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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"
},
{
... | 0.090742 |
[
{
"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"
},
{
... | 0.149143 |
[
{
"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"
},
{
... | 0.10002 |
[
{
"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"
},
{
... | 0.197863 |
[
{
"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"
},
{
... | -0.027334 |
[
{
"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"
},
{
... | 0.069854 |
[
{
"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"
},
{
... | -0.253053 |
[
{
"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"
},
{
... | -0.034343 |
[
{
"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... | -0.740701 |
[
{
"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... | -1.237672 |
[
{
"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... | -0.807234 |
[
{
"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... | 1.145619 |
[
{
"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... | 1.010445 |
[
{
"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... | -0.913973 |
[
{
"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... | 1.294302 |
[
{
"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... | 1.199687 |
[
{
"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}.\\... | 0.386317 |
[
{
"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}.\\... | 0.133935 |
[
{
"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}.\\... | -0.218569 |
[
{
"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}.\\... | -0.736273 |
[
{
"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}.\\... | 0.04417 |
[
{
"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}.\\... | 0.032531 |
[
{
"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}.\\... | 0.157694 |
[
{
"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}.\\... | 1.915284 |
[
{
"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 $$(x+r)^... | -0.077387 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.226982 |
[
{
"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 $$(x+r)^... | -0.080926 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.090151 |
[
{
"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 $$(x+r)^... | -0.52339 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.017392 |
[
{
"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 $$(x+r)^... | 0.276253 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.028331 |
[
{
"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... | -0.578304 |
[
{
"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... | -0.070435 |
[
{
"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... | 0.093604 |
[
{
"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... | 0.741418 |
[
{
"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... | 0.153324 |
[
{
"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... | 0.860414 |
[
{
"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... | -0.779749 |
[
{
"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... | -0.155857 |
[
{
"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... | 0.16212 |
[
{
"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... | 0.210108 |
[
{
"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... | 0.054353 |
[
{
"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... | 0.170185 |
[
{
"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... | -0.198163 |
[
{
"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... | -0.02087 |
[
{
"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... | 0.16346 |
[
{
"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... | 0.444906 |
[
{
"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... | -0.011438 |
[
{
"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... | -0.117054 |
[
{
"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... | 0.197772 |
[
{
"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... | 0.157705 |
[
{
"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... | 0.040034 |
[
{
"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... | -0.062009 |
[
{
"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... | 0.107621 |
[
{
"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... | 0.084893 |
[
{
"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... | 0.366219 |
[
{
"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... | -0.109546 |
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