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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"
},
{
... | -0.012939 |
[
{
"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"
},
{
... | 0.08121 |
[
{
"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"
},
{
... | 0.001653 |
[
{
"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"
},
{
... | 0.16407 |
[
{
"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"
},
{
... | -0.187292 |
[
{
"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"
},
{
... | -0.045301 |
[
{
"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"
},
{
... | -0.468384 |
[
{
"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"
},
{
... | -0.197009 |
[
{
"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... | -1.194231 |
[
{
"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... | -1.834622 |
[
{
"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... | -1.299485 |
[
{
"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... | 1.489455 |
[
{
"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... | 1.304922 |
[
{
"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... | -1.454376 |
[
{
"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... | 1.723418 |
[
{
"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... | 1.570537 |
[
{
"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}.\\... | 0.765563 |
[
{
"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}.\\... | 0.163004 |
[
{
"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}.\\... | -0.460516 |
[
{
"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}.\\... | -1.056723 |
[
{
"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}.\\... | -0.017192 |
[
{
"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}.\\... | -0.039395 |
[
{
"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}.\\... | 0.213468 |
[
{
"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}.\\... | 2.610188 |
[
{
"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 $$(x+r)^... | -0.255209 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | 0.215557 |
[
{
"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 $$(x+r)^... | -0.259884 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.013864 |
[
{
"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 $$(x+r)^... | -0.731332 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.123659 |
[
{
"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 $$(x+r)^... | 0.306162 |
[
{
"content": "You are a helpful assistant. You should think step-by-step and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^... | -0.188677 |
[
{
"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... | -0.848544 |
[
{
"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... | -0.247275 |
[
{
"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... | 0.021877 |
[
{
"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... | 1.643208 |
[
{
"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... | 0.131356 |
[
{
"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... | 2.072895 |
[
{
"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... | -1.014321 |
[
{
"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... | -0.370896 |
[
{
"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... | 0.084039 |
[
{
"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... | 0.156977 |
[
{
"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... | -0.067548 |
[
{
"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... | 0.096053 |
[
{
"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... | -0.364832 |
[
{
"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... | -0.164083 |
[
{
"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... | 0.086028 |
[
{
"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... | 0.568965 |
[
{
"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... | -0.165002 |
[
{
"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... | -0.30667 |
[
{
"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... | 0.163911 |
[
{
"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... | 0.09548 |
[
{
"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... | -0.090344 |
[
{
"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... | -0.234703 |
[
{
"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... | 0.013708 |
[
{
"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... | -0.02207 |
[
{
"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... | 0.483572 |
[
{
"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... | -0.297085 |
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