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- pathpen = black; pair O = (3.5,3.5); D(O); fill(shift(4,3)*unitsquare,rgb(1,1,.4));fill(shift(4,5)*unitsquare,rgb(1,1,.4));4 KB (551 words) - 11:44, 26 June 2020
- The probability that one team wins all games is <math>5\cdot \left(\frac{1}{2}\right)^4=\frac{5}{16}</math>. Similarity, the probability that one team loses all games is <math>\frac{5}{16}</math>.3 KB (461 words) - 00:33, 16 May 2024
- ...</math>, <math>b</math>, and <math>c</math>, and that the roots of <math>x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</m ...^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11</math>. Then3 KB (585 words) - 22:08, 19 November 2022
- .... The area of the shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed <math (Figure not to scale) The area of the square shadow base is <math>48 + 1 = 49</math>, and so the sides of the shadow are <math>7</math>2 KB (257 words) - 17:50, 4 January 2016
- ...square after <math>1996</math> is <math>2025 = 45^2</math>, so our answer is <math>45 - 1 = \boxed{044}</math>. ...</math> terms. Therefore, we need to find the smallest perfect square that is greater than <math>1996</math>. From trial and error, we get <math>44^2 = 13 KB (515 words) - 04:29, 27 November 2023
- ...that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer? ...4</math> for the third, and <math>256</math> for the fourth, so the answer is <math>4+16+64+256=\boxed{340}</math>.1 KB (163 words) - 19:31, 4 July 2013
- ...c [[square]], the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Fin \multicolumn{3}{c}{\text{Table}}\\\hline2 KB (332 words) - 11:28, 4 August 2021
- ...ath>q</math>, and <math>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...\left(\frac {11}{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.4 KB (609 words) - 22:49, 17 July 2023
- ...t <math>\frac{m}{n}</math> be the [[probability]] that <math>\sqrt{2+\sqrt{3}}\le\left|v+w\right|</math>, where <math>m</math> and <math>n</math> are [[ ...rt{3}</math>, which simplifies to <cmath>\cos((m-n)\theta) \ge \frac{\sqrt{3}}{2}</cmath>Thus, <cmath>|m - n| \le \frac{\pi}{6} \cdot \frac{1997}{2 \pi}5 KB (875 words) - 01:25, 19 June 2024
- ...<math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>. | <math>f(3) = 0</math> || <math>f(1.1) = 0.1</math>7 KB (1,225 words) - 19:56, 4 August 2021
- ...r all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. The only value that is not in the range of this function is <math>\frac {a}{c}</math>. To find <math>\frac {a}{c}</math>, we use the tw11 KB (2,063 words) - 22:59, 21 October 2023
- ...{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}</math>. What is the greatest integer that does not exceed <math>100x</math>? We want to pair up <math>[1, 44]</math>, <math>[2, 43]</math>, <math>[3, 42]</math>, etc. from the numerator and <math>[46, 89]</math>, <math>[47,10 KB (1,512 words) - 17:16, 18 June 2024
- ...pe-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true: *'''Case 1''': All three attributes are the same. This is impossible since sets contain distinct cards.3 KB (585 words) - 19:37, 25 April 2022
- ...ve, <math>\langle a^{-1}\rangle=\langle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>. ...+\sqrt{5}}2</math> (the [[phi|golden ratio]]) is the answer. The following is the way to derive that:4 KB (586 words) - 21:53, 30 December 2023
- ...of the entries in each row is 0 and the sum of the entries in each column is 0? The problem is asking us for all configurations of <math>4\times 4</math> grids with 2 1's4 KB (638 words) - 16:41, 22 January 2024
- ...</math> mile per minute. At time <math>t=0</math>, the center of the storm is <math>110</math> miles due north of the car. At time <math>t=t_1</math> min ...ar is at <math>\left(\frac 23t,0\right)</math> and the center of the storm is at <math>\left(\frac{t}{2}, 110 - \frac{t}{2}\right)</math>. Using the dist4 KB (617 words) - 18:47, 17 July 2022
- ...A_n</math>, and <math>A_1A_2B</math> is an [[equilateral triangle]]. What is the largest value of <math>n</math> for which <math>A_1</math>, <math>A_n</ Clearly <math>n</math> is maximized when <math>m = 7, n = \boxed{042}</math>.3 KB (497 words) - 00:39, 22 December 2018
- ...<math>5, x, 5+r</math> and <math>5, 8+r+x, 13</math>, wher <math>x</math> is the distance between the center of the circle in question and the segment c NOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been2 KB (354 words) - 22:33, 2 February 2021
- ...number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? ...and <math>y=112</math>, which satisifies the conditions. Hence the answer is <math>112 + 14 = \boxed{126}</math>.2 KB (375 words) - 19:34, 4 August 2021
- ...is pattern can be easily generalized and we see that the number of squares is just <math>\sum^8_{i=1}{i^2}</math>. This can be simplified by using the we ...{i=1}{i}}</math>. This gets us <math>{(\frac{9\cdot8}{2})}^2,</math> which is just <math>1296.</math>3 KB (416 words) - 21:09, 27 October 2022
