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[ { "meta": {}, "loss_mask": 1, "name": "", "content": "Find a closed-form for $\\sum_{k\\ge0}\\binom nk\\binom{n-k}{\\left\\lfloor\\frac{m-k}2\\right\\rfloor}(-2)^k$ as a function of $n$ and $m$.", "role": "user" }, { "meta": {}, "loss_mask": 1, "name": "", "content": "The sum...
[ { "meta": {}, "loss_mask": 1, "name": "", "content": "Given the operation * defined by \\( a * (b * c) = (a * b)c \\), solve the equation \\( 2016 * (6 * x) = 100 \\).", "role": "user" }, { "meta": {}, "loss_mask": 1, "name": "", "content": "The operation \\(*\\) satisfies \\...
[ { "meta": {}, "loss_mask": 1, "name": "", "content": "A single-mode field is prepared in a number state \\(|n\\rangle\\) with precisely \\(n\\) photons. Calculate the standard deviation (uncertainty) of the field operator \\(\\hat{E} = E_0(\\hat{a} + \\hat{a}^\\dagger) \\sin kz\\) and interpret the ...
[ { "meta": {}, "loss_mask": 1, "name": "", "content": "Find the greatest common divisor of 9,009 and 14,014.", "role": "user" }, { "meta": {}, "loss_mask": 1, "name": "", "content": "The greatest common divisor of 9,009 and 14,014 is \\(\\boxed{1001}\\).", "role": "assista...
[{"meta":{},"loss_mask":1,"name":"","content":"Find the number of ordered tuples of natural numbers (...TRUNCATED)
[{"meta":{},"loss_mask":1,"name":"","content":"A black hole with mass $ M = 10 M_\\odot $ emits Hawk(...TRUNCATED)
[{"meta":{},"loss_mask":1,"name":"","content":"Let $A$, $B$, $C$, and $D$ be four distinct points in(...TRUNCATED)
[{"meta":{},"loss_mask":1,"name":"","content":"Find the general solution of the differential equatio(...TRUNCATED)
[{"meta":{},"loss_mask":1,"name":"","content":"How many solutions are there to the equation \\(x^2 =(...TRUNCATED)
[{"meta":{},"loss_mask":1,"name":"","content":"Find a positive constant \\( c \\) such that the norm(...TRUNCATED)
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