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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"
},
{
... | 0.05964 |
[
{
"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"
},
{
... | 0.14629 |
[
{
"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"
},
{
... | 0.073405 |
[
{
"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"
},
{
... | 0.218576 |
[
{
"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"
},
{
... | -0.115551 |
[
{
"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"
},
{
... | 0.028647 |
[
{
"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"
},
{
... | -0.450453 |
[
{
"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"
},
{
... | -0.125951 |
[
{
"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... | -1.173981 |
[
{
"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... | -1.911342 |
[
{
"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... | -1.272697 |
[
{
"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... | 1.624772 |
[
{
"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... | 1.424211 |
[
{
"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... | -1.431066 |
[
{
"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... | 1.845374 |
[
{
"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... | 1.704992 |
[
{
"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}.\\... | 0.498187 |
[
{
"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}.\\... | 0.123725 |
[
{
"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}.\\... | -0.399289 |
[
{
"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}.\\... | -1.167411 |
[
{
"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}.\\... | -0.00946 |
[
{
"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}.\\... | -0.026729 |
[
{
"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}.\\... | 0.158977 |
[
{
"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}.\\... | 2.766731 |
[
{
"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 $$(x+r)^... | -0.189815 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.261781 |
[
{
"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 $$(x+r)^... | -0.195067 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.058763 |
[
{
"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 $$(x+r)^... | -0.851555 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.049191 |
[
{
"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 $$(x+r)^... | 0.334884 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.11703 |
[
{
"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... | -0.933032 |
[
{
"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... | -0.179501 |
[
{
"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... | 0.063886 |
[
{
"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... | 1.025054 |
[
{
"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... | 0.152492 |
[
{
"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... | 1.201609 |
[
{
"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... | -1.231917 |
[
{
"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... | -0.306242 |
[
{
"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... | 0.165544 |
[
{
"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... | 0.236743 |
[
{
"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... | 0.005649 |
[
{
"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... | 0.17751 |
[
{
"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... | -0.369011 |
[
{
"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... | -0.105961 |
[
{
"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... | 0.167532 |
[
{
"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... | 0.585116 |
[
{
"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... | -0.091967 |
[
{
"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... | -0.24867 |
[
{
"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... | 0.218441 |
[
{
"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... | 0.158993 |
[
{
"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... | -0.015597 |
[
{
"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... | -0.166998 |
[
{
"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... | 0.084683 |
[
{
"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... | 0.050961 |
[
{
"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... | 0.468367 |
[
{
"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... | -0.237529 |
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