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- ...rsecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by re ...uad \text{(C)}\ 28-4\pi \qquad \text{(D)}\ 28-2\pi \qquad \text{(E)}\ 32-2\pi</math>2 KB (268 words) - 09:28, 17 June 2024
- ...m two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. Wh ...s <math>\dfrac{1}{3} \times 4^2\pi \times 8\sqrt{2}=\dfrac{128\sqrt{2}}{3}\pi</math>.2 KB (364 words) - 00:51, 17 January 2021
- <math>a_k = 2\sin ( \frac{k\pi}{12} )</math>. <math>2a_2 = 2 = a_6</math>. Thus any two segments with at least one them longer than <math>a_2</math> have a sum gre3 KB (395 words) - 15:54, 8 November 2022
- ...on <math>20\%</math> of her three-point shots and <math>30\%</math> of her two-point shots. Shenille attempted <math>30</math> shots. How many points did ...operty that every term beginning with the third is the sum of the previous two. That is, <cmath> S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3. </cmath> S14 KB (2,206 words) - 19:31, 15 May 2024
- ...One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5-year-old ...on <math>20\%</math> of her three-point shots and <math>30\%</math> of her two-point shots. Shenille attempted <math>30</math> shots. How many points di12 KB (1,894 words) - 15:59, 3 January 2024
- ...ath> P </math>, sweeping out a region whose area is <math> \frac{1}{c} (a \pi + b) </math>, where <math>a </math>, <math>b</math>, and <math> c </math> a ...h>. It must be that <math> \overline{AP} </math> divides the diagonal into two segments in the ratio <math>\sqrt{3}</math> to <math>1</math>. It is not di4 KB (603 words) - 16:51, 3 April 2020
- ...e is the diameter, the opposite angle is a right angle. Also, because the two adjacent angles are one sixth of the circle apart, the angle opposite them ...ch arc is <math>\frac{\pi}{3}</math>, because the circumference is <math>2\pi</math>. Once we draw the triangle (as is explained in solution 1), we see t3 KB (417 words) - 20:21, 12 January 2024
- ...into two squares and two nonsquare rectangles. The sum of the areas of the two squares is <math>\frac{9}{10}</math> of the area of square <math>ABCD.</mat ...uted in equal numbers to each student, there are exactly three blocks left over.9 KB (1,580 words) - 13:07, 24 February 2024
- For <math>\pi \le \theta < 2\pi</math>, let ...>\sin \theta = -17/19, 1/3.</math> Since we're given <math>\pi\leq\theta<2\pi,</math> <math>\sin\theta</math> is nonpositive. We therefore use the negati10 KB (1,654 words) - 13:24, 13 June 2024
- <math>|3-\pi|=</math> ...4 \qquad \textbf{(C) }3-\pi \qquad \textbf{(D) }3+\pi \qquad \textbf{(E) }\pi-3 </math>16 KB (2,451 words) - 04:27, 6 September 2021
- ...here we require a clockwise rotation, so we multiply by <math>e^{-\frac{i\pi}{3}}</math> to obtain C. Upon averaging the coordinates of A, B, and C, we We construct point <math>C</math> by drawing two circles with radius <math>r = AB = \sqrt{13}</math>. One circle will be ce8 KB (1,268 words) - 14:10, 31 January 2024
- A=expi(pi); B=expi(pi-a); C=expi(a); D=expi(0); E=expi(-a); F=expi(pi+a); ...342x^2-2420x=0</cmath>, it is easy to see that <math>x=-10, x=0</math> are two solutions for the equation, so we can factorize that into <cmath>x(x+10)(x^8 KB (1,357 words) - 09:23, 11 March 2024
- ...de by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be construc real r = 5/dir(54).x, h = 5 tan(54*pi/180);1 KB (237 words) - 23:06, 3 February 2020
- Let <math>f(x)=(x^2+3x+2)^{\cos(\pi x)}</math>. Find the sum of all positive integers <math>n</math> for which First, let's split it into two cases to get rid of the absolute value sign3 KB (436 words) - 02:10, 12 December 2023
- ...ted to put <math>z</math> in any other location.) Then there are precisely two possible distinct locations for <math>w</math>; one is obtained by going 12 Let the two possible locations for <math>w</math> be <math>W_1</math> and <math>W_2</ma6 KB (1,045 words) - 13:08, 21 January 2024
- We see that both the radii are the two shorter sides of the triangle, making this a isosceles 45-45-90 triangle. We also see that the height that we drew is half the hypotenuse(note the two smaller 45-45-90 isosceles triangles).23 KB (3,182 words) - 12:30, 5 April 2014
- <cmath>P = AI \cap \Omega \implies PB = PC = PI \implies</cmath> It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at19 KB (3,292 words) - 13:04, 13 May 2024
- Kelvin the frog creates an infinite sequence of rational numbers. He chooses two starting terms <math>a_1=\tfrac{n_1}{d_1}</math> and <math>a_2=\tfrac{n_2}{ Kelvin the frog wants to find all positive integers <math>n</math> not over <math>9000</math> such that the order of <math>n \pmod{1000}</math> is <mat10 KB (1,710 words) - 23:23, 10 January 2020
- ...thmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is <math>1</math> foot wide on all four sides. What is the l ...lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie h14 KB (2,104 words) - 21:35, 7 June 2024
- Let <math>A_0B_0C_0</math> and <math>A_1B_1C_1</math> be any two acute-angled triangles. Consider all triangles <math>ABC</math> that are si ...nt <math>P</math> inside <math>A_0B_0C_0</math> s.t. <math>\angle X_0PY_0=\pi-\angle X_1Z_1Y_1</math>, where <math>X,Y,Z</math> are a permutation of <mat2 KB (368 words) - 13:15, 29 January 2021
