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{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | -0.013193 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | 0.07827 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | 0.000983 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | 0.158766 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | -0.182571 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | -0.044631 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | -0.455644 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $n$ such that\n\\[(\\cos 84^\\circ + i \\sin 84^\\circ)^n\\]is a real number.",
"role": "user"
},
{
... | -0.192011 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | -1.160782 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | -1.782902 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | -1.263033 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | 1.446337 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | 1.267069 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | -1.413504 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | 1.673625 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars?",
"role": "user"
},
{
"content": "Alright, I... | 1.525106 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | 0.743098 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | 0.15773 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | -0.448 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | -1.027197 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | -0.017324 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | -0.038894 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | 0.206755 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two non-zero vectors such that\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = -3 \\mathbf{b}.\\... | 2.535094 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.24855 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.208784 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.253093 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.014092 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.711089 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.120754 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.296804 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.183917 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | -0.824958 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | -0.240843 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | 0.02063 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | 1.595704 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | 0.126985 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | 2.013131 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | -0.986004 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let us have a triangle $\\triangle ABC$ and a point $D$ on $BC$ such that $BD = DC = DA.$ If $\\angle ABC = 50^\\circ,$ then how many degrees are i... | -0.360937 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | 0.081018 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | 0.151875 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | -0.066244 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | 0.09269 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | -0.355046 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | -0.160024 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | 0.08295 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\Join$ is an operation defined by $x \\Join y = (x+2)(y-3)$. What is $((t) \\Join (t+2)) - ((t+1) \\Join (t+1))$?",
"role": "use... | 0.552109 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | -0.160917 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | -0.298544 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | 0.158612 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | 0.092133 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | -0.08839 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | -0.22863 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | 0.012694 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $s$ is an integer and the root(s) of the quadratic expression $\\frac{1}{2}x^2+sx-\\frac{1}{2}$ are integers, find the sum of all possible value... | -0.022063 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is th... | 0.469152 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is th... | -0.289232 |
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