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- What is the largest positive integer that is not the sum of a positive integral multiple of <math>42</math> and a positi ...h>\mod {42}</math> must be a [[composite]] number. Also, every number that is a multiple of <math>42</math> greater than that prime number must also be p3 KB (436 words) - 19:26, 2 September 2023
- ...on <math>\overline{AM}</math> with <math>AD=10</math> and <math>\angle BDC=3\angle BAC.</math> Then the perimeter of <math>\triangle ABC</math> may be ...defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3;7 KB (1,181 words) - 13:47, 3 February 2023
- ...tory positive integers for all integers <math>y \le 100</math>. The answer is ...1}^{99} \left\lfloor\frac{100-y}{y(y+1)} \right\rfloor = 49 + 16 + 8 + 4 + 3 + 2 + 1 + 1 + 1 = \boxed{085}.</cmath>4 KB (646 words) - 17:37, 1 January 2024
- ...s t - \sin t - \cos t + 1 = \frac{13}{4} - \sqrt{10}</math>, so the answer is <math>13 + 4 + 10 = \boxed{027}</math>. ...ath>. Therefore, <math>x = \frac{13}{4} - \sqrt{10}</math>, and the answer is <math> 13 + 4 + 10 = \boxed{027}</math>.3 KB (427 words) - 09:23, 13 December 2023
- Let <math>n=2^{31}3^{19}.</math> How many positive [[integer]] [[divisor]]s of <math>n^2</math ...h>) into pairs that multiply to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} =2 KB (407 words) - 08:14, 4 November 2022
- ...ese roots is <math>13+i</math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math> ...pairs. Let the first two roots be <math>m,n</math>. Since <math>m+n</math> is not real, <math>m,n</math> are not conjugates, so the other pair of roots m3 KB (451 words) - 15:02, 6 September 2021
- ...adius <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the ...i(acos(1/3)), F=B+3*expi(acos(1/3)), P=IP(F--F+3*(D-F),CR(A,9)), Q=IP(F--F+3*(F-D),CR(A,9));3 KB (605 words) - 11:30, 5 May 2024
- ...coordinate plane]] via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let <math>p</math> be t ...reach <math>(2,2)</math>, so the number of steps the object may have taken is either <math>4</math> or <math>6</math>.3 KB (602 words) - 23:15, 16 June 2019
- ...2}</math>. Thus, the product of the two roots (both of which are positive) is <math>1995^{1+\sqrt{2}/2} \cdot 1995^{1 - \sqrt{2}/2} = 1995^2</math>, maki ...od{1000}\equiv (-5)^2\pmod{1000}\equiv 25\pmod{1000}</math>, so our answer is <math>\boxed{025}</math>.2 KB (362 words) - 00:40, 29 January 2021
- ...</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>m/n,</math> where <m The sum of the areas of the [[square]]s if they were not interconnected is a [[geometric sequence]]:2 KB (302 words) - 19:29, 4 July 2013
- ...each twice as large as angle <math>DBA</math>, and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find <math>\lfloor pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D);5 KB (710 words) - 21:04, 14 September 2020
- A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is made by gluing together <math>1\times 1\times 1</math> cubes. An internal [ ...point on the diagonal with coordinates <math>(ma, mb, mc)</math>. We have 3 key observations as this point moves from <math>(0,0,0)</math> towards <mat5 KB (923 words) - 21:21, 22 September 2023
- .../math> [[bisect]]s <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio <math>\frac{[ADB]}{[ABC]}</math> can be written in ...he problem asks for a ratio, we can divide each side length by <math>\sqrt{3}</math> to make things simpler. We now have a triangle with sides <math>\sq3 KB (521 words) - 01:18, 25 February 2016
- ...utation]] <math>a_1,a_2,a_3,\cdots,a_{10}</math> of the integers <math>1,2,3,\cdots,10</math>, form the sum + |2 - 10| + \ldots + |2 - 3| + |2 - 1|\\5 KB (879 words) - 11:23, 5 September 2021
- ...math>\mathrm {P}</math> be the product of the [[root]]s of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive [[imaginary]] part, and suppose that <m 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\6 KB (1,022 words) - 20:23, 17 April 2021
- ...ent|tangent]] function is <math>180^\circ</math>, and the tangent function is [[one-to-one]] over each period of its domain. Therefore, the smallest positive solution is <math>x = \boxed{159}</math>.4 KB (503 words) - 15:46, 3 August 2022
- ...s wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens? ...>, leaving lockers <math>86, 342, 598</math>, and <math>854</math>, and he is at where he started again. He then opens <math>86</math> and <math>598</mat3 KB (525 words) - 23:51, 6 September 2023
- ...rdered pairs of positive integers <math>(x,y)</math> with <math>x<y</math> is the harmonic mean of <math>x</math> and <math>y</math> equal to <math>6^{20 ...h>x<y</math>, the answer is half of the remaining number of factors, which is <math>\frac{1599-1}{2}= \boxed{799}</math>.1 KB (155 words) - 19:32, 4 July 2013
- 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
