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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"
},
{
... | 0.094987 |
[
{
"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"
},
{
... | 0.160839 |
[
{
"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"
},
{
... | 0.105193 |
[
{
"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"
},
{
... | 0.218796 |
[
{
"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"
},
{
... | -0.026964 |
[
{
"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"
},
{
... | 0.072351 |
[
{
"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"
},
{
... | -0.223573 |
[
{
"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"
},
{
... | -0.03376 |
[
{
"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... | -0.731265 |
[
{
"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... | -1.179185 |
[
{
"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... | -0.804885 |
[
{
"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... | 1.145833 |
[
{
"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... | 1.016762 |
[
{
"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... | -0.913223 |
[
{
"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... | 1.309478 |
[
{
"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... | 1.202546 |
[
{
"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}.\\... | 0.639509 |
[
{
"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}.\\... | 0.21805 |
[
{
"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}.\\... | -0.21807 |
[
{
"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}.\\... | -0.635086 |
[
{
"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}.\\... | 0.092012 |
[
{
"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}.\\... | 0.076483 |
[
{
"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}.\\... | 0.253347 |
[
{
"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}.\\... | 1.929728 |
[
{
"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 $$(x+r)^... | -0.074468 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.254808 |
[
{
"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 $$(x+r)^... | -0.077739 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.09434 |
[
{
"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 $$(x+r)^... | -0.407491 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.017544 |
[
{
"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 $$(x+r)^... | 0.318182 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.027933 |
[
{
"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... | -0.489475 |
[
{
"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... | -0.068919 |
[
{
"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... | 0.119339 |
[
{
"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... | 1.253376 |
[
{
"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... | 0.195913 |
[
{
"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... | 1.55392 |
[
{
"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... | -0.605427 |
[
{
"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... | -0.155385 |
[
{
"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... | 0.162818 |
[
{
"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... | 0.213834 |
[
{
"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... | 0.056791 |
[
{
"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... | 0.171221 |
[
{
"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... | -0.151144 |
[
{
"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... | -0.01073 |
[
{
"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... | 0.164209 |
[
{
"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... | 0.501998 |
[
{
"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... | -0.011373 |
[
{
"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... | -0.110463 |
[
{
"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... | 0.218684 |
[
{
"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... | 0.170821 |
[
{
"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... | 0.040846 |
[
{
"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... | -0.060125 |
[
{
"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... | 0.113625 |
[
{
"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... | 0.0886 |
[
{
"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... | 0.442271 |
[
{
"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... | -0.103759 |
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