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- ...s in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from <math>-12</math> to <math>0</math>, when written in modulo 5, are This just relates each integer to its [[remainder]] from the [[Division Theorem]]. While this may not seem all that useful at first16 KB (2,406 words) - 07:56, 10 July 2024
- #redirect [[AoPS Wiki:What AoPSWiki is not]] {{R from shortcut}}64 bytes (10 words) - 16:30, 29 July 2020
- #redirect [[AoPS Wiki:Tutorial]] {{R from shortcut}}52 bytes (7 words) - 14:40, 8 December 2020
- #redirect [[AoPS Wiki:Sandbox]] {{R from shortcut}}51 bytes (7 words) - 06:44, 15 July 2020
- #redirect [[AoPSWiki:Community portal]] {{R from shortcut}}59 bytes (7 words) - 22:22, 11 July 2011
- #redirect [[AoPSWiki:Community portal]] {{R from shortcut}}59 bytes (7 words) - 22:17, 11 July 2011
- #redirect [[AoPS_Wiki:Allowlisting]] {{R from shortcut}}56 bytes (7 words) - 17:07, 16 January 2024
- ...e diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly <math>12</math> feet b On the way home from their sixth game, Michael notes to his father that the attendance seems to71 KB (11,749 words) - 11:39, 20 November 2024
- <cmath>R=(b+1)(b^2+2a+1)</cmath> In order for <math>h(z)=z</math>, we must have <math>R=0</math>, <math>Q=0</math>, and <math>P=S</math>.3 KB (519 words) - 14:49, 5 November 2023
- Indeed, we know from the [[Law of Sines]] that Denote this ratio by <math>r</math>; we have <math>\sin A = ra, \sin B = rb, \sin C = rc</math>.11 KB (1,832 words) - 18:24, 10 November 2024
- where <math>-p\in\mathbb{R}</math> is the fourth root of <math>f(x)</math>. (Using <math>(x+p) = (x-r))</math> instead of <math>(x-r)</math> makes the following computations less messy.)10 KB (1,726 words) - 19:55, 11 October 2024
- .../math>. Therefore, the [[answer]] is <cmath>\frac{r^2 \sqrt{3} - \frac{\pi r^2}3}{\frac{3r^2 \sqrt{3}}4} = \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt === Multiple Choice Shortcut ===6 KB (1,007 words) - 00:47, 1 November 2024
- ...qrt{74}</math>. However, by Power of a Point, <math>(EC)(EC + CA) = EO^2 - R^2 = (2+3)^2 + 5^2 - 2^2 = 25 + 25 - 4 = 46 \implies EC = \frac{46}{\sqrt{74 == Solution 2b: Area shortcut ==7 KB (986 words) - 20:44, 30 September 2024
- ...ath>, <math>20</math>, and <math>24</math>. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have ...which we will call <math>O</math>. Note that <math>O</math> is equidistant from each of <math>A</math>, <math>B</math>, and <math>C</math>. Then,7 KB (1,177 words) - 14:55, 5 January 2024
- ...umber of solutions will be <math>\tbinom{6}{x} \cdot \tbinom{6}{y}</math>. From here we continue our casework, and our subcases will be coming out quickly ...ng the corresponding values of <math>x, y</math> of each subcase in Case 3 from 6 ( For example, the subcase 0, 6 in Case 5 corresponds to the subcase 6, 026 KB (4,044 words) - 12:58, 24 January 2024
- ...raws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four ite ==Solution 1 (The Shortcut Passed Every Barrier)==12 KB (2,082 words) - 18:01, 27 November 2024
- real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q;9 KB (1,503 words) - 14:09, 1 August 2023