832 |
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