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2015 AIME I Problems
...the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is <math>\frac{m}{n}</math>, wher ...2</math> of the remaining <math>8</math> socks at random, and on Wednesday two of the remaining <math>6</math> socks at random. The probability that Wedn

10 KB (1,615 words) - 21:48, 13 January 2024
2012 AIME I Problems
...= z,</math> and the imaginary part of <math>z</math> is <math>\sin{\frac{m\pi}{n}}</math>, for relatively prime positive integers <math>m</math> and <mat ...he plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form <math>\tfrac{p}{q},</math> where <math>p<

10 KB (1,617 words) - 14:49, 2 June 2023
2012 AIME II Problems/Problem 13
...ath> and <math>\sqrt{11}</math> from point <math>B</math>, so they are the two points where a circle centered at <math>A</math> with radius <math>\sqrt{11 There are two equilateral triangles with <math>\overline{AD_1}</math> as a side; let <mat

13 KB (2,052 words) - 18:02, 5 February 2024
2017 AMC 8 Problems
...ves but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are tru ...with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?

12 KB (1,771 words) - 21:13, 20 January 2024
Mock AIME I 2012 Problems
...math>, <math>\cdots</math>, <math>9\}</math> is called ''golden'' if every two consecutive numbers in the permutation are relatively prime. How many ''gol ...ing (possibly the same string) chosen at random. He continues on by tying two randomly selected untied ends together until there are no more untied ends,

7 KB (1,309 words) - 11:13, 8 April 2012
2015 IMO Problems/Problem 1
if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math> regular polygons <math>P_n</math> with <math>n</math> sides: given any two vertices <math>A</math> and

4 KB (773 words) - 08:14, 19 July 2016
1988 USAMO Problems/Problem 5
...{a_n}</math> are all <math>n</math>th roots of unity. If <math>\omega=e^{2\pi i/n}</math>, then the sum of <math>32</math>nd powers of these roots will b ...mials of degree <math>n</math> over the field <math>\mathbb{F}_2</math> of two elements, and on the left, we have the factorisation of this generating fun

8 KB (1,348 words) - 09:44, 25 June 2022
2006 SMT/General Problems
A geometric series is one where the ratio between each two consecutive terms is constant (ex. <math> 3, 6, 12, 24,\cdots </math>). The Given two <math> 2 </math>'s, "plus" can be changed to "times" without changing the r

10 KB (1,477 words) - 16:02, 27 May 2012
2006 SMT/Team Problems
...e <math> M </math> minus the volume fo cone <math> N </math> is <math> 140\pi </math>, find the length of <math> \overline{BC} </math>. ...ath> S_n=\sum_{i=1}^{n-1}i\cdot S_{n-i} </math>. For example, if the first two elements are <math> 2 </math> and <math> 3 </math>, respectively, the third

6 KB (910 words) - 17:32, 27 May 2012
2006 SMT/Advanced Topics Problems/Problem 1
...s the square root of himself, or the square root of <math> i </math>. What two answers should he give? ...math> i=\cos\left(\frac{\pi}{2}\right)+i\sin\left(\frac{\pi}{2}\right)=e^{\pi i/2} </math>.

1 KB (170 words) - 20:12, 27 May 2012
1952 AHSME Problems
Two high school classes took the same test. One class of <math> 20 </math> stud Two equal circles in the same plane cannot have the following number of common

23 KB (3,556 words) - 15:35, 30 December 2023
1953 AHSME Problems
\textbf{(B)}\ \frac{440}{\pi} \qquad \textbf{(C)}\ \frac{880}{\pi} \qquad

21 KB (3,123 words) - 14:24, 20 February 2020
1955 AHSME Problems
The ratio of the areas of two concentric circles is <math>1: 3</math>. If the radius of the smaller is <m Two numbers whose sum is <math>6</math> and the absolute value of whose differe

22 KB (3,509 words) - 21:29, 31 December 2023
1998 USAMO Problems/Problem 6
..., we can easily alternate them, but if <math>n</math> is odd, we must have two adjacent sides be the same length, and that length must be the larger side

5 KB (871 words) - 18:59, 10 May 2023
2012 AMC 10B Problems/Problem 16
...xtbf{(C)}\ 12\pi+\sqrt{3}\qquad\textbf{(D)}\ 10\pi+9\qquad\textbf{(E)}\ 13\pi</math> ...<math>\dfrac{1}{2}</math>, we get <math>2\sqrt{3}</math>. Since there are two 30-60-90 triangles in the equilateral triangle, we multiply the area of the

4 KB (675 words) - 03:58, 23 January 2023
1986 USAMO Problems
Two distinct circles <math>K_1</math> and <math>K_2</math> are drawn in the pla ...lar to a line through a point on or off the line), find a construction for two points <math>C</math> and <math>D</math> on <math>K_1</math> such that <mat

3 KB (500 words) - 19:00, 23 October 2019
2008 AMC 8 Problems/Problem 21
...(6) = 96\pi</math>. The volume of the wedge is half this which is <math>48\pi \approx \boxed{\textbf{(C)}\ 151}</math>.

1 KB (198 words) - 20:54, 29 October 2023
2008 AMC 8 Problems
In <math>2005</math> Tycoon Tammy invested <math>100</math> dollars for two years. During the the first year investment showed a <math>20\%</math> gain. Over the two-year period, what was the change

14 KB (2,035 words) - 15:23, 26 January 2024
2012 AMC 8 Problems
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inche ...ass. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she r

13 KB (1,835 words) - 08:51, 8 March 2024
2012 AMC 8 Problems/Problem 24
...D)}\hspace{.05in}\frac{\pi-1}{\pi}\qquad\textbf{(E)}\hspace{.05in}\frac{3}\pi </math> ...two areas, we find the answer is <math> \boxed{\textbf{(A)}\ \frac{4-\pi}{\pi}} </math>.

2 KB (310 words) - 16:18, 10 November 2023
2004 AMC 8 Problems/Problem 25
...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 (346 words) - 21:43, 2 January 2023
2012 AMC 10B Problems/Problem 17
...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
2013 AMC 12A Problems/Problem 24
<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 gre

3 KB (395 words) - 15:54, 8 November 2022
2013 AMC 12A Problems
...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> S

14 KB (2,206 words) - 19:31, 15 May 2024
2013 AMC 10A Problems
...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 di

12 KB (1,894 words) - 15:59, 3 January 2024
2013 AMC 12A Problems/Problem 23
...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 di

4 KB (603 words) - 16:51, 3 April 2020
2013 AMC 10B Problems/Problem 7
...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 t

3 KB (417 words) - 20:21, 12 January 2024
2013 AIME I Problems
...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
2013 AIME I Problems/Problem 14
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 negati

10 KB (1,641 words) - 20:03, 3 January 2024
1991 AHSME Problems
<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
2013 AIME II Problems/Problem 4
...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 ce

8 KB (1,268 words) - 14:10, 31 January 2024
2013 AIME II Problems/Problem 8
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
2003 AMC 12B Problems/Problem 10
...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
2014 AIME II Problems/Problem 7
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 sign

3 KB (436 words) - 02:10, 12 December 2023
2014 AIME II Problems/Problem 10
...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</ma

6 KB (1,045 words) - 13:08, 21 January 2024
Geometry Solutions
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
Symmetry
<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 at

19 KB (3,292 words) - 13:04, 13 May 2024
ASIA TEAM Problems
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 <mat

10 KB (1,710 words) - 23:23, 10 January 2020
2016 AMC 10A Problems
...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 h

14 KB (2,104 words) - 22:26, 16 September 2022
1967 IMO Problems/Problem 4
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 <mat

2 KB (368 words) - 13:15, 29 January 2021
1971 IMO Problems/Problem 5
...2,\ldots, w_m</math>. we claim that for some choice of <math>0\leq\theta<2\pi</math>, <math>v_i+e^{i\theta}w_j</math> will do the job(a suitable rotation ...y set and adding vectors to it, one by one. We just need to make sure that two thing are respected:

4 KB (749 words) - 14:09, 29 January 2021
1972 IMO Problems/Problem 6
...is the <math>3</math>-dimensional version of this), and then complete the two regular tetrahedra <math>A_1A_2A_3P_1,A_1A_2A_3P_2</math> having <math>A_1A ..._1,P_2</math>, these will be continuous functions of the angle that <math>\pi</math> makes with <math>\pi_i</math>, and for one of the points <math>P_1,P

7 KB (1,370 words) - 15:42, 29 January 2021
1962 AHSME Problems
<math>\textbf{(A)}\ \pi + 2 \qquad \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad

17 KB (2,459 words) - 22:40, 10 April 2023
2013 AMC 8 Problems
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, <math>30 = 6\times5</mat A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?

15 KB (2,162 words) - 20:05, 8 May 2023
1994 AHSME Problems
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectang each term is the sum of the two terms to its left. Find <math>a</math>.

14 KB (2,124 words) - 13:39, 19 February 2020
1981 AHSME Problems
...D</math>. If <math>EB</math> has length one and <math>EC</math> has length two, then the area of the square is ...the smaller and the difference between the numbers is 8, the the larger of two numbers is

17 KB (2,633 words) - 15:44, 16 September 2023
1968 AHSME Problems
...increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter. Then <math>P</math> equals: ...pi\quad\text{(C) } \frac{\pi^2}{2}\quad\text{(D) } \pi^2\quad\text{(E) } 2\pi</math>

16 KB (2,571 words) - 14:13, 20 February 2020
2014 AMC 10A Problems
...morning for <math>\textdollar 2.50</math> each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges on The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</

15 KB (2,190 words) - 15:21, 22 December 2020
1994 AHSME Problems/Problem 17
...qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 </math> ...length <math>\sqrt{2}</math>. We deduce that the two triangles formed by two radii of circle <math>O</math> and the segment of the rectangle are 45-45-9

2 KB (287 words) - 03:07, 28 May 2021
1960 AHSME Problems
...r{10,000}</math> the difference between a discount of <math>40</math>% and two successive discounts of <math>36</math>% and <math>4</math>%, Each of two angles of a triangle is <math>60^{\circ}</math> and the included side is <m

21 KB (3,242 words) - 21:27, 30 December 2020
2014 AMC 12A Problems/Problem 12
Two circles intersect at points <math>A</math> and <math>B</math>. The minor a <cmath>\dfrac{\pi x^2}{\pi y^2} = \dfrac{x^2}{y^2} </cmath>

7 KB (1,191 words) - 23:37, 23 June 2022
2014 AMC 12B Problems
...consists of triangles whose sides have integer lengths less than 5, and no two elements of <math>S</math> are congruent or similar. What is the largest nu ...itive solutions of <cmath>2\cos(2x) \left(\cos(2x) - \cos\left( \frac{2014\pi^2}{x} \right)\right) = \cos(4x) - 1</cmath>

13 KB (2,066 words) - 14:08, 1 November 2022
Mock AIME 3 2006-2007 Problems/Problem 11
(one value of <math>a</math> which produces this maximum is <math>a=\frac{\pi}{4}</math>) ...^2+y^2</math>. Adding and subtracting the first two equations, dividing by two, and setting <math>\mu = 2\lambda</math> for convenience yields:

4 KB (699 words) - 01:53, 30 April 2022
2014 AIME II Problems
...ee factors, the probability that a randomly selected man has exactly these two risk factors (but not the third) is 0.14. The probability that a randomly s ...es of length <math>a</math> can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the

8 KB (1,410 words) - 00:04, 29 December 2021
2014 AIME I Problems/Problem 10
Let <math>F</math> be the new tangency point of the two disks. The smaller disk rolled along minor arc <math>\overarc{AF}</math> on ...the smaller disk, so <math>6\alpha = 2\pi</math>, or <math>\alpha = \frac{\pi}{3}</math>.

5 KB (854 words) - 20:02, 4 September 2021
1973 AHSME Problems
...ations of the even number 126, the largest possible difference between the two primes is Two congruent 30-60-90 are placed so that they overlap partly and their hypoten

18 KB (2,788 words) - 13:55, 20 February 2020
1992 AHSME Problems
If <math>3(4x+5\pi)=P</math> then <math>6(8x+10\pi)=</math> A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there

16 KB (2,548 words) - 13:40, 19 February 2020
1991 AHSME Problems/Problem 22
Two circles are externally tangent. Lines <math>\overline{PAB}</math> and <math <math>\text{(A) } 1.44\pi\quad

2 KB (295 words) - 19:09, 11 October 2016
1990 AHSME Problems/Problem 22
This equation is <math>r^6e^{6\theta i}=2^6e^{(\pi\pm 2k\pi) i}</math>. Solving in the usual way, <math>r=2</math> and <math>\theta\in\ Thus there are only two solutions with positive real part, and they are conjugates, so their produc

798 bytes (133 words) - 13:17, 4 February 2016
1990 AHSME Problems/Problem 14
draw((0,1)--(cos(pi/14),-sin(pi/14))--(-cos(pi/14),-sin(pi/14))--cycle,dot); draw((-cos(pi/14),-sin(pi/14))--(0,-1/cos(3pi/7))--(cos(pi/14),-sin(pi/14)),dot);

1 KB (221 words) - 14:09, 2 April 2023
1990 AHSME Problems
draw((0,1)--(cos(pi/14),-sin(pi/14))--(-cos(pi/14),-sin(pi/14))--cycle,dot); draw((-cos(pi/14),-sin(pi/14))--(0,-1/cos(3pi/7))--(cos(pi/14),-sin(pi/14)),dot);

14 KB (2,099 words) - 01:15, 10 September 2021
1969 AHSME Problems/Problem 6
The area of the ring between two concentric circles is <math>12\tfrac{1}{2}\pi</math> square inches. The length of a chord of the larger circle tangent to <cmath>\pi a^2 - \pi b^2 = \frac{25 \pi}{2}</cmath>

1 KB (187 words) - 03:29, 7 June 2018
1969 AHSME Problems
The area of the ring between two concentric circles is <math>12\tfrac{1}{2}\pi</math> square inches. The length of a chord of the larger circle tangent to The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is:

16 KB (2,662 words) - 14:12, 20 February 2020
1952 AHSME Problems/Problem 37
Two equal parallel chords are drawn <math>8</math> inches apart in a circle of <math>\textbf{(A)}\ 21\frac{1}{3}\pi-32\sqrt{3}\qquad

2 KB (333 words) - 20:23, 15 April 2020
1970 AHSME Problems
<math>\textbf{(A) }\frac{4}{\pi}\qquad \textbf{(B) }\frac{\pi}{\sqrt{2}}\qquad

15 KB (2,366 words) - 07:52, 26 December 2023
1987 AHSME Problems
...eal numbers have the property that each number is the product of the other two? <math>\textbf{(A)}\ 36\pi \qquad

16 KB (2,291 words) - 13:45, 19 February 2020
1986 AHSME Problems
Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length <math>r</ (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.)

17 KB (2,512 words) - 18:30, 12 October 2023
1983 AHSME Problems
Consider the two functions <math>f(x) = x^2+2bx+1</math> and <math>g(x) = 2a(x+b)</math>, wh \textbf{(B)} \ \pi \qquad

15 KB (2,309 words) - 23:43, 2 December 2021
1982 AHSME Problems
If a number eight times as large as <math>x</math> is increased by two, then one fourth of the result equals <math>\textbf{(A)} \ \pi \qquad

17 KB (2,500 words) - 19:05, 11 September 2023
1979 AHSME Problems
<math>\textbf{(A) }\frac{\pi}{4}\qquad \textbf{(B) }\frac{3\pi}{4}\qquad

17 KB (2,664 words) - 01:34, 19 March 2022
1978 AHSME Problems
<math>\textbf{(A) }\frac{1}{\pi^2}\qquad \textbf{(B) }\frac{1}{\pi}\qquad

15 KB (2,432 words) - 01:06, 22 February 2024
1976 AHSME Problems
\textbf{(C) }\text{two}\qquad\\ \textbf{(D) }\text{a finite number greater than two}\qquad

17 KB (2,835 words) - 14:36, 8 September 2021
1972 AHSME Problems
A contractor estimated that one of his two bricklayers would take <math>9</math> hours to build a certain wall and the There are two positive numbers that may be inserted between <math>3</math> and <math>9</m

17 KB (2,732 words) - 13:54, 20 February 2020
1965 AHSME Problems
\textbf{(C) }\ 1: \pi \qquad \textbf{(D) }\ 3: \pi \qquad

19 KB (2,873 words) - 18:57, 16 August 2023
1957 AHSME Problems
The area of a circle inscribed in an equilateral triangle is <math>48\pi</math>. The perimeter of this triangle is: In the binary system, which has base two, the first five positive integers are <math>1,\,10,\,11,\,100,\,101</math>.

26 KB (3,950 words) - 21:09, 31 August 2020
2013 UNCO Math Contest II Problems
In the diagram, the two circles are tangent to the two parallel lines. The small circle is <math>9\pi</math>. Find the area of the shaded <math>\textit{lune}</math>, the region

11 KB (1,648 words) - 09:55, 20 December 2021
2013 UNCO Math Contest II Problems/Problem 1
In the diagram, the two circles are tangent to the two parallel lines. The (8)(6) - 2\left(\dfrac12 \cdot 3^2\pi\right) &= 48 - 2\left(\dfrac12 \cdot 9\pi\right) \\

1 KB (162 words) - 14:13, 24 February 2022
2012 UNCO Math Contest II Problems/Problem 7
region that lies between the two circles and the line. ...ac{1}{3} \cdot \pi - \frac{1}{6} \cdot 9\pi = \boxed {4\sqrt{3} - \frac{11\pi}{6}}</math>

2 KB (336 words) - 12:57, 2 January 2020
2007 UNCO Math Contest II Problems
...hat every odd number <math>2n+1</math> can be expressed as a difference of two squares. (b) Demonstrate which even numbers can be expressed as a difference of two squares.

6 KB (890 words) - 22:14, 7 November 2014
2007 UNCO Math Contest II Problems/Problem 9
your answer as an exact multiple of <math>\pi</math> (and not as a ...triangle and let <math>A_1B_1</math> be the internal common tangent of the two largest circles, with the points <math>A_1</math> and <math>B_1</math> layi

3 KB (412 words) - 18:49, 29 January 2018
1986 AHSME Problems/Problem 21
pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10); ...for the equality of the two shaded areas, given <math>0 < \theta < \frac{\pi}{2}</math>, is

2 KB (301 words) - 18:50, 1 April 2018
2014 AMC 8 Problems
The sum of two prime numbers is 85. What is the product of these two prime numbers? ...ike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue ei

13 KB (1,957 words) - 12:08, 13 January 2024
2014 AMC 8 Problems/Problem 25
...}\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad\textbf{(E) }\frac{2\pi}{3} </math> ...h> and time = distance/rate) to arrive at <math>\boxed{\textbf{(B) }\frac{\pi}{10}}</math> hours.

2 KB (395 words) - 12:16, 17 January 2024
2015 AMC 10A Problems
The sum of two positive numbers is <math> 5 </math> times their difference. What is the ra Two years ago Pete was three times as old as his cousin Claire. 2 years before

12 KB (1,887 words) - 18:52, 14 May 2024
2015 AMC 12A Problems
Two of the three sides of a triangle are 20 and 15. Which of the following numb The sum of two positive numbers is 5 times their difference. What is the ratio of the larg

13 KB (2,117 words) - 12:33, 24 August 2023
2015 AMC 10A Problems/Problem 9
...e radius of the first. What is the relationship between the heights of the two cylinders? ...uting the first equation into the second and dividing both sides by <math>\pi</math>, we get <cmath>r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{12

2 KB (258 words) - 23:04, 26 June 2023
2015 AMC 10A Problems/Problem 25
...tween the points is at least <math>\dfrac{1}{2}</math> is <math>\dfrac{a-b\pi}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are po ...0.25 - x^2} dx = 4\left( \frac{1}{4} - \frac{\pi}{16} \right) = 1 - \frac{\pi}{4}.</cmath>

8 KB (1,406 words) - 19:14, 9 October 2023
2015 AMC 10A Problems/Problem 12
Points <math>( \sqrt{\pi} , a)</math> and <math>( \sqrt{\pi} , b)</math> are distinct points on the graph of <math>y^2 + x^4 = 2x^2 y + ...(C)} \ 2 \qquad\textbf{(D)} \ \sqrt{1+\pi} \qquad\textbf{(E)} \ 1 + \sqrt{\pi} </math>

2 KB (418 words) - 15:25, 1 August 2022
2015 AMC 10A Problems/Problem 14
...t 6:00. It would already have completed three 180 degree turns. Therefore, two 180 degree turns would be completed at 4:00. ...s <math>20\pi</math>, so that is <math>20\pi</math> traveled on a <math>60\pi</math> arrow path. This is a ratio of 1/3, so the angle it carves is 120 de

9 KB (1,380 words) - 09:00, 1 December 2023
2015 AMC 12A Problems/Problem 24
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath> Let <math>\cos(a\pi) = x</math> and <math>\sin(b\pi) = y</math>. Consider the binomial expansion of the expression:

5 KB (793 words) - 20:32, 26 May 2022
2015 AMC 12A Problems/Problem 23
...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) - 18:33, 3 September 2023
2015 AMC 10B Problems
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is <math ...week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the

14 KB (2,247 words) - 11:38, 26 March 2024
2015 AMC 12B Problems
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, ...c{\sqrt{2}}{2} \qquad\textbf{(D)}\; 4(\pi-\sqrt{3}) \qquad\textbf{(E)}\; 2\pi-\dfrac{\sqrt{3}}{2}</math>

13 KB (2,064 words) - 13:39, 1 October 2022
2015 AMC 12B Problems/Problem 14
...c{\sqrt{2}}{2} \qquad\textbf{(D)}\; 4(\pi-\sqrt{3}) \qquad\textbf{(E)}\; 2\pi-\dfrac{\sqrt{3}}{2}</math> ...ction), so the answer is <math>4\pi-4\sqrt{3} = \boxed{\textbf{(D)}\; 4(\pi-\sqrt{3})}</math>.

1 KB (197 words) - 15:02, 4 March 2015
2015 AIME I Problems/Problem 15
...ce on each half. The area of one of these unpainted faces is <math>a\cdot\pi + b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math ...\ + 12 \pi</math>. Thus, the area of section <math>ABCD</math> is <math>20\pi + 30\sqrt{3}</math>, and so our answer is <math>20+30+3=\boxed{053}</math>.

9 KB (1,407 words) - 19:37, 17 February 2024
Searching the community
The first of these two options is shown in the image to the right. It is primarily useful for quic ...hat approximates how well a post correlates to the search query. The other two options, "Newest first" and "Oldest first", are self-explanatory.

21 KB (3,334 words) - 23:53, 23 March 2022
2015 UNCO Math Contest II Answer Key
2) <math>3\pi</math> meets a blue triangle along the edge where the two faces meet. Consider the two triangles

2 KB (310 words) - 00:19, 6 November 2015
2019 AMC 12A Problems
...at <math>(2,2)</math>. What is the area of the triangle enclosed by these two lines and the line <math>x+y=10</math>? ...s in a plane, there are exactly <math>N</math> distinct points that lie on two or more of the lines. What is the sum of all possible values of <math>N</ma

14 KB (2,180 words) - 22:25, 25 April 2024
2016 AMC 10B Problems
Isaac added two three-digit positive integers. All six digits in these numbers are differen ...the complement of the other. What is the sum of the degree measures of the two angles?

12 KB (1,868 words) - 17:50, 30 December 2023
2016 AMC 12B Problems
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The ...the complement of the other. What is the sum of the degree measures of the two angles?

14 KB (2,037 words) - 19:09, 29 July 2023
Gmaas
...te dreadful. EDIT: He now eats gnats again. He thinks that they taste like pi(e). EDIT: He ate 3,141,592,653,589,793,238,462,643,383 so far. 4. Gmaas wants every fact to have a pi reference. Gmaas created pi. EDIT: He made it 3.1415926 times.

69 KB (11,805 words) - 20:49, 18 December 2019
2006 iTest Problems
...face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a <math>.5</math> probability o \text{(N) }\pi\qquad

31 KB (4,811 words) - 00:02, 4 November 2023
2016 USAJMO Problems/Problem 5
...2} - \gamma.</math> Similarly, one can show that <math>\angle OAC = \frac{\pi}{2} - \beta.</math> (One could probably cite this as well-known, but I have ...\frac{AO}{AQ}.</math> We also have <math>\angle PAH = \angle OAQ = \frac{\pi}{2} - \beta,</math> so <math>\triangle PAH\sim\triangle OAQ</math> by SAS s

10 KB (1,733 words) - 19:15, 14 June 2020
1976 AHSME Problems/Problem 18
\textbf{(B) }15/\pi\qquad ...s, draw the diameter of the circle that contains <math>OD</math>). Let the two endpoints of this diameter be <math>P</math> and <math>Q,</math> where <mat

2 KB (361 words) - 08:05, 9 April 2023
1953 AHSME Problems/Problem 11
...centric circles. It is <math>10</math> feet wide. The circumference of the two circles differ by about: ...in the circles' diameters is simply <math>20\pi </math> feet. Using <math>\pi \approx 3</math>, the answer is <math>\fbox{C}</math>.

814 bytes (124 words) - 12:23, 22 April 2020
1953 AHSME Problems/Problem 12
The diameters of two circles are <math>8</math> inches and <math>12</math> inches respectively. ...a of the smaller to the area of the larger circle is <math>\frac{16\pi}{36\pi}=\boxed{\textbf{(C) }\frac{4}{9}}</math>.

850 bytes (134 words) - 20:49, 1 April 2017
1990 IMO Problems/Problem 6
...that there exists a convex <math>1990\text{-gon}</math> with the following two properties: ...1,a_2,\cdots,a_{1990}\}=\{1,2,\cdots,1990\}</math> and <math>\theta=\frac{\pi}{995}</math>.

3 KB (522 words) - 13:54, 30 January 2021
1975 Canadian MO Problems
...less than or equal to <math>r</math>, <math>e.g.,</math> <math>[6] = 6, [\pi] = 3, [-1.5] = -2.</math> Indicate on the <math>(x,y)</math>-plane the set ...in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated. </div>

3 KB (499 words) - 12:17, 11 August 2016
1979 AHSME Problems/Problem 20
<math>\textbf{(A) }\frac{\pi}{2}\qquad \textbf{(B) }\frac{\pi}{3}\qquad

2 KB (363 words) - 12:46, 10 May 2022
2017 AMC 12A Problems/Problem 8
...th> units of line segment <math>\overline{AB}</math> has volume <math>216 \pi</math>. What is the length <math>AB</math>? ...central axis <math>\overline{AB}</math> and radius <math>3</math> and the two hemispheres connected to each face of the cylinder (also with radius <math>

1 KB (159 words) - 14:59, 10 June 2023
2017 AMC 10A Problems/Problem 11
...ath> units of line segment <math>\overline{AB}</math> has volume <math>216\pi</math>. What is the length <math>\textit{AB}</math>? ...of our cylinder + the volume of our two hemispheres will equal <math>216 \pi</math>):

2 KB (320 words) - 14:24, 13 April 2024
2017 AMC 10A Problems/Problem 22
...{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}</math> ...}{\frac{3r^2 \sqrt{3}}4} = \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}</cmath>

6 KB (993 words) - 20:28, 23 February 2024
2017 AMC 10A Problems
Tamara has three rows of two <math>6</math>-feet by <math>2</math>-feet flower beds in her garden. The b ...th>4</math> times their product. What is the sum of the reciprocals of the two numbers?

15 KB (2,285 words) - 18:02, 28 October 2023
2017 AMC 12A Problems/Problem 15
Between <math>1</math> and <math>\frac{\pi}{2}</math>, <math>f(x)</math> starts at a positive number and increases to Between <math>\frac{\pi}{2}</math> and 3, <math>f(x)</math> starts at <math>-\infty</math> and incr

3 KB (564 words) - 14:12, 23 October 2021
2017 AMC 12A Problems/Problem 25
...e of odd multiples of <math>\pi/4</math>, i.e. <math>\pi/4,3\pi/4,5\pi/4,7\pi/4</math>. When we draw these 6 complex numbers out on the complex plane, we ...\sum\theta\equiv\pi \mod 2\pi</math>. Letting <math>z</math> be <math>e^{i\pi/4}</math>, we see that the angles we have available are <math>\{z^1,z^2,z^3

18 KB (2,878 words) - 01:47, 16 December 2023
2017 AMC 12A Problems
...umbers is 4 times their product. What is the sum of the reciprocals of the two numbers? ...th> units of line segment <math>\overline{AB}</math> has volume <math>216 \pi</math>. What is the length <math>AB</math>?

15 KB (2,418 words) - 16:58, 7 November 2022
2017 AMC 12B Problems
...(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math> ...i}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)}</math> It's not periodic.

15 KB (2,343 words) - 18:26, 25 December 2020
2017 AMC 12B Problems/Problem 12
...has base <math>\frac{\sqrt{6}}{2}</math>. Adding these and multiplying by two, we get the sum of the four roots as <math>\sqrt{2} + \sqrt{6}</math>. Howe ...are <math>\cos(0), \pm\cos(\frac{\pi}{6}),</math> and <math>\pm\cos(\frac{\pi}{3})</math>. Their sum is <math>1+2(\frac{\sqrt{3}}{2}+\frac{1}{2})=1+\sqrt

3 KB (449 words) - 01:54, 11 February 2019
2017 AIME I Problems/Problem 8
...he interval <math>(0, 75)</math>. Let <math>O</math> and <math>P</math> be two points on the plane with <math>OP = 200</math>. Let <math>Q</math> and <mat ...</math> by <math>75</math> box in the <math>ab</math> plane. This cuts off two isosceles right triangles from opposite corners with side lengths <math>45<

9 KB (1,539 words) - 15:47, 17 February 2024
2017 AIME I Problems/Problem 10
...subtract two vectors, the geometric result is the line segment between the two endpoints of the vectors. Thus we can fill in <math>z_3 - z_1, z_2 - z_1, z ...ath>, so let’s try to prove that they are congruent. We can show this in two ways;

13 KB (2,252 words) - 15:46, 6 January 2024
2017 AIME I Problems/Problem 15
...>(0,0),(5,0),(0,2\sqrt{3}),</math> and let <math>(a,0),(0,b)</math> be the two vertices of the equilateral triangle on the legs of the right triangle. The We know that <math>(-a+bi)\cdot\cos(-\frac{\pi}{3})=(-\frac{a+\sqrt{3}b}{2}+\frac{\sqrt{3}a+b}{2}i)</math>. We know that t

22 KB (3,622 words) - 17:11, 6 January 2024
2017 USAMO Problems/Problem 5
...labels occur, and for each label <math>i</math>, the distance between any two points labeled <math>i</math> is at least <math>c^i</math>. ...oints on the coordinate plane, but with the minimum separation between any two points equaling <math>x</math> (as opposed to <math>1</math>).

8 KB (1,495 words) - 12:19, 17 July 2023
2020 AMC 12A Problems
...ditional unit squares that must be shaded so that the resulting figure has two lines of symmetry<math>?</math> ...>\tan{(2x)} = \cos{(\tfrac{x}{2})}</math> have on the interval <math>[0, 2\pi]?</math>

14 KB (2,073 words) - 15:15, 21 October 2021
2020 AMC 12B Problems
...e. What is the greatest possible value of the product of the slopes of the two lines? ...the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

16 KB (2,417 words) - 01:03, 28 April 2022
2017 AMC 8 Problems/Problem 25
...{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}</math> ...th> Thus, our final answer is <math>\boxed{\textbf{(B)}\ 4\sqrt{3}-\frac{4\pi}{3}}.</math>

4 KB (711 words) - 21:00, 16 January 2024
MIE 2016
...th>. With <math>\beta</math> being a real number such that <math>0< \beta<\pi/8</math> and <math>x\neq0</math>, the value of <math>\beta</math> is: (a) <math>\frac{\pi}{9}</math>

8 KB (1,278 words) - 09:46, 11 January 2018
British Mathematical Olympiad
...a national math competition held in the United Kingdom. Solvers who score over a certain threshold in the Senior Mathematical Challenge are automatically The British Mathematical Olympiad is divided into two rounds. In the first round (BMO 1), solvers have 3.5 hours to solve 6 prob

6 KB (1,034 words) - 10:12, 7 June 2023
2018 AMC 12A Problems/Problem 22
...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> <p> ...ath> or <math>\theta\in\biggl(0,\frac{\pi}{4}\biggr)\cup\biggl(\pi,\frac{5\pi}{4}\biggr).</math> We conclude that <math>(r,\operatorname{cis}\theta)=\lef

10 KB (1,662 words) - 12:45, 13 September 2021
2018 AMC 12A Problems
...gebra, geometry, and number theory -- in a <math>6</math>-period day if no two mathematics courses can be taken in consecutive periods? (What courses the ...st subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which

15 KB (2,380 words) - 18:52, 7 April 2022
2018 AMC 12B Problems
...math>. What is the distance between the <math>x</math>-intercepts of these two lines? \textbf{(A) }25\pi \qquad

14 KB (2,118 words) - 15:36, 28 October 2021
2018 AMC 10B Problems/Problem 22
...The triangle inequality tells us that <math>a + b > c</math>. So, we have two inequalities: So, the area is <math>\frac{\pi}{4} - \frac12</math> which is approximately <math>\boxed{\textbf{(C)} ~0.29

3 KB (509 words) - 22:17, 25 February 2024
2018 AMC 12B Problems/Problem 16
...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 (1,708 words) - 12:01, 18 March 2023
2018 AMC 12B Problems/Problem 25
...placed in the plane so that each circle is externally tangent to the other two. Points <math>P_1</math>, <math>P_2</math>, and <math>P_3</math> lie on <m ...e each <math>2\pi/3</math>. Also, the distances between the centers of any two of the <math>3</math> given circles are each <math>8</math>.

13 KB (2,080 words) - 19:09, 21 October 2023
2018 AMC 10B Problems/Problem 12
...r of mass) of <math>\triangle ABC</math> traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bou ...n below, <math>\triangle ABC_1</math> and <math>\triangle ABC_2</math> are two shapes of <math>\triangle ABC</math> with centroids <math>G_1</math> and <m

4 KB (647 words) - 14:19, 5 June 2023
1973 IMO Problems/Problem 1
...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

1 KB (227 words) - 15:45, 29 January 2021
1960 AHSME Problems/Problem 23
The volume <math>V = \pi R^2H</math> is to be increased by the same fixed positive amount when <math \textbf{(E)}\ \text{two real values of x} </math>

1 KB (245 words) - 19:07, 17 May 2018
2018 AIME I Problems/Problem 6
...sin \theta = \sin (\theta + 2\pi)</math>. We can use these facts to create two types of solutions: <cmath>\sin \theta = \sin ((2m + 1)\pi - \theta)</cmath>

7 KB (1,211 words) - 00:23, 20 January 2024
2018 AIME I Problems/Problem 12
...s in the case as <math>x, y</math> respectively. Then in this case we have two subcases; ...ut the remaining cases, we can use this symmetry. The total number of ways over all the cases is <math>\sum_{k=0}^{6} W_{3k} = 2 \cdot (1+56+336)+580 = 136

26 KB (4,044 words) - 13:58, 24 January 2024
2018 AIME I Problems/Problem 10
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a c ...e moves be <math>C</math> and clockwise moves be <math>C'</math>. The only two feasible solutions for <math>(C,C')</math> are <math>(4,9)</math> and <math

11 KB (1,934 words) - 12:18, 29 March 2024
2018 AIME II Problems/Problem 5
...hat <math>y=10e^{\frac{5\pi i}{4}}\sqrt{2}</math> and <math>z=-3e^{\frac{7\pi i}{4}}\sqrt{2}</math>. Then plug <math>z</math> into <math>zx</math>, you c ...>\frac{\pi}{4}</math>, and the argument of <math>z</math> is <math>-\frac{\pi}{4}</math>. We need to convert the polar form to a standard form. Simple tr

11 KB (2,077 words) - 20:15, 12 January 2024
2018 AIME II Problems/Problem 12
...ight)^2\end{align}</cmath>Eliminating <math>\cos\theta</math> in the above two equations and solving for <math>\cos\phi</math> we get<cmath>\cos\phi = \fr ...is clear that <cmath>x^2+y^2-100=2 xy \cdot \cos{APB}=-(2 xy \cdot \cos{(\pi-CPB)})=-(x^2+y^2-196) </cmath> or <cmath>\begin{align}x^2+y^2=148\end{align

18 KB (2,912 words) - 13:12, 24 January 2024
2008 iTest Problems/Problem 56
...wormhole is 2400 units. Your wormhole distance allotment is <math>\textit{two}</math>."' ...<math>3840</math> units. Your wormhole distance allotment is <math>\textit{two}</math>."

7 KB (1,092 words) - 19:05, 17 December 2021
Bibhorr Formula
...se of two [[constant|constants]] - <math>90^{\circ}</math> or <math>\frac{\pi}{2}</math> and <math>1.5</math> makes the [[equation|formula]] more legible ...ee (geometry)|degrees]] but if Bibhorr sthiron is in the form <math>\frac{\pi}{2}</math> then Bibhorr [[angle]] results in [[radian|radians]].

2 KB (245 words) - 06:45, 9 September 2018
2018 AMC 8 Problems
...he edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use? Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.

14 KB (2,191 words) - 03:19, 2 April 2024
2018 AMC 8 Problems/Problem 15
...f each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of <math>1</math> square unit, then wh ...extbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}</math>

3 KB (454 words) - 10:15, 10 April 2024
2019 AMC 10A Problems
...at <math>(2,2)</math>. What is the area of the triangle enclosed by these two lines and the line <math>x+y=10 ?</math> ...s in a plane, there are exactly <math>N</math> distinct points that lie on two or more of the lines. What is the sum of all possible values of <math>N</ma

13 KB (2,024 words) - 16:07, 22 April 2024
2019 AIME II Problems/Problem 15
...ath>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</ma ...h>, and its angles are <math>\pi-2A</math>,<math>\pi-2B</math>, and <math>\pi-2C</math>. We thus have the three sides of the orthic triangle now.

7 KB (1,115 words) - 03:11, 7 January 2024
1993 IMO Problems
...nteger. Prove that <math>f(x)</math> cannot be expressed as the product of two nonconstant polynomials with integer coefficients. ...ide acute triangle <math>ABC</math> such that <math>\angle ADB=\angle ACB+\pi/2</math> and <math>AC\cdot BD=AD\cdot BC</math>.

2 KB (452 words) - 17:59, 24 March 2019
2020 AMC 10B Problems
...lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? ...f{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math>

16 KB (2,496 words) - 20:09, 19 May 2024
2015 UNM-PNM Statewide High School Mathematics Contest II Problems
...points are on the same line), then there is a line which contains exactly two of the Let <math>A</math> and <math>B</math> be two points in the plane. Describe the set <math>S</math> of all points in the p

4 KB (604 words) - 03:48, 12 January 2019
1983 AHSME Problems/Problem 22
Consider the two functions <math>f(x) = x^2+2bx+1</math> and <math>g(x) = 2a(x+b)</math>, wh \textbf{(B)} \ \pi \qquad

2 KB (267 words) - 00:59, 20 February 2019
2019 AMC 10A Problems/Problem 21
surface s=surface(f,(0,0),(pi,2pi),70,Spline); .../math>, so <math>\triangle{BIE} \sim \triangle{BDA}</math> since they have two equal angles.

7 KB (1,000 words) - 15:03, 23 October 2021
2019 AMC 10A Problems/Problem 7
...at <math>(2,2)</math>. What is the area of the triangle enclosed by these two lines and the line <math>x+y=10 ?</math> ...other solutions, solve the systems of equations to see that the triangle's two other vertices are at <math>(4, 6)</math> and <math>(6, 4)</math>. Then app

7 KB (1,079 words) - 22:24, 10 November 2023
2019 AMC 10B Problems
Alicia had two containers. The first was <math>\tfrac{5}{6}</math> full of water and the s Two jars each contain the same number of marbles, and every marble is either bl

15 KB (2,458 words) - 23:52, 12 November 2023
2019 AMC 10B Problems/Problem 20
<cmath>\frac{a}{b}\cdot\pi-\sqrt{c}+d,</cmath> The area of <math>A</math> is <math>\frac{1}{2} \pi \cdot 2^2 = 2\pi</math>.

6 KB (984 words) - 23:52, 11 November 2023
2019 AMC 10B Problems/Problem 23
<math>\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) } \frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}</math>

6 KB (967 words) - 10:25, 20 December 2023
2019 AMC 12B Problems
Alicia had two containers. The first was <math>\tfrac{5}{6}</math> full of water and the s ...gle <math>ABC</math> with leg length <math>1</math>, as shown, so that the two triangles have equal perimeters. What is <math>\sin(2\angle BAD)</math>?

16 KB (2,477 words) - 15:41, 9 September 2023
2019 AMC 12B Problems/Problem 17
...< \theta < \pi</math>. By symmetry, the interval <math>\pi \leq \theta < 2\pi</math> will also give <math>2</math> solutions. The answer is thus <math>2 ...rm an equilateral triangle, their difference in angle must be <math>\frac{\pi}{3}</math>, so

6 KB (987 words) - 19:19, 12 November 2022
2019 AMC 12B Problems/Problem 24
...c{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</math> Notice that <math>\omega=e^{\frac{2i\pi}{3}}</math>, which is one of the cube roots of unity. We wish to find the s

4 KB (729 words) - 21:23, 15 November 2022
2019 AIME I Problems/Problem 12
...ex numbers you subtract the angle of one from the other. Therefore, if the two complex numbers are perpendicular, the difference between their arguments w According to the question <math>\arg{Z_1}=\frac{\pi}{2}</math>.

8 KB (1,534 words) - 22:17, 28 December 2023
Euler's formula and It's history
π (the famous number pi that turns up in many interesting areas) And here is the miracle ... the two groups are quite neatly the Taylor Series for cos and sin

3 KB (543 words) - 15:24, 13 June 2019
What is the definition of Pure Mathematics?
...ve number would have an additive inverse. ... The fact that the product of two negatives is positive is therefore related to the fact that the inverse of A fraction a number that can be expressed as two numbers divided. For example, five divided by four is <math>\frac{5}{4}</ma

35 KB (5,882 words) - 18:08, 28 June 2021
2006 SMT/Team Problems/Problem 15
(Hint: <math> \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} </math>) <cmath>\prod_{primes} (1-p^{-2}) = \frac{1}{\xi(2)} = \frac{6}{\pi^2}</cmath>.

3 KB (445 words) - 18:37, 14 January 2020
2006 SMT/General Problems/Problem 14
...math>\frac{\pi \cdot r^2}{2} = \frac{\pi \cdot (\pi)^2}{2} = \boxed{\frac{\pi^3}{2}}</cmath>

544 bytes (99 words) - 18:36, 13 January 2020
Set of Complex Numbers
The set of complex numbers is a set of all numbers ever existed, from pi to 1. The set of all real numbers can be classified into two major sub sets, the set of irrational and rational numbers.

2 KB (270 words) - 16:28, 13 June 2022
Mock AMC 10B Problems
...or a boating trip. How many ways are there to split up the six boys if the two groups are indistinguishable? ...\sqrt{3})^{2019} \qquad \textbf{(C) }(3+\sqrt{2})^2 \qquad \textbf{(D) }(2\pi)^2 \qquad \textbf{(E) }(3+\sqrt{2})(3-\sqrt{2}) \qquad</math>

13 KB (2,059 words) - 02:59, 21 January 2021
2000 Pan African MO Problems/Problem 1
...h>a = -b</math> or <math>a^2 + b^2 + a - b = 0</math>. We can divide into two cases. ...ath> to get <math>\tan{x} = -1</math>. Thus, <math>x = \frac{3 \pi}{4} + \pi n</math>, where <math>n</math> is an integer.

2 KB (305 words) - 06:07, 23 February 2023
GMAAS
4. The Infinity Numeral, PI, is the second deity of the Almighty Gmaas's heaven. ...te dreadful. EDIT: He now eats gnats again. He thinks that they taste like pi(e). EDIT: He ate 3,141,592,653,589,793,238,462,643,383 so far.

85 KB (13,971 words) - 18:02, 16 May 2024
2000 SMT/Team Problems
How many times per day do at least two of the three hands on a clock coincide? ...eed <math>v_b</math>. What is the distance of closest approach between the two?

4 KB (618 words) - 13:33, 21 January 2020
2008 Indonesia MO Problems
...each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of <cmath> \frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}.</cmath>

3 KB (478 words) - 14:09, 23 June 2021
2020 AMC 12A Problems/Problem 12
In the isosceles right triangle, the two legs are congruent. We can therefore construct an isosceles right triangle ...</math> and equal to the original line segment. The difference between the two vectors is <math><2,8></math>, which is the slope <math>4</math>, and that

7 KB (1,145 words) - 20:27, 5 November 2023
2020 AMC 12B Problems/Problem 25
<cmath>\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1</cmath> <cmath>\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1</cmath>

8 KB (1,412 words) - 06:17, 30 December 2023
2020 AMC 10B Problems/Problem 10
...lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? ...f{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math>

3 KB (493 words) - 16:14, 11 September 2023
2020 AMC 10B Problems/Problem 14
...quad \textbf{(D) } 3\sqrt3 - \pi \qquad \textbf{(E) } \frac{9\sqrt3}{2} - \pi</math> ...ing vertex, and let point <math>C</math> be the second intersection of the two semicircles that pass through point <math>A</math>. Then, <math>BC = 1</mat

17 KB (2,392 words) - 12:36, 24 December 2023
2021 AMC 10A Problems
...chool has <math>3</math> times as many students as Lara's high school. The two high schools have a total of <math>2600</math> students. How many students ...r is erased, the other number is obtained. What is the difference of these two numbers?

15 KB (2,302 words) - 23:41, 14 April 2024
2020 AMC 12B Problems/Problem 17
...rt{3}}{2}</math> = <math>\cos(120^\circ) + i \cdot \sin(120^\circ) = e^{ 2\pi i / 3}</math>. ...p)</math> which are <math>(1,2,2)</math> and <math>(3,1,1)</math> yielding two possible polynomials. The answer is thus <math>\boxed{\textbf{(C) } 2}</mat

4 KB (726 words) - 16:55, 11 September 2023
2020 AMC 12B Problems/Problem 23
...th> to be equal to <math>e^{i\frac{2\pi}{3}}</math> and <math>e^{-i\frac{2\pi}{3}}</math>, meaning that all three are equally spaced along the unit circl ...e condition in the problem holds for some <math>n=k</math>. We can now add two points <math>z_{k+1}</math> and <math>z_{k+2}</math> anywhere on the unit c

2 KB (306 words) - 17:45, 28 January 2024
2020 AIME I Problems/Problem 1
A = (0, tan(3 * pi / 7)); ...{BAC}</math> to <math>x</math>, we can find all other angles through these two properties:

6 KB (968 words) - 15:01, 24 January 2024
2021 HMC 10
Dr.Jeck is arranging books on a bookshelf. How many ways are there to arrange two identical math books, three identical science books and four identical Engl ...4</math>, a quarter circle arc of radius <math>\sqrt{10}</math> intersects two sides that divided the side length into ratio <math>3:1</math>, as shown in

11 KB (1,745 words) - 16:44, 6 October 2021
2022 AMC 10A Problems
...o opposite corners of the index card and measures the distance between the two closest vertices of these squares to be <math>4\sqrt{2}</math> centimeters, ...</math> What is the sum of all real numbers <math>m</math> for which these two expressions have the same value?

18 KB (2,662 words) - 02:08, 9 March 2024
AMC 12C 2020 Problems
A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours ...\pi}{13}</math>, <math>\cos \frac{5\pi}{13}</math>, and <math>\cos \frac{7\pi}{13}</math>. What is the least possible sum of the coefficients of <math>P(

8 KB (1,223 words) - 15:02, 27 November 2022
Mathcounts Week
...[[Mathcounts Week Countdown Round]]. Most of the times of the activity are pi time (3:14 PM). ...not saving for the mock sprint and target competitions. There are at least two incidents of incorrect answers being given points in the "World Largest Cou

1 KB (207 words) - 17:51, 22 May 2020
1989 IMO Problems/Problem 5
...'true' powers and the primes. Then this would give that <math>\frac{p(n)+\pi(n)}{n} \geq \frac{1}{k}</math> for all <math>n</math> in contrary to the ab Edit: to elementarize the <math>\lim_{n \to \infty} \frac{\pi(n)}{n} = 0</math> part:

2 KB (390 words) - 21:58, 4 July 2021
2012 JBMO Problems/Problem 2
Let the circles <math>k_1</math> and <math>k_2</math> intersect at two points <math>A</math> and <math>B</math>, and let <math>t</math> be a commo ...>, <math>\angle PMB \cong \boxed {\angle NMB \cong 45^{\circ} \cong \frac{\pi}{4}}</math>.

2 KB (392 words) - 15:47, 11 January 2021
2021 AIME I Problems/Problem 7
...a</math> is <math>1</math>, is achieved at <math>\theta = \frac{\pi}{2}+2k\pi</math> for some integer <math>k</math>. ...that <math>mx = \frac{\pi}{2}+2a\pi</math> and <math>nx = \frac{\pi}{2}+2b\pi</math>, for integers <math>a, b</math>.

9 KB (1,523 words) - 09:12, 3 December 2023
2021 AMC 12A Problems
...r is erased, the other number is obtained. What is the difference of these two numbers? Two right circular cones with vertices facing down as shown in the figure below

15 KB (2,383 words) - 09:49, 25 June 2023
2021 AMC 12A Problems/Problem 13
...eta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> \textbf{(A)} & & 2 & & & \pi & & & &32\cos{(5\pi)}&=&32\cos\pi&=&32(-1)& & \\ [2ex]

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2023 AIME II Problems
...the set of all positive rational numbers <math>r</math> such that when the two numbers <math>r</math> and <math>55r</math> are written as fractions in low ...region formed by three unit squares joined at their sides, as shown below. Two points <math>A</math> and <math>B</math> are chosen independently and unifo

8 KB (1,370 words) - 21:34, 28 January 2024
2021 AMC 10B Problems
How many integer values of <math>x</math> satisfy <math>|x| < 3\pi</math>? ...he cousins' ages multiplied together give <math>24</math>, while the other two multiply to <math>30</math>. What is the sum of the ages of Jonie's four co

17 KB (2,418 words) - 12:52, 5 November 2023
1955 AHSME Problems/Problem 34
...xtbf{(C)}\ 12\sqrt{3}+14\pi\\ \textbf{(D)}\ 12+15\pi\qquad\textbf{(E)}\ 24\pi </math> We can separate the total length of the wire into two sections: the straight and the curve.

2 KB (360 words) - 22:07, 16 August 2020
2022 AIME I Problems/Problem 11
...th> and <math>CY=20</math>. We can compute the area of <math>ABC</math> in two ways: Equating the two expressions for <math>[ABC]</math> and solving for <math>r</math> yields <m

16 KB (2,517 words) - 20:22, 31 January 2024
2022 AIME I Problems/Problem 14
...passes through <math>P</math> and divides <math>\triangle ABC</math> into two polygons of equal perimeter. Let <math>\triangle ABC</math> be a triangle w ...th>r^2+r+1=(r-\omega)(r-{\omega}^2)</math>, where <math>\omega=e^{i\frac{2\pi}{3}}</math>. Thus,

16 KB (2,730 words) - 02:56, 4 January 2023
2022 AIME I Problems/Problem 15
We can symmetrically apply this to the two other equations/triangles. Taking the inverse sine (<math>0\leq\theta\frac{\pi}{2}</math>) of each equation yields a simple system:

15 KB (2,208 words) - 01:25, 1 February 2024
2019 CIME I Problems/Problem 12
...<math>75\pi+50</math>. The greatest integer less than or equal to <math>75\pi+50</math> is <math>\boxed{285}</math>.

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2022 AMC 12B Problems
...constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots? How many of these sets contain exactly two multiples of <math>7</math>?

15 KB (2,233 words) - 13:02, 10 November 2023
2021 Fall AMC 10B Problems
...has risen by <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possibl ...s <math>15</math>. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

16 KB (2,450 words) - 00:13, 12 November 2023
2021 Fall AMC 10A Problems
Mr. Lopez has a choice of two routes to get to work. Route A is <math>6</math> miles long, and his averag ...sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

14 KB (2,162 words) - 21:33, 2 November 2023
1978 IMO Problems/Problem 2
...> We construct three segments <math>PA, PB,PC</math>, perpendicular two by two<math>,</math> with the vertexes <math>A, B, C</math> on the sphere<math>.</ ...| PB \right|^{2} - 2 \left| OP \right| \left| PB \right| cos \left( \frac{\pi}{2}-\theta \right)</math>

6 KB (1,039 words) - 13:00, 13 March 2024
2020 IMO Problems/Problem 1
...dicular bisector of <math>AP,BP</math> meet at point <math>O</math>, those two lines meet at <math>AD,BC</math> at <math>N,M</math> respectively. <cmath>\angle APE = \alpha, \angle PEA = \pi - 2\alpha, \angle ABP = 2\alpha \implies</cmath>

3 KB (464 words) - 08:29, 28 September 2023
2021 AMC 12A Problems/Problem 10
Two right circular cones with vertices facing down as shown in the figure below \textbf{Narrow Cone} & 3 & h_1 & & \frac13\pi(3)^2h_1=3\pi h_1 & \\ [2ex]

9 KB (1,503 words) - 15:09, 1 August 2023
2021 AMC 12A Problems/Problem 19
...ft( \frac{\pi}2 \sin x\right)</math> have in the closed interval <math>[0,\pi]</math>? ...<math>\frac{\pi}2 \cos x</math> are both <math>\left[-\frac{\pi}2, \frac{\pi}2 \right],</math> which is included in the range of <math>\arcsin,</math> s

7 KB (1,110 words) - 20:10, 5 November 2022
2020 INMO Problems/Problem 5
[*] also suppose two consecutive lines are one unit apart to each other . ...that <math>l_k \equiv \ Im ({P_k})=y \cos \frac{2\pi k}{n} +x \sin \frac{2\pi k}{n}</math> .

3 KB (607 words) - 03:12, 22 September 2023
2021 AMC 12B Problems
How many integer values of <math>x</math> satisfy <math>|x|<3\pi?</math> Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is <ma

15 KB (2,302 words) - 12:31, 27 October 2023
2021 April MIMC 10
A worker cuts a piece of wire into two pieces. The two pieces, <math>A</math> and <math>B</math>, enclose an equilateral triangle ...s into <math>3</math> different boxes such that each box contains at least two balls, and that no box can contain <math>7</math> or more balls. Find the n

15 KB (2,366 words) - 17:45, 19 September 2021
2021 MECC Mock AMC 10
...they just had a fight. Find the number of ways the coach can put them into two such groups. ...f{(C)} ~\frac{(2-\sqrt{2})\pi}{8-2\pi} \qquad\textbf{(D)} ~\frac{\pi}{32-8\pi}\qquad</math>

12 KB (1,915 words) - 17:38, 29 April 2021
2021 CMC 12A Problems
...of <math>7\pi</math> and the total area covered by the circles is <math>25\pi</math>. What is the value of <math>r</math>? There are <math>8</math> scoops of ice cream, two of each flavor: vanilla, strawberry, cherry, and mint. If scoops of the sam

5 KB (776 words) - 09:35, 8 August 2023
2021 CMC 12A Problems/Problem 2
...of <math>7\pi</math> and the total area covered by the circles is <math>25\pi</math>. What is the value of <math>r</math>? ...ac{25\pi+7\pi}{2}=16\pi.</cmath> The radius of a circle with area <math>16\pi</math> is <math>\boxed{\textbf{(C) } 4}</math>.

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2022 AMC 12A Problems
has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written ...erings of the cards will the <math>13</math> cards be picked up in exactly two passes?

15 KB (2,250 words) - 00:32, 9 March 2024
2021 AMC 10B Problems/Problem 7
...\qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi</math> ...ius <math>1,</math> in which we do not need to subtract more. That is, the two smallest circles are on the same side of <math>\ell,</math> but can be on e

2 KB (363 words) - 18:22, 16 August 2022
2021 AMC 12B Problems/Problem 1
How many integer values of <math>x</math> satisfy <math>|x|<3\pi</math>? Since <math>3\pi\approx9.42</math>, we multiply <math>9</math> by <math>2</math> for the int

4 KB (581 words) - 23:48, 18 July 2023
2021 AMC 12B Problems/Problem 6
...the volume of the water is <math>\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi</math>. ...> so the volume of the water in the cylinder would be <math>24\cdot24\cdot\pi\cdot h</math>.

2 KB (405 words) - 23:51, 18 July 2023
2021 AMC 12B Problems/Problem 20
z &= e^{i 0}, e^{i \frac{2\pi}{3}}, e^{i\left(-\frac{2\pi}{3}\right)}. Obviously the right two solutions are the roots of <math>z^2 + z + 1 = 0.</math>

8 KB (1,315 words) - 11:43, 24 October 2023
2021 AMC 12B Problems/Problem 13
How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy <cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath> We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </

2 KB (250 words) - 18:57, 19 September 2023
1976 AHSME Problems/Problem 8
from the point to the origin is at most two units? \textbf{(D) }\frac{\pi}{16}\qquad

519 bytes (83 words) - 19:07, 15 March 2024
Laurent Series
<center><math>C_m=\mathcal{C}(t;z_0+r_me^{it},0,2\pi)</math></center> <center><math>2\pi f(w)=\oint_{C_2}\frac{f(z)}{z-w}\,dz-\oint_{C_1}\frac{f(z)}{z-w}\,dz</math>

8 KB (1,471 words) - 22:02, 12 April 2022
2021 GCIME Problems
...ss than or equal to <math>n</math>. Suppose <math>\pi(a)^{\pi(b)}=\pi(b)^{\pi(a)}=c</math>. For some fixed <math>c</math> what is the maximum possible nu ...the two are related by the formula <math>F=\tfrac{9}{5}C+32</math>. When a two-digit integer degree temperature <math>n</math> in Celcius is converted to

4 KB (651 words) - 20:18, 6 March 2021
2021 Fall MIMC 10
A worker cuts a piece of wire into two pieces. The two pieces, <math>A</math> and <math>B</math>, enclose an equilateral triangle ...s into <math>3</math> different boxes such that each box contains at least two balls, and that no box can contain <math>7</math> or more balls. Find the n

14 KB (2,226 words) - 23:39, 12 September 2021
2021 April MIMC 10 Problems/Problem 23
...> and line segments <math>BO</math> and <math>GH</math>) has area <math>25\pi</math>. Furthermore, another quarter circle <math>DOF</math> formed by arc ...extbf{(E)} ~\frac{50\pi^2+700\pi\sqrt{2}+3001\pi-70\sqrt{2}+60}{2\pi^2+240\pi+6920}\qquad</math>

3 KB (520 words) - 14:04, 26 April 2021
2021 April MIMC 10 Problems/Problem 24
...i} \qquad\textbf{(D)} ~\frac{13}{28+6\pi} \qquad\textbf{(E)} ~\frac{13}{30\pi}\qquad</math> ...atio would be <math>\frac{26}{56+12\pi}=\boxed{\textbf{(E)} \frac{13}{28+6\pi}}</math>.

3 KB (599 words) - 14:05, 26 April 2021
2021 April MIMC 10 Problems/Problem 25
<math>\textbf{(A)} ~4 \qquad\textbf{(B)} ~\frac{12}{\pi} \qquad\textbf{(C)} ~\frac{2+3\sqrt{3}}{2} \qquad\textbf{(D)} ~\frac{4+\sqr ...d intersection of <math>\frac{6}{\pi}\cdot2=\boxed{\textbf{(B)} \frac{12}{\pi}}</math>.

1 KB (228 words) - 14:06, 26 April 2021
2021 April MIMC 10 Problems/Problem 18
<math>\textbf{(A)}</math> Two overlapping circles with each area <math>2\pi</math>. <math>\textbf{(B)}</math> Four not overlapping circles with each area <math>4\pi</math>.

1 KB (223 words) - 13:54, 26 April 2021
2021 GMC 12
Given that the two roots of polynomial <math>x^{2}-ax+\frac{1}{2}</math> are <math>\sec(n)</ma ...} \qquad\textbf{(D)} ~\frac{\pi+2\sqrt{3}-3}{4} \qquad\textbf{(E)} ~\frac{\pi+3-2\sqrt{2}}{4}</math>

13 KB (2,097 words) - 17:38, 29 April 2021
2021 GMC 12B
...LL</math> civilization, each number digits of a number will be replaced by two times of the digit. For example: <math>1234</math> in <math>MICWELL</math> Statement 4: The length of a diagonal is the product of two adjacent sides.

11 KB (1,691 words) - 18:56, 25 April 2022
2022 AIME II Problems/Problem 11
<math>\triangle AMB' \sim \triangle MDC'</math> by two angles. ...</math>. Then <math>\angle{AMD}=\pi-(a+b), \angle{ABM}=\pi-c, \angle{DCM}=\pi-c</math>, implying that <math>\angle{BMA}+\angle{CMD}=a+b</math>, implying

14 KB (2,254 words) - 18:26, 8 February 2024
2022 AIME II Problems/Problem 15
Two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> <math>\angle B'O_2 D > \frac{\pi}{2},</math> hence <math>\cos \angle B'O_2 D = \cos 2\alpha = -\frac{3}{5}.<

14 KB (2,217 words) - 00:28, 29 June 2023
Cavalieri's principle
If the cross sections of two 2D objects at each height have the same length, the areas of the 2D objects If the cross sections of two 3D objects at each height have the same area, the volumes of the 3D objects

1 KB (167 words) - 22:01, 27 August 2021
2022 AMC 10B Problems
...constant <math>k</math> will the polynomial <math>x^{2}+kx+36</math> have two distinct integer roots? How many of these sets contain exactly two multiples of <math>7</math>?

15 KB (2,224 words) - 13:10, 20 February 2024
2021 JMPSC Sprint Problems
What two-digit even number has digits that sum to <math>17</math>? How many positive two-digit numbers exist such that the product of its digits is not zero?

7 KB (1,100 words) - 18:40, 11 July 2021
2021 IMO Problems/Problem 4
...h>. Connect <math>IM,IT,IT,IZ,IX,IN,IJ,IF</math> separately, we can create two pairs of congruent triangles. In <math>\triangle{XIF},\triangle{YIM}</math> <cmath>\overset{\Large\frown} {XY}= 2\angle XAY = 2\pi - 4\alpha,</cmath>

7 KB (1,196 words) - 10:30, 18 June 2023
Jadhav Sin Function Theorem
'''then <math>y = (2n-1)\pi^c + x </math> or <math>y = (2n-1)180^\circ - x </math>''' ...nstant in the theorem <math>\pi^c </math>/<math>180^\circ </math>. <math>\pi^c </math> is used if the angle <math>x </math> is in [https://en.wikipedi

2 KB (374 words) - 03:28, 29 July 2021
2021 Fall AMC 12B Problems/Problem 9
...\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi</math> The area is therefore <math> \pi r^2 = \boxed{\textbf{(B)}\ 12\pi}</math>.

5 KB (765 words) - 22:33, 10 July 2023
2021 Fall AMC 10A Problems/Problem 19
...h>2A</math>. The value of <math>s</math> can be written as <math>a+\frac{b\pi}{c}</math>, where <math>a,b</math>, and <math>c</math> are positive integer real s = 5 + pi/4;

3 KB (475 words) - 17:45, 30 October 2023
2021 Fall AMC 10B Problems/Problem 11
determined by a side of the hexagon is reflected over that side. What is the area of the region ...\qquad(\textbf{D}) \: \pi - \frac{\sqrt{3}}{2}\qquad(\textbf{E}) \: \frac{\pi + \sqrt{3}}{2}</math>

6 KB (1,021 words) - 19:40, 1 November 2023
2021 Fall AMC 12A Problems
Mr. Lopez has a choice of two routes to get to work. Route A is <math>6</math> miles long, and his averag Consider two concentric circles of radius <math>17</math> and <math>19.</math> The large

15 KB (2,452 words) - 19:37, 7 June 2023
2021 Fall AMC 12B Problems
...has risen by <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possibl ...s <math>15</math>. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

14 KB (2,191 words) - 19:57, 12 November 2023
2021 WSMO Speed Round Problems
...c}</math>. If the volume the square sweeps out can be expressed as <math>m\pi</math>, find <math>m</math>. The number of ways to arrange the characters in "delicious greenbeans" into two separate strings of letters can be expressed as <math>a\cdot b!,</math> whe

4 KB (707 words) - 12:38, 6 June 2022
Chebyshev polynomials of the first kind
==Proof of equivalence of the two definitions== ...gives <math>2mk\pi \pm mny</math>; taking the cosine gives <math>\cos (2mk\pi \pm mny) = \cos mny = \cos ( mn (\arccos x)) = T_{mn}(x)</math>.

10 KB (1,919 words) - 15:24, 26 June 2023
Chebyshev polynomials of the second kind
==Proof of equivalence of the two definitions== ...</math> and the fact that the range of <math>\arccos x</math> is <math>[0,\pi]</math>, on which <math>\sin y</math> is nonnegative.

2 KB (392 words) - 22:12, 11 March 2022
2022 AMC 10A Problems/Problem 23
...h> over the perpendicular bisector of <math>\overline{BC}</math>, creating two new isosceles trapezoids <math>DAPP^{\prime}</math> and <math>CBPP^{\prime} ...}\cdot AD=15</math> and <math>PP^{\prime}\cdot BC=5</math>; dividing these two equations and taking the reciprocal yields <math>\frac{BC}{AD}=\boxed{\text

12 KB (1,806 words) - 12:52, 26 March 2024
2022 AMC 12A Problems/Problem 22
...th> be a real number, and let <math>z_1</math> and <math>z_2</math> be the two complex numbers satisfying the equation ...ce between the two lines, is the difference between the real values of the two bases <math>\implies h= a-\frac{a}{10}=\frac{9a}{10}</math>.

7 KB (1,164 words) - 11:20, 1 January 2024
2022 AMC 10A Problems/Problem 15
...but exterior to the quadrilateral can be written in the form <math>\frac{a\pi-b}{c},</math> where <math>a,b,</math> and <math>c</math> are positive integ ...2\cdot AD\cdot DC = \frac{625\pi}{4}-\frac{168}{2}-\frac{300}{2}=\frac{625\pi-936}{4}.</cmath>

5 KB (768 words) - 21:00, 21 February 2024
2022 AMC 12A Problems/Problem 17
has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written Since <math>\sin{x} \ne 0</math> as it is on the open interval <math>(0, \pi)</math>, we can divide out <math>\sin{x}</math> from both sides, leaving us

6 KB (1,001 words) - 21:27, 17 March 2024
2022 AMC 10A Problems/Problem 21
//Comment two lines below to remove red edges ...tersection by symmetry. Therefore, the triangle that is wedged between the two hexagons has the same angle as the square at the bottom wedged between the

13 KB (2,080 words) - 11:27, 25 October 2023
2022 AMC 12A Problems/Problem 13
...o semicircles will be formed with a radius of 1 for a total area of <math>\pi \approx 3</math>. Therefore, the total area is <math>5(2) + \pi \approx 10 + 3 = \boxed{\textbf{(A) } 13}</math>.

2 KB (344 words) - 15:41, 25 October 2023
2022 AMC 10B Problems/Problem 22
...\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad</math> There are four cases with two circles belonging to each:

7 KB (1,202 words) - 01:15, 10 June 2023
2022 AMC 12B Problems/Problem 11
-1 + i\sqrt{3} &= 2e^{\frac{2\pi i}{3}}, \\ -1 - i\sqrt{3} &= 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}.

5 KB (866 words) - 22:17, 27 October 2023
2021 EGMO Problems
(ii) no two points in <math>P</math> lie on a line through the origin. ...real number <math>x</math>. Thus <math>\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,</math> and <math>

3 KB (492 words) - 14:07, 24 December 2022
2023 AMC 12B Problems
...) of a <math>3 \times 3</math> grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is cover ...ath>-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

13 KB (2,107 words) - 22:19, 20 April 2024
2023 AMC 8 Problems/Problem 12
...he total area of the radius <math>3</math> circle is simply just <math>9* \pi</math> when using our area of a circle formula. ...\frac{1}{4} \pi + 4 \pi -\pi - \pi = \frac{3}{4} \pi + 2 \pi = \frac{11 * \pi}{4}</math>.

3 KB (453 words) - 17:31, 18 May 2024
Random Problem
...maximum possible volume of this cylinder in the form of <math>\frac{a}{b}\pi</math>? <math>V=\pi{r^2}h</math>

4 KB (697 words) - 17:07, 24 March 2023

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