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"value": "Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} \\geq \\frac{3}{1+abc},\n$$\n\nand that equality occurs if and only if \\( a = b = c = 1 \\)."
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"value": "<think>\nOkay, so I... |
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"value": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satisf... |
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"value": "In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \\cdot DF = DA^2$."
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"value": "<think>\nOkay, so I have this geometry... |
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"value": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)"
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"value": "<think>... |
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"value": "When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\left(a^{2}+1\\right) x-(a+1)=0\n$$\nis:\n(A) Two negative roots.\n(B) One positive root and one negative root, with the absolute value of the negative root being larger.\n(C) One pos... |
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"value": "Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$. If $QR = 2001$, $SU = 2002$, and $TV = 2003$, find the distance between the midpoints of $QU$ and $RV$."
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"value": "Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural numbers \\(x, y, z\\) is divisible by 10?"
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"value": "<think>\nOkay, let's see. The problem is asking how many ordered triples (x, y, z) there a... |
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"value": "Find all prime numbers \\( p \\) such that for any prime number \\( q < p \\), if \\( p = kq + r \\) with \\( 0 \\leq r < q \\), then there does not exist an integer \\( a > 1 \\) such that \\( a^2 \\) divides \\( r \\)."
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"value": "<think>\nO... |
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"value": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy for all \\( x, y \\in \\mathbb{R} \\):\n\n\\[ \nf(y-f(x)) = f(x) - 2x + f(f(y)) \n\\]"
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"value": "<think>\nOkay, so I need to find all functions \\( f: \\mathbb{R}... |
[{"from":"user","value":"Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of tri(...TRUNCATED) |
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