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Number theory/Advanced
...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the

5 KB (849 words) - 15:14, 18 May 2021
Chain Rule
The theorem states that if <math>h(x) = f(g(x))</math>, then <math>h'(x)=f'(g(x))\cdot g'(x)</math> wherever those expressions make sense. ...ath>g(x)=x^2</math>, and <math>h(x)=f(g(x))=\sin{(x^2)}</math>, then <math>h'(x) = \cos{(x^2)}\cdot(2x)</math>.

12 KB (2,377 words) - 10:48, 22 July 2009
Mock AMC
...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==

51 KB (6,135 words) - 05:44, 2 November 2024
2006 AIME I Problems
{{AIME Problems|year=2006|n=I}} In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat

7 KB (1,173 words) - 02:31, 4 January 2023
2006 AIME I Problems/Problem 14
...f the tripod from the ground when the broken tripod is set up. Then <math> h </math> can be written in the form <math> \frac m{\sqrt{n}}, </math> where triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0);

6 KB (980 words) - 20:45, 31 March 2020
2006 AIME I Problems/Problem 8
...area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math ..., D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);

5 KB (730 words) - 14:05, 15 January 2024
2006 AMC 10A Problems
{{AMC10 Problems|year=2006|ab=A}} Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \tex

13 KB (2,028 words) - 15:32, 22 March 2022
2005 AIME I Problems/Problem 5
...e, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the numbe ...ifferent positions, and then there is also the possibility that it doesn’t occur at all, for <math>9</math> total configurations. Thus, the answer is

5 KB (830 words) - 00:51, 1 March 2023
2002 AIME I Problems
{{AIME Problems|year=2002|n=I}} ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-

8 KB (1,374 words) - 20:09, 27 July 2023
2001 AIME II Problems
...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb

8 KB (1,282 words) - 20:12, 19 February 2019
1984 AIME Problems/Problem 9
...math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the [[volume]] of the tetrahedron in <mat path3 rightanglemark(triple A, triple B, triple C, real s=8)

6 KB (947 words) - 19:44, 26 November 2021
1995 AIME Problems/Problem 15
...in, one will encounter a run of <math>5</math> heads before one encounters a run of <math>2</math> tails. Given that <math>p_{}</math> can be written i ...h>3/2</math> of the probability given that the sequence starts with an <tt>H</tt>.

7 KB (1,087 words) - 12:09, 17 November 2024
1995 AIME Problems/Problem 12
// projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C)

8 KB (1,172 words) - 20:57, 22 September 2022
2014 USAJMO Problems
...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...ngle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</mat

3 KB (600 words) - 15:42, 5 August 2023
Dedekind domain
A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].

9 KB (1,648 words) - 15:36, 14 October 2017
Mock AIME 1 2006-2007 Problems/Problem 14
...</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>. Let <math>P</math> be the point such that circle <math>O</math> is ...ee below an attempted solution to understand why this problem doesn't have a solution:

3 KB (541 words) - 16:32, 22 November 2023
2007 iTest Problems
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>

30 KB (4,794 words) - 22:00, 8 May 2024
1987 IMO Problems/Problem 6
..., then we can conclude that <math>m</math> is a prime. Since there must be a factor of <math>m</math> less than <math>\sqrt{m}</math>. ...en we can write <math>n=3(r+\epsilon)^2(0\leq\epsilon< 1)</math>, so <math>h=6r\epsilon+3\epsilon^2\leq 6r+3</math>).

2 KB (430 words) - 12:03, 24 February 2024
2005 AMC 10A Problems
{{AMC10 Problems|year=2005|ab=A}} <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \

14 KB (2,026 words) - 10:45, 12 July 2021
Euler line
...fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]s hold among these points. In particular, <math>\overline{OGNH}</math> and < ...s examine what else this transformation, which we denote as <math>\mathcal{S}</math>, will do.

59 KB (10,203 words) - 03:47, 30 August 2023

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