conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Let $f(x)=x^4+ax^3+bx^2+cx+d$ be a quartic polynomial with real coefficients and leading coefficient $1$.\nSuppose that for some real number $k,$\nthe equation $f(x)=k$ has exactly $3$ distinct real roots for any value of $k.$\nFind the maximum value of $b.$"
},
{
"from"... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be real numbers such that $0 < a < \\frac{\\pi}{2}$ and $0 < b < \\frac{\\pi}{2}.$ Find the smallest possible value of $a + b$ such that the function\n\\[f(x) = \\cos (x - a) + \\cos (x - b)\\]achieves its maximum value."
},
{
"from": "gpt",
"value":... |
[
{
"from": "human",
"value": "A certain positive integer has the property that if any one of its digits is changed, the resulting number is always divisible by 3. What is the smallest number that has this property?"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the smallest positi... |
[{"from":"human","value":"Find the function $g(x)$ defined on all positive real numbers $x$ that sat(...TRUNCATED) |
[{"from":"human","value":"The Cybermen have again outwitted the Daleks! Unfortunately, this time the(...TRUNCATED) |
[{"from":"human","value":"Potassium ferrocyanide is diamagnetic (no unpaired electron) while potassi(...TRUNCATED) |
[{"from":"human","value":"Express $5\\sqrt{6}+12\\sqrt{2}$ in the form $\\sqrt{a}+\\sqrt{b}+\\sqrt{c(...TRUNCATED) |
[{"from":"human","value":"Warning: this is a complex challenge.\nOverview\nA lightbox, in the contex(...TRUNCATED) |
[{"from":"human","value":"While Grisha was celebrating New Year with Ded Moroz, Misha gifted Sasha a(...TRUNCATED) |
[{"from":"human","value":"A jar contains a gas and a few drops of water at T K. The pressure in the (...TRUNCATED) |
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