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- (which is well-defined by this formula for <math>\Re s>0</math>) admits an to some [[open set | open]] domain <math>E</math> containing the closed half-plane6 KB (1,034 words) - 07:55, 12 August 2019
- ...e case <math> b=a^2 </math>, note that <math> 44^2=1936 </math> and <math> 45^2=2025 </math>. Thus, all values of <math>a</math> from <math>2</math> to < ...re <math> 44-2+1=43 </math> possibilities for the square case and <math> 12-2+1=11 </math> possibilities for the cube case. Thus, the answer is <math> 43 KB (547 words) - 19:15, 4 April 2024
- ...math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math> m. </math> ...> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </math> Given that <math> BF=p+q\sqrt{r},7 KB (1,119 words) - 21:12, 28 February 2020
- ...> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </math> Given that <math> BF=p+q\sqrt{r}, ...0)); pair A=(0,9), B=(9,9), C=(9,0), D=(0,0), E=(2.5-0.5*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), G=(4.5,9), O=(4.5,4.5); draw(A--B--C--D--A);draw(E--O--F);dr13 KB (2,080 words) - 21:20, 11 December 2022
- ...math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>. T4 KB (686 words) - 12:52, 13 June 2024
- ...semicircle is tangent to only one side of the square, we will have "wiggle-room" to increase its size. Once it is tangent to two adjacent sides of the We can just look at a quarter circle inscribed in a <math>45-45-90</math> right triangle. We can then extend a radius, <math>r</math> to one4 KB (707 words) - 11:11, 16 September 2021
- ...05 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math> It is well-known that <math>\tau(n)</math> is odd if and only if <math>n</math> is a [[4 KB (647 words) - 02:29, 4 May 2021
- <cmath>s_{82, 9} = 2s_{82, 8} = 4s_{82, 7} = 8s_{127 - 82, 6} = 8s_{45, 6}</cmath> <cmath>s_{45, 6} = 2s_{63 - 45, 5} + 1 = 2s_{18, 5} + 1 = 4s_{31 - 18, 4} + 1 = 4s_{13, 4} + 1</cmath>6 KB (899 words) - 20:58, 12 May 2022
- ...CR(E, 45/7)), A=D+ (5+(75/7))/(75/7) * (F-D), C = E+ (3+(45/7))/(45/7) * (F-E), B=IP(CR(A,3), CR(C,5)); == Additional Trigonometry-Free Alternative ==3 KB (486 words) - 22:15, 7 April 2023
- Let <math>f(x)=|x-p|+|x-15|+|x-p-15|</math>, where <math>0 < p < 15</math>. Determine the [[minimum]] value t ...the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>?7 KB (1,104 words) - 03:13, 27 May 2024
- Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>. <center><math>\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0</math></center>6 KB (870 words) - 10:14, 19 June 2021
- ...h> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</math> ...h>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d=0</math> has four non-real roots. The product of two of these roots is <math>13+i</math> and the6 KB (1,000 words) - 00:25, 27 March 2024
- Consider the parallelogram with vertices <math>(10,45),</math> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</ Find the sum of all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.7 KB (1,094 words) - 13:39, 16 August 2020
- ...is, and let <math>E</math> be the reflection of <math>D</math> across the y-axis. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <mat The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle ar7 KB (1,204 words) - 03:40, 4 January 2023
- Find the sum of all positive two-digit integers that are divisible by each of their digits. ...the roots, real and non-real, of the equation <math>x^{2001}+\left(\frac 12-x\right)^{2001}=0</math>, given that there are no multiple roots.7 KB (1,212 words) - 22:16, 17 December 2023
- ...ht distinguishable rings, let <math>n</math> be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order The equation <math>2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n6 KB (947 words) - 21:11, 19 February 2019
- ...th>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? ...to the axis of the cylinder, and the plane of the second cut forms a <math>45^\circ</math> angle with the plane of the first cut. The intersection of the7 KB (1,127 words) - 09:02, 11 July 2023
- ...] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? ...The second root is extraneous since <math>2\sqrt{y+15}</math> is always non-negative (and moreover, plugging in <math>y=-6</math>, we get <math>-6=6</ma3 KB (532 words) - 05:18, 21 July 2022
- A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of A=r*dir(45),B=(A.x,A.y-r);11 KB (1,741 words) - 22:40, 23 November 2023
- .../math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answer is <math>\boxed{038}</math>. ...could possibly work by Chicken McNugget is <math>9 \cdot 25 - 9 - 25 = 225-34 = 191</math>. We then bash from top to bottom:8 KB (1,346 words) - 01:16, 9 January 2024
