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Consider the number
N = 123456789101112 … 9899100 ‾ , N = \overline{123456789101112\ldots 9899100}, N=123456789101112…9899100,
which is obtained from writing all the integers from 1 to 100 and removing the spaces that are in between. What are the first 3 digits of N 2 N^2 N2?
Evaluate
⌊ ( 23 + 17 ) 4 ⌋ . \lfloor \left( \sqrt{ 23 } + \sqrt{17 } \right)^4 \rfloor . ⌊(23 +17 )4⌋.
If the quartic x 4 + 9 x 3 + 11 x 2 + 36 x + A x^4 + 9x^3 + 11 x^2 + 36 x + A x4+9x3+11x2+36x+Ahas roots k , l , m k, l, m k,l,mand n n nsuch that k l = m n kl = mn kl=mn, determine A A A.
Integers N N Nand M M Msatisfy 0 ≤ M ≤ 21 ! 0 \leq M \leq 21! 0≤M≤21!and
∑ i = 1 21 ( i 2 + i + 1 ) i ! = N × 21 ! − M . \sum_{i=1}^{21} (i^2 + i + 1) i ! = N \times 21 ! - M. i=1∑21(i2+i+1)i!=N×21!−M.
What is N + M N + M N+M?
If x = 11 + 7 x = \sqrt{ 11 + \sqrt{ 7 } } x=11+7 and y = 11 − 7 y = \sqrt{ 11 - \sqrt{ 7 } } y=11−7 , evaluate
x 6 + y 6 . x^6 + y^6. x6+y6.
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