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- ...7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? <math>PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}</math>709 bytes (103 words) - 03:32, 6 December 2019
- &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|.971 bytes (150 words) - 21:56, 11 February 2020
- ...math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. import three; currentprojection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");4 KB (518 words) - 15:01, 31 December 2021
- ...{2}i</math> and <math>\text{cis\,} {8}\theta = -\frac{1}{2}- \frac{\sqrt{3}}{2}i</math> ...}}{2}i</math> and <math>\text{cis\,} {8}\theta = -\frac{1}{2}+\frac{\sqrt{3}}{2}i</math>2 KB (380 words) - 15:03, 22 July 2018
- ...Hence <math>\angle ADB = \angle DEC</math>, and <math>\triangle BDE</math> is [[isosceles triangle|isosceles]]. Then <math>BD = BE = 10</math>. The answer is <math>m+n = \boxed{069}</math>.4 KB (743 words) - 03:32, 23 January 2023
- ...has the midpoint triangle as a face. The [[volume]] of <math>P_{3}</math> is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...ath>\left(\frac 12\right)^3 = \frac 18</math>. The total volume added here is then <math>\Delta P_1 = 4 \cdot \frac 18 = \frac 12</math>.2 KB (380 words) - 00:28, 5 June 2020
- ...ty]] that Club Truncator will finish the season with more wins than losses is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...and losses. Thus, by the [[complement principle]], the desired probability is half the probability that Club Truncator does not have the same number of w3 KB (415 words) - 23:25, 20 February 2023
- ...0^{j-i} - 1</math>. From the factorization <math>10^6 - 1 = (10^3 + 1)(10^{3} - 1)</math>, we see that <math>j-i = 6</math> works; also, <math>a-b | a^n ...^{0},\dots\implies 94 - 6 = 88</math>, and so forth. Therefore, the answer is <math>94 + 88 + 82 + \dots + 4\implies 16\left(\dfrac{98}{2}\right) = \boxe4 KB (549 words) - 23:16, 19 January 2024
- ...[[probability]] of obtaining a grid that does not have a 2-by-2 red square is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are [[re ...easy: 4 ways to choose which the side the squares will be on, and <math>2^3</math> ways to color the rest of the squares, so 32 ways to do that. For th8 KB (1,207 words) - 20:04, 5 September 2023
- ...<math>x</math>, and that <math>f(x) = 1-|x-2|</math> for <math>1\le x \le 3</math>. Find the smallest <math>x</math> for which <math>f(x) = f(2001)</ma ...h>f(2001) = 3^kf\left(\frac{2001}{3^k}\right),\ 1 \le \frac{2001}{3^k} \le 3 \Longrightarrow k = 6</math>. Indeed,3 KB (545 words) - 23:41, 14 June 2023
- ...h> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>? ...> and <math>b</math> are the legs of the right triangle and <math>c</math> is the hypotenuse. (This formula should be used ''only for right triangles''.)7 KB (1,112 words) - 02:15, 26 December 2022
- x_{3}&=420,\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}2 KB (300 words) - 01:28, 12 November 2022
- ...5</math> percent of the school population, and the number who study French is between <math>30</math> percent and <math>40</math> percent. Let <math>m</m Therefore, the answer is <math>M - m = 499 - 201 = \boxed{298}</math>.2 KB (252 words) - 00:54, 10 January 2024
- ...ac{1 - 0}{\sin 1} = \frac{1}{\sin 1^{\circ}}</math>. Therefore, the answer is <math>\boxed{001}</math>. The average term is around the 60's which gives <math>\frac{4}{3}</math>.3 KB (469 words) - 21:14, 7 July 2022
- ..., and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+20 ...k+1)\cdot k!- k!)} = 1+\sum_{k=1}^{n-1} {((k+1)!- k!)} = 1 + ((2! - 1!) + (3! - 2!) + \cdots + (n! - (n-1)!)) = n!</math>.7 KB (1,131 words) - 14:49, 6 April 2023
- ...2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n}}r</math>, where <math>m</math>, <math>n</math> and < 2000x^6+100x^5+10x^3+x-2&=0\\6 KB (1,060 words) - 17:36, 26 April 2024
- ...the absolute values of all possible slopes for <math>\overline{AB}</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively pr ...ction of <math>D</math> across that perpendicular. Then <math>ABCD'</math> is a [[parallelogram]], and <math>\overrightarrow{AB} = \overrightarrow{D'C}</4 KB (750 words) - 22:55, 5 February 2024
- A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tangent]] to <math> == Solution 3 (Smart algebra to make 2 less annoying) ==2 KB (399 words) - 17:37, 2 January 2024
- ...ch that <math>z+\frac 1z=2\cos 3^\circ</math>, find the least integer that is greater than <math>z^{2000}+\frac 1{z^{2000}}</math>. ...ac{2\cos 3 \pm \sqrt{4\cos^2 3 - 4}}{2} = \cos 3 \pm i\sin 3 = \text{cis}\,3^{\circ}</math>.4 KB (675 words) - 13:42, 4 April 2024
- In [[trapezoid]] <math>ABCD</math>, leg <math>\overline{BC}</math> is [[perpendicular]] to bases <math>\overline{AB}</math> and <math>\overline{C ...<math>\overline{CD}</math>. Then <math>AE = x</math>, and <math>ADE</math> is a [[right triangle]]. By the [[Pythagorean Theorem]],4 KB (584 words) - 19:35, 7 December 2019
