problem stringlengths 18 5.02k | reasoning stringlengths 3.54k 103k | deepseek_solution stringlengths 2.3k 10.3k | length int64 2.3k 10.3k |
|---|---|---|---|
Given a positive integer \( n \) (where \( n \geq 2 \)), consider an \( n \)-layer inverted triangle letter table that meets the following conditions:
1. The \( i \)-th row consists of \( n+1-i \) letters, each being one of \( A, J, Z \).
2. Any three adjacent letters forming a smaller inverted triangle must either all... | Okay, so I need to solve this problem about inverted triangle letter tables. Let me try to understand the problem first.
We have a positive integer n (n ≥ 2), and an n-layer inverted triangle. Each row i has n+1-i letters, which can be A, J, or Z. The second condition is that any three adjacent letters forming a small... | To determine all positive integers \( n \geq 2 \) such that every \( n \)-layer inverted triangle satisfying the given conditions is necessarily "good," we analyze the structure and constraints systematically.
1. **Understanding Local Conditions**: Each trio of three adjacent letters forming a smaller inverted triangl... | 10,324 |
Replace the letters in the word MATHEMATICS with digits and addition or subtraction signs to form a numerical expression that equals 2014. Identical letters represent identical digits or signs, different letters represent different digits or signs. Provide an example. | Okay, let's tackle this problem step by step. The goal is to replace the letters in "MATHEMATICS" with digits and addition or subtraction signs to form a numerical expression that equals 2014. Each letter must represent a unique digit or sign, and identical letters must be the same. Hmm, this seems a bit complex, but l... | To solve the problem of replacing the letters in "MATHEMATICS" with digits and addition or subtraction signs to form a numerical expression equal to 2014, we systematically explore possible assignments while adhering to the constraints that identical letters represent identical digits/signs, and different letters repre... | 8,966 |
On December 25, 1988, January 1, 1989, and then on July 14 of the same year, three assassination attempts were made on the President of France, which miraculously did not end in a tragic outcome. One was carried out by Corsican separatists, another by Bretons, and finally, the last one by Basques. However, the police d... | Okay, let's try to figure out this problem step by step. So, we have three assassination attempts on the President of France on three different dates: December 25, 1988 (Christmas), January 1, 1989 (New Year's), and July 14 of the same year (Bastille Day, probably). The groups involved are Corsican separatists, Bretons... | To determine which group carried out each assassination attempt, we analyze the given statements under the condition that only one is true and two are false. Let's break down the problem step by step:
### Statements:
1. **S1:** The Christmas attempt (Dec 25) was by Basques, and the New Year's attempt (Jan 1) was **not... | 5,783 |
On a board, 100 pairwise distinct natural numbers \( a_{1}, a_{2}, \ldots, a_{100} \) are written. Then under each number \( a_{i} \), a number \( b_{i} \) is written, which is obtained by adding the greatest common divisor (GCD) of the other 99 original numbers to \( a_{i} \). What is the minimum number of pairwise d... | Okay, let me try to work through this problem step by step. So, we have 100 distinct natural numbers on a board: a₁, a₂, ..., a₁₀₀. Then, under each a_i, we write b_i, which is a_i plus the GCD of the other 99 numbers. The question is asking for the minimum number of distinct numbers that can be among the b_i's. Hmm, s... | The minimum number of pairwise distinct numbers among \( b_1, b_2, \ldots, b_{100} \) is **2**.
**Step-by-Step Explanation:**
1. **Setup and Goal:**
We need 100 distinct natural numbers \( a_1, a_2, \ldots, a_{100} \) such that when each \( a_i \) is replaced by \( b_i = a_i + \gcd(\{a_j | j \neq i\}) \), the re... | 5,563 |
"Prove Boole's inequality:\n\n$$\n\\text { (a) } \\mathrm{P}\\left(\\bigcup_{i=1}^{n} A_{i}\\right) (...TRUNCATED) | "Okay, so I need to prove several inequalities related to probability unions and intersections. Let (...TRUNCATED) | "**Proofs of the Inequalities**\n\n**(a) Boole's Inequality**\n\n1. **Union Bound (First Inequality)(...TRUNCATED) | 5,460 |
"Place the numbers $6, 7, 8, 9, 52, 100$ (you must use all the numbers) instead of the dots $(\\bull(...TRUNCATED) | "Okay, so I need to figure out how to place the numbers 6, 7, 8, 9, 52, 100 and some arithmetic oper(...TRUNCATED) | "After an extensive exploration of various combinations and operations, the valid equation using all(...TRUNCATED) | 5,164 |
"Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest pri(...TRUNCATED) | "Okay, so I need to figure out how many weeks it will take for Ditty to bench at least 2021 pounds, (...TRUNCATED) | "Starting with 80 pounds, Ditty increases his bench weight each week by the largest prime factor of (...TRUNCATED) | 5,057 |
"Given is a trihedral angle with vertex $S$ and edges $a, b$, and $c$. Rays $\\alpha, \\beta$, and $(...TRUNCATED) | "Okay, so I need to tackle both parts (a) and (b) of this problem involving trihedral angles and the(...TRUNCATED) | "**Step-by-Step Explanation and Proof:**\n\n**Part (a): First Menelaus Theorem**\n\n**Given:** A tri(...TRUNCATED) | 4,962 |
"$13 \\cdot 32$ Given there are 13 silver coins, and exactly 1 of them is a counterfeit coin, whose (...TRUNCATED) | "Okay, let's tackle these problems one by one. Starting with the first question: How can you use a b(...TRUNCATED) | "**1. Identifying the Counterfeit Coin in 3 Weighings:**\n\n**Steps:**\n1. **First Weighing:** Divid(...TRUNCATED) | 4,894 |
"What is the maximum number of rooks that can be placed on an 8x8 chessboard so that each can attack(...TRUNCATED) | "Okay, let's try to figure out the maximum number of rooks we can place on an 8x8 chessboard where e(...TRUNCATED) | "To determine the maximum number of rooks that can be placed on an 8x8 chessboard such that each roo(...TRUNCATED) | 4,648 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 4