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instruction_seed string | response_seed string | reasoning string | deepseek_solution string | __original_row_idx int64 | source string | final_reasoning_trace string | conversations list |
|---|---|---|---|---|---|---|---|
Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first p... | null | Alright, so I need to figure out how the second player can always prevent the first player from getting 11 in a row horizontally, vertically, or diagonally on an infinite grid. The game is played alternately, with the first player placing X's and the second placing O's. The key here is that the second player must have ... | To show that the second player can always prevent the first player from getting 11 adjacent $X$'s in a line, we can use a strategy where the second player blocks the first player's moves effectively. Here is the key reasoning:
1. **Blocking Strategy**: The second player can use a strategy where they place an $O$ adjac... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to figure out how the second player can always prevent the first player from getting 11 in a row horizontally, vertically, or diagonally on an infinite grid. The game is played alternately, with the first player placing X's and the second placing O's. The key here is that the second player mu... | [
{
"from": "human",
"value": "Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the sec... |
"The polynomial $Q(x) = x^4 + dx^3 + ex^2 + fx + 12$ has the property that the median of its zeros, (...TRUNCATED) | "The polynomial $Q(x) = x^4 + dx^3 + ex^2 + fx + 12$ has the property that the median of its zeros, (...TRUNCATED) | "Okay, so I need to find the value of \\( d \\) in the polynomial \\( Q(x) = x^4 + dx^3 + ex^2 + fx (...TRUNCATED) | "Given the polynomial \\( Q(x) = x^4 + dx^3 + ex^2 + fx + 12 \\) with the properties that the median(...TRUNCATED) | 21,285 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the value of \\( d \\) in the polynomial \\( Q(x) = x^4 + dx^3 + e(...TRUNCATED) | [{"from":"human","value":"The polynomial $Q(x) = x^4 + dx^3 + ex^2 + fx + 12$ has the property that (...TRUNCATED) |
"406. Rationalize $y=\\sqrt{(a+b x)(c+d x)}$ by the substitution $y=(a+b x) z$.\n\nTranslate the abo(...TRUNCATED) | null | "Okay, let me try to figure out how to rationalize the equation y = sqrt((a + bx)(c + dx)) using the(...TRUNCATED) | "406. Rationalize \\( y = \\sqrt{(a + b x)(c + d x)} \\) by the substitution \\( y = (a + b x) z \\)(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let me try to figure out how to rationalize the equation y = sqrt((a + bx)(c + dx)) (...TRUNCATED) | [{"from":"human","value":"406. Rationalize $y=\\sqrt{(a+b x)(c+d x)}$ by the substitution $y=(a+b x)(...TRUNCATED) |
"Given the function $f(x)= \\frac{ \\sqrt{3}}{2}\\sin 2x- \\frac{1}{2}(\\cos ^{2}x-\\sin ^{2}x)-1 $,(...TRUNCATED) | null | "Okay, so I need to solve part (1) and part (2) of this problem. Let me start with part (1). \n\nFir(...TRUNCATED) | "Given the function \\( f(x) = \\frac{\\sqrt{3}}{2} \\sin 2x - \\frac{1}{2} (\\cos^2 x - \\sin^2 x) (...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I need to solve part (1) and part (2) of this problem. Let me start with part (1)(...TRUNCATED) | [{"from":"human","value":"Given the function $f(x)= \\frac{ \\sqrt{3}}{2}\\sin 2x- \\frac{1}{2}(\\co(...TRUNCATED) |
"A rectangular garden has an area that can be expressed as the product of three distinct prime numbe(...TRUNCATED) | "A rectangular garden has an area that can be expressed as the product of three distinct prime numbe(...TRUNCATED) | "Okay, so there's this rectangular garden with an area that's the product of three distinct prime nu(...TRUNCATED) | "The area of the rectangular garden is the product of three distinct prime numbers \\( p \\), \\( q (...TRUNCATED) | 29,910 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so there's this rectangular garden with an area that's the product of three distinct(...TRUNCATED) | [{"from":"human","value":"A rectangular garden has an area that can be expressed as the product of t(...TRUNCATED) |
If $x$ is real, compute the maximum integer value of
\[\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\] | If $x$ is real, compute the maximum integer value of
\[\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\] | "Okay, so I need to find the maximum integer value of the expression (3x² + 9x + 17)/(3x² + 9x + 7(...TRUNCATED) | "To find the maximum integer value of the expression \\(\\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}\\) whe(...TRUNCATED) | 20,185 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the maximum integer value of the expression (3x² + 9x + 17)/(3x²(...TRUNCATED) | [{"from":"human","value":"If $x$ is real, compute the maximum integer value of\n\\[\\frac{3x^2 + 9x (...TRUNCATED) |
If the polynomial $x^3 - 5x^2 + 7x - 12$ is divided by $(x-2)(x-3)$, what is the remainder? | If the polynomial $x^3 - 5x^2 + 7x - 12$ is divided by $(x-2)(x-3)$, what is the remainder? | "Okay, so I need to find the remainder when the polynomial \\( x^3 - 5x^2 + 7x - 12 \\) is divided b(...TRUNCATED) | "To find the remainder when the polynomial \\( x^3 - 5x^2 + 7x - 12 \\) is divided by \\( (x-2)(x-3)(...TRUNCATED) | 23,073 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the remainder when the polynomial \\( x^3 - 5x^2 + 7x - 12 \\) is (...TRUNCATED) | [{"from":"human","value":"If the polynomial $x^3 - 5x^2 + 7x - 12$ is divided by $(x-2)(x-3)$, what (...TRUNCATED) |
The monotonic increasing interval of the function $f(x) = \lg(\cos x)$ is __________. | null | "Okay, so I need to find the monotonic increasing interval of the function f(x) = lg(cos x). Hmm, le(...TRUNCATED) | "To find the monotonic increasing interval of the function \\( f(x) = \\lg(\\cos x) \\), we need to (...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I need to find the monotonic increasing interval of the function f(x) = lg(cos x)(...TRUNCATED) | [{"from":"human","value":"The monotonic increasing interval of the function $f(x) = \\lg(\\cos x)$ i(...TRUNCATED) |
"Annie went to a liquidation sale and bought 3 televisions of Brand A, 2 televisions of Brand B, 4 t(...TRUNCATED) | null | "Okay, let's see. Annie bought a bunch of stuff at a liquidation sale, right? Televisions of differe(...TRUNCATED) | "Annie bought 3 televisions of Brand A, 2 televisions of Brand B, 4 televisions of Brand C, and a se(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's see. Annie bought a bunch of stuff at a liquidation sale, right? Televisions o(...TRUNCATED) | [{"from":"human","value":"Annie went to a liquidation sale and bought 3 televisions of Brand A, 2 te(...TRUNCATED) |
"Given the exponential function $f(x)=(a-1)^x$ is monotonically decreasing on $\\mathbb{R}$, the ran(...TRUNCATED) | null | "Okay, let me try to figure out this problem. So, we have the function f(x) = (a - 1)^x, and we need(...TRUNCATED) | "To determine the range of the real number \\( a \\) such that the exponential function \\( f(x) = ((...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let me try to figure out this problem. So, we have the function f(x) = (a - 1)^x, an(...TRUNCATED) | [{"from":"human","value":"Given the exponential function $f(x)=(a-1)^x$ is monotonically decreasing (...TRUNCATED) |
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