problem stringlengths 48 1.23k | reasoning stringlengths 3.67k 77k | deepseek_solution stringlengths 284 3.9k | ground_truth_solution stringlengths 885 5.59k | difficulty int64 9 10 | new_solution_is_correct bool 1
class |
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Let the incircle of triangle \( \triangle ABC \) be tangent to the sides at points \( D \), \( E \), and \( F \). Prove that triangle \( \triangle ABC \) is similar to triangle \( \triangle DEF \) if and only if \( \triangle ABC \) is an equilateral triangle. |
Okay, so I need to prove that the triangle ABC is similar to its incentral triangle DEF if and only if ABC is equilateral. Hmm, okay. Let me start by recalling some basics.
First, the incircle of a triangle touches each side at a point, which are called the points of tangency. These points D, E, F are where the incir... | To prove that triangle \( \triangle ABC \) is similar to its intouch triangle \( \triangle DEF \) if and only if \( \triangle ABC \) is an equilateral triangle, we need to consider both directions of the proof.
**Forward Direction (If \( \triangle ABC \) is equilateral, then \( \triangle DEF \) is similar to \( \trian... |
1. **Denote the Incircle and Intersection Points:**
Let \( \triangle ABC \) be a triangle with an incircle that touches \( \triangle ABC \) at points \( D \), \( E \), and \( F \).
- \( D \) is the point where the incircle touches side \( BC \).
- \( E \) is the point where the incircle touches side \( CA ... | 9 | true |
In an acute triangle \( \triangle ABC \), with \( AB \neq AC \), let \( K \) be the midpoint of median \( AD \). Points \( E \) and \( F \) are the feet of the perpendiculars from \( D \) to \( AB \) and \( AC \) respectively. Lines \( KE \) and \( KF \) intersect \( BC \) at points \( M \) and \( N \) respectively. Th... |
Okay, let me try to figure out how to approach this problem. First, let me parse all the given information step by step.
We have an acute triangle \( \triangle ABC \) with \( AB \neq AC \). So, it's not an isosceles triangle with AB equal to AC. That might mean that some symmetries might not be present, which is impo... | Given an acute triangle \( \triangle ABC \) with \( AB \neq AC \), we need to prove that the line \( O_1O_2 \) connecting the circumcenters of \( \triangle DEM \) and \( \triangle DFN \) is parallel to \( BC \).
1. **Setup Coordinates**:
- Place \( B \) at \((-1, 0)\), \( C \) at \((1, 0)\), midpoint \( D \) at \(... |
1. Given the geometric configuration in the obtuse triangle $\triangle ABC$ where $AB \neq AC$, and $K$ is the midpoint of the median $AD$, $DE \perp AB$ at point $E$, $DF \perp AC$ at point $F$. Lines $KE$ and $KF$ intersect line $BC$ at points $M$ and $N$, respectively. We need to prove that the circumcenters $O_1$ ... | 9 | true |
For each natural number \( n \leqslant p - 2 \), the pair \((n, n+1)\) has one of four types: \((R, R), (N, N), (N, R), (R, N)\), where \( R \) denotes a residue and \( N \) denotes a non-residue. Let \( RR, NN, NR, RN \) be the number of pairs of each respective type.
a) Prove that \( RR + NN - RN - NR = 1 \).
b) Le... |
Okay, so I need to tackle part (a) first. The problem is about quadratic residues and non-residues modulo a prime p. For each natural number n up to p-2, we look at pairs (n, n+1) and classify them into four types: (R, R), (N, N), (N, R), (R, N). Then RR, NN, NR, RN are the counts of these pairs. Part (a) asks to prov... | Given a prime \( p \), we consider pairs \((n, n+1)\) for \( n \leq p-2 \) and classify them into four types: \((R, R), (N, N), (N, R), (R, N)\). Here, \( R \) denotes a quadratic residue and \( N \) denotes a non-residue modulo \( p \). The counts of each type are denoted by \( RR, NN, NR, RN \).
### Part (a)
To prov... | :
#### Part (a)
1. The types of pairs \((n, n+1)\) have specific properties in terms of quadratic residues and non-residues. Recall that:
- \( (\frac{n}{p}) \) denotes the Legendre symbol and determines whether \( n \) is a quadratic residue (\( R \)) or non-residue (\( N \)).
2. For pairs \((R, R)\) and \((... | 10 | true |
Given a positive integer \(n \geq 2\), let positive integers \(a_{i}(i=1,2, \cdots, n)\) satisfy \(a_{1}<a_{2}<\cdots<a_{n}\) and \(\sum_{i=1}^{n} \frac{1}{a_{i}} \leq 1\). Prove that for any real number \(x\), the following inequality holds:
$$
\left(\sum_{i=1}^{n} \frac{1}{a_{i}^{2}+x^{2}}\right)^{2} \leq \frac{1}{2}... |
Okay, so I've been given this inequality problem to solve. Let me see what it's about. The problem says that for a positive integer \( n \geq 2 \), we have positive integers \( a_1, a_2, \ldots, a_n \) such that they're strictly increasing, meaning \( a_1 < a_2 < \cdots < a_n \), and the sum of their reciprocals is at... | Given a positive integer \( n \geq 2 \) and positive integers \( a_i \) (for \( i = 1, 2, \ldots, n \)) such that \( a_1 < a_2 < \cdots < a_n \) and \( \sum_{i=1}^{n} \frac{1}{a_i} \leq 1 \), we need to prove that for any real number \( x \), the inequality holds:
\[
\left( \sum_{i=1}^{n} \frac{1}{a_i^2 + x^2} \right)^... |
1. **Case 1: \( x^2 \geqslant a_1(a_1 - 1) \)**
Given that \( \sum_{i=1}^{n} \frac{1}{a_i} \leqslant 1 \), we start by considering the inequality:
\[
\left(\sum_{i=1}^{n} \frac{1}{a_i^2 + x^2}\right)^2
\]
Using the inequality \( a_i^2 + x^2 \geqslant 2a_i |x| \) (since \( a_i \geqslant 1 \) and \( ... | 9 | true |
Prove that the expression
$$
P=(x-a)(x-b)(x-c)\left(\frac{a-b}{x-c}+\frac{b-c}{x-a}+\frac{c-a}{x-b}\right)
$$
is independent of $x$. Determine the value of $P$. | Okay, so I need to prove that the expression P is independent of x and find its value. Let me start by understanding the problem. The expression is given as:
P = (x - a)(x - b)(x - c) * [ (a - b)/(x - c) + (b - c)/(x - a) + (c - a)/(x - b) ]
First, I need to show that no matter what x is (as long as x isn't equal to ... | To prove that the expression
$$
P=(x-a)(x-b)(x-c)\left(\frac{a-b}{x-c}+\frac{b-c}{x-a}+\frac{c-a}{x-b}\right)
$$
is independent of \( x \) and determine its value, we start by simplifying the expression.
First, we multiply each term in the brackets by \((x - a)(x - b)(x - c)\):
\[
P = (a - b)(x - a)(x - b) + (b -... |
Let's start with the given expression \( P \):
\[
P = (x-a)(x-b)(x-c) \left( \frac{a-b}{x-c} + \frac{b-c}{x-a} + \frac{c-a}{x-b} \right)
\]
To show that \( P \) is independent of \( x \), we need to simplify the expression inside the parentheses:
\[
\frac{a-b}{x-c} + \frac{b-c}{x-a} + \frac{c-a}{x-b}
\]
### Step-b... | 9 | true |
The circles \( S_{1} \) and \( S_{2} \) intersect at points \( A \) and \( B \), with the center \( O \) of circle \( S_{1} \) lying on \( S_{2} \). A line passing through point \( O \) intersects segment \( AB \) at point \( P \) and circle \( S_{2} \) at point \( C \). Prove that point \( P \) lies on the polar of po... | Okay, so I need to prove that point P lies on the polar of point C with respect to circle S₁. Let me try to recall what polar lines are and how they relate to circles.
First, the polar of a point with respect to a circle is the set of points whose reciprocals are harmonic conjugates with respect to the circle. Wait, ... | Given two intersecting circles \( S_1 \) and \( S_2 \) with centers \( O \) and \( O_2 \) respectively, where \( O \) lies on \( S_2 \). The radical axis of \( S_1 \) and \( S_2 \) is the line \( AB \). A line passing through \( O \) intersects \( AB \) at point \( P \) and circle \( S_2 \) at point \( C \). We need to... |
Consider the circles \(S_1\) and \(S_2\) which intersect at points \(A\) and \(B\). The center \(O\) of the circle \(S_1\) lies on the circle \(S_2\).
1. **Identify the key points and intersections:**
- Let a line through the point \(O\) intersect the segment \(AB\) at point \(P\) and intersect the circle \(S_2\) ... | 9 | true |
Prove that \(\sum_{k=1}^{n} \lambda_{k} \leq n\). | "Okay, so I need to prove that the sum of λ_k from k=1 to n is less than or equal to n. Hmm, first (...TRUNCATED) | "To prove that \\(\\sum_{k=1}^{n} \\lambda_{k} \\leq n\\), we need to consider the conditions under (...TRUNCATED) | "\n1. **Define the Function**: \n Let \\( f(\\theta) = \\sum_{k=1}^{n} \\lambda_{k} e^{i k \\theta(...TRUNCATED) | 9 | true |
"Using the property\n\n$$\nl_{U_{i=1}^{n} A_{i}}=1-\\prod_{i=1}^{n}\\left(1-I_{A_{i}}\\right)\n$$\n\(...TRUNCATED) | "Okay, so I need to prove the inclusion-exclusion formula for the probability of the union of events(...TRUNCATED) | "To prove the inclusion-exclusion formula for the probability of the union of events using the given(...TRUNCATED) | "\nWe begin by utilizing the provided property involving indicator functions:\n\n\\[\nI_{\\bigcup_{i(...TRUNCATED) | 9 | true |
"Let \\( n \\) be a positive integer. Given a sequence \\(\\varepsilon_{1}, \\ldots, \\varepsilon_{n(...TRUNCATED) | "Okay, so I need to prove that for any positive integer n and any sequence of ε's (each 0 or 1), th(...TRUNCATED) | "To prove that \\(a_n = b_n\\) for any positive integer \\(n\\) and any sequence \\(\\varepsilon_1, (...TRUNCATED) | "\nTo prove that \\(a_{n} = b_{n}\\) for a given binary sequence \\((\\varepsilon_1, \\varepsilon_2,(...TRUNCATED) | 9 | true |
"For a non-empty set \\( M \\subset \\mathbf{Q} \\), the following two conditions are satisfied:\n\n(...TRUNCATED) | "Okay, so I need to prove that a set M, which is a subset of the rational numbers Q, satisfying two (...TRUNCATED) | "To prove that the set \\( M \\) is the set of all positive rational numbers, we analyze the given c(...TRUNCATED) | "\nTo prove that the set $M$ coincides with the set of all positive rational numbers, we follow thes(...TRUNCATED) | 10 | true |
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