for (int i = 0; i < 7; ++i) {
for (int j = 0; j < i; ++j) {
...
22 KB (3,378 words) - 20:29, 9 November 2025
<cmath>\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1</cmath>
<cmath>\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1</cmath>
...
9 KB (1,546 words) - 17:33, 27 October 2025
...mega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},</math> where <math>i = \sqrt{-1}.</math> Find the value of the product<cmath>\prod_{k=0}^6 \left
Man, What can I say? Mamba never out!
...
10 KB (1,507 words) - 19:46, 15 August 2025
...lems already on this page to contribute, please put it below. If you would like, put your user name to the list of contributors of this page (at the near-e
These are problems designed to look like and prepare you for problems on the AMC 8, 10, and 12.
...
76 KB (12,720 words) - 20:43, 1 November 2025
...circle, and line <math>\ell</math> can be written as <math>a \sqrt{b} - c \pi</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positi
...is <imath>24\pi</imath>, so the area of the region is <imath>48\sqrt{3}-24\pi</imath>
...
26 KB (3,978 words) - 00:32, 11 November 2025
-1. GMAAS looks like this when he is mad: https://cdn.artofproblemsolving.com/images/7/4/b/74b21
...eorem. You wouldn't waste that much money to solve one problem. Therefore, I proved that you cannot use the Games theorem to solve problems. But using t
...
69 KB (11,812 words) - 20:42, 19 October 2025
-1. GMAAS looks like this when he is mad: https://cdn.artofproblemsolving.com/images/7/4/b/74b21
...eorem. You wouldn't waste that much money to solve one problem. Therefore, I proved that you cannot use the Games theorem to solve problems. But using t
...
69 KB (11,844 words) - 21:54, 20 October 2025
...} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n </math>, where <math>i = \sqrt{-1}</math>. What is <math>f(2022)</math>?
-1 + i\sqrt{3} &= 2e^{\frac{2\pi i}{3}}, \\
...
5 KB (888 words) - 19:34, 7 November 2025
...or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>?
...al number]]s <math>t</math> and all [[integer]]s <math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply
...
6 KB (1,154 words) - 02:30, 11 January 2024
...\left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)</math>. The intersection of the domain of <math>f(x)</math> with
...align*}\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x) &= 0\\
...
9 KB (1,434 words) - 16:54, 17 August 2022
{{title restriction|<math>\pi</math>|romanized}}
...mber]], as proved by Lindemann in 1882) denoted by the Greek letter <math>\pi</math>.
...
8 KB (1,424 words) - 14:02, 24 July 2025
...nce between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integer
...t{0.25 - x^2} dx = 4\left(\frac{1}{4} - \frac{\pi}{16}\right) = 1 - \frac{\pi}{4}.</cmath>
...
12 KB (1,981 words) - 17:33, 3 September 2023
...ere <math>z</math> is a complex number with <math>|z|=4</math>. Here <math>i = \sqrt{-1}</math>.
&=(75a-117b)+(117a+75b)i+48\left(\dfrac{2+3i}{a+bi}\right) \\
...
17 KB (2,684 words) - 11:05, 26 July 2025
for(int i=0;i<=5;i=i+1)
draw(circle(7/2+d*i,3/2));
...
71 KB (11,749 words) - 11:39, 20 November 2024
...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\
pi.</math>
...{4},\frac{\
pi}{2},\frac{3\
pi}{4},\
pi,\frac{5\
pi}{4},\frac{3\
pi}{2},\frac{7\
pi}{4}.</math></li><p>
...
11 KB (1,719 words) - 15:01, 31 October 2025
...ot, so is <imath>\frac{-1+i\sqrt{3}}{2} \cdot r</imath>? (Note that <imath>i=\sqrt{-1}</imath>)
...}{2}</imath> = <imath>\cos(120^\circ) + i \cdot \sin(120^\circ) = e^{ 2\pi i / 3}</imath>.
...
8 KB (1,272 words) - 00:36, 12 November 2025
...We have our <math>z^{2021}</math> term but we have these unnecessary terms like <math>z^{2020}</math>. We can get rid of these terms by adding <math>-z^{20
z^3 &= e^{i 0} \\
...
8 KB (1,407 words) - 22:54, 30 September 2025
...>, <math>\sqrt{y}</math>, or <math>\sqrt{z}</math>, the system should look like this:
Taking the inverse sine (<math>0\leq\theta\frac{\pi}{2}</math>) of each equation yields a simple system:
...
15 KB (2,208 words) - 00:25, 1 February 2024
...of the two vectors of norm <math>1</math> makes an angle of <math>\le\frac\pi 2</math> with the vector of norm <math>\ge 1</math>, so their sum has norm
Below, I will fill in the gaps in the first "solution". Then,
...
11 KB (1,861 words) - 17:31, 9 July 2025
...aw(0.9*dir(90-30*i)--dir(90-30*i));label("$"+(string) i+"$",0.78*dir(90-30*i));}
...a total of <math> \frac{40\pi + 20\pi}{20\pi} \Rightarrow \frac{60\pi}{20\pi} \Rightarrow \frac{60}{20} \Rightarrow 3 </math>.
...
11 KB (1,714 words) - 18:34, 6 October 2025
