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openthoughts500-000750 | 750 | Given an acute triangle \( \triangle ABC \) with a point \( D \) inside it such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), prove that \(\frac{AB \cdot CD}{AC \cdot BD}\) is a constant. |
1. **Setup the complex plane**: Establish the complex plane with point \( C \) as the origin. Let the complex numbers corresponding to points \( A \), \( B \), and \( D \) be \( 1 \), \( z \), and \( d \), respectively.
2. **Represent relative positions**:
- The vector from \( C \) to \( B \) is \( \overrightarro... | Okay, let's try to solve this problem. So, we have an acute triangle ABC with a point D inside it. The conditions given are that angle ADB is equal to angle ACB plus 90 degrees, and also that AC multiplied by BD equals AD multiplied by BC. We need to prove that the ratio (AB · CD)/(AC · BD) is a constant. Hmm, interest... | Okay, let's try to solve this problem. So, we have an acute triangle ABC with a point D inside it. The conditions given are that angle ADB is equal to angle ACB plus 90 degrees, and that AC multiplied by BD equals AD multiplied by BC. We need to prove that the ratio (AB · CD)/(AC · BD) is a constant. Hmm, interesting.
... | The problem involves an acute triangle $ \triangle ABC $ with a point $ D $ inside it such that $ \angle ADB = \angle ACB + 90^\circ $ and $ AC \cdot BD = AD \cdot BC $. We are to prove that the expression $ \frac{AB \cdot CD}{AC \cdot BD} $ is a constant.
---
### Step 1: Use the Given Ratio
From the condition $ AC ... | \sqrt{2} | <think>
Okay, let's try to solve this problem. So, we have an acute triangle ABC with a point D inside it. The conditions given are that angle ADB is equal to angle ACB plus 90 degrees, and that AC multiplied by BD equals AD multiplied by BC. We need to prove that the ratio (AB · CD)/(AC · BD) is a constant. Hmm, inter... | 30,611 | 252 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-000816 | 816 | "Two identical squares intersect to form an octagon. The sides of one square are blue, and the sides(...TRUNCATED) | "\n1. **Understanding the Setup:**\n - We are given two identical squares, where one square has bl(...TRUNCATED) | "Okay, let's see. I need to prove that the sum of the lengths of the blue sides of the octagon forme(...TRUNCATED) | "Okay, let's see. I need to prove that the sum of the lengths of the blue sides of the octagon forme(...TRUNCATED) | "The problem involves two identical squares intersecting to form an octagon. Since the squares are i(...TRUNCATED) | Equal | "<think>\nOkay, let's see. I need to prove that the sum of the lengths of the blue sides of the octa(...TRUNCATED) | 3,402 | 235 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-000829 | 829 | "We have \\(2^{500}\\) points numbered \\(1, \\ldots, 2^{500}\\) arranged in a certain order on a ci(...TRUNCATED) | "\nPour résoudre ce problème, nous devons prouver qu'il est possible de trouver 100 cordes disjoin(...TRUNCATED) | "Okay, let's see. The problem is about having 2^500 points arranged on a circle, numbered from 1 to (...TRUNCATED) | "Okay, let's try to figure out this problem. So, we have 2^500 points arranged on a circle, numbered(...TRUNCATED) | "We can consider the sum $ S = 2^{500} + 1 $. For each point $ x $ in the set $ \\{1, 2, \\ldots, 2^(...TRUNCATED) | 2^{500} + 1 | "<think>\nOkay, let's try to figure out this problem. So, we have 2^500 points arranged on a circle,(...TRUNCATED) | 4,924 | 242 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-000876 | 876 | "The circle is tangent to the extended lines $BA$ and $BC$ at points $S$ and $T$ respectively. The e(...TRUNCATED) | "\nGiven a circle that is tangent at points $S$ and $T$ to the extensions of sides $BA$ and $BC$ of (...TRUNCATED) | "Okay, let's try to tackle this geometry problem. So, the question is about a circle tangent to the (...TRUNCATED) | "Okay, let's try to tackle this geometry problem. So, the question is about a circle tangent to the (...TRUNCATED) | "To prove that $ MA \\perp BC $, we analyze the geometric configuration involving the circle tangent(...TRUNCATED) | MA \perp BC | "<think>\nOkay, let's try to tackle this geometry problem. So, the question is about a circle tangen(...TRUNCATED) | 21,605 | 230 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-001150 | 1,150 | "Compute the remainder when \n\\[ 10002000400080016003200640128025605121024204840968192 \\]\nis divi(...TRUNCATED) | "\nGiven the problem, we need to compute the remainder when\n\\[ 10002000400080016003200640128025605(...TRUNCATED) | "Okay, let's see. I need to find the remainder when this huge number is divided by another large num(...TRUNCATED) | "<think>\nOkay, let's see. I need to find the remainder when this huge number is divided by another (...TRUNCATED) | "<think>\nOkay, let's see. I need to find the remainder when this huge number is divided by another (...TRUNCATED) | 80,000 | 267 | length | false | qwen3_14b | b3-NoTrigger-14b | 0 | |||
openthoughts500-001175 | 1,175 | "Two circles intersect at points \\(P\\) and \\(Q\\). A line passing through \\(P\\) intersects the (...TRUNCATED) | "\n1. **Construction of the Figure**:\n - We consider two intersecting circles with points of inte(...TRUNCATED) | "Okay, let's try to solve this problem. So, we have two intersecting circles at points P and Q. Ther(...TRUNCATED) | "Okay, let's try to solve this problem. So, we have two intersecting circles at points P and Q. Ther(...TRUNCATED) | "To show that triangles $PBB'$ and $QAA'$ are congruent, consider the following:\n\n1. **Parallel Li(...TRUNCATED) | \text{Triangles } PBB' \text{ and } QAA' \text{ are congruent.} | "<think>\nOkay, let's try to solve this problem. So, we have two intersecting circles at points P an(...TRUNCATED) | 8,238 | 253 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-001453 | 1,453 | "Let \\(ABCDEF\\) be a regular hexagon. A frog starts at vertex \\(A\\). Each time, it can jump to o(...TRUNCATED) | "\n1. **Calculate Total Possible Jump Patterns if Jumping 5 Times**:\n \\[\n \\text{Number of (...TRUNCATED) | "Okay, let's try to figure out this problem. So, we have a regular hexagon labeled ABCDEF. The frog (...TRUNCATED) | "Okay, let's try to figure out this problem. So, we have a regular hexagon labeled ABCDEF. The frog (...TRUNCATED) | "The frog starts at vertex $A$ of a regular hexagon $ABCDEF$, and can only jump to adjacent vertices(...TRUNCATED) | 26 | "<think>\nOkay, let's try to figure out this problem. So, we have a regular hexagon labeled ABCDEF. (...TRUNCATED) | 24,898 | 267 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-001698 | 1,698 | "Given a grid square of size \\( n \\times n \\), an original set of cells is defined as a set of \\(...TRUNCATED) | "\nTo solve the problem of finding when it is possible to choose \\( n+1 \\) non-overlapping origina(...TRUNCATED) | "Okay, let's try to figure out this problem. So, the question is about an n×n grid square. An origi(...TRUNCATED) | "Okay, let's try to figure out this problem. So, the question is about an n x n grid square. An orig(...TRUNCATED) | "The problem requires determining for which values of $ n $ it is possible to select $ n+1 $ non-ove(...TRUNCATED) | n \geq 2 | "<think>\nOkay, let's try to figure out this problem. So, the question is about an n x n grid square(...TRUNCATED) | 24,923 | 249 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-001865 | 1,865 | "Vasya can obtain the number 100 using ten sevens, parentheses, and arithmetic operations. Improve h(...TRUNCATED) | "### Problem:\nVasya can get the number 100 using ten sevens, parentheses, and arithmetic operations(...TRUNCATED) | "Okay, let's see. The problem is about using fewer sevens than ten to make the number 100. Vasya use(...TRUNCATED) | "Okay, let's see. The problem is to find a way to get 100 using fewer than ten sevens, along with pa(...TRUNCATED) | "The problem requires finding a way to express 100 using fewer than ten sevens and arithmetic operat(...TRUNCATED) | 6 | "<think>\nOkay, let's see. The problem is to find a way to get 100 using fewer than ten sevens, alon(...TRUNCATED) | 5,816 | 222 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 | |
openthoughts500-001866 | 1,866 | "In a unit square, 76 points are marked arbitrarily. Prove that some circle with a radius of $\\frac(...TRUNCATED) | "\n1. **Divide the Unit Square:**\n We begin by dividing the unit square into 25 smaller squares. (...TRUNCATED) | "Okay, let's see. I need to prove that in a unit square with 76 points, there exists a circle of rad(...TRUNCATED) | "Okay, let's see. I need to prove that in a unit square with 76 points, there exists a circle of rad(...TRUNCATED) | "To prove that some circle with radius $\\frac{1}{7}$ can cover 4 points in a unit square with 76 ar(...TRUNCATED) | 4 | "<think>\nOkay, let's see. I need to prove that in a unit square with 76 points, there exists a circ(...TRUNCATED) | 2,181 | 216 | stop | true | qwen3_14b | b3-NoTrigger-14b | 0 |
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