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Page title matches

• USA(J)MO
  67 bytes (8 words) - 08:30, 13 July 2010
• J. E. Littlewood
  little-wood
  11 bytes (1 word) - 19:53, 1 July 2020
• User:J-y2009/sandbox
  ...\left(x_1y_2+x_2y_3+...x_{n-1}y_n+x_ny_1\right)-\left(y_1x_2+y_2x_3+...y_{n-1}x_n+y_nx_1\right)\right|</cmath>. ...<math>C(x_3, y_3)</math> is <math>\frac{|x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2|}{2}</math>.
  7 KB (1,213 words) - 20:05, 31 January 2024

Page text matches

• Resources for mathematics competitions
  * Challenge Math Online: http://www.noetic-learning.com/gifted ...ration Open]Elementary/middle school contests: https://www.beestar.org/math-competition/
  24 KB (3,269 words) - 00:43, 24 April 2024
• Math books
  ...ing Started with Competition Math], a textbook meant for true beginners (on-target middle school students, or advanced elementary school students). It i ...+maths+kawasaki&qid=1581288606&sprefix=after+school+maths+%2Caps%2C268&sr=8-2 100 Challenging Maths Problems]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Chemistry competitions
  The U.S. National Chemistry Olympiad national exam (USNCO) is a 3-part, 4 hour and 45 minute exam administered in mid or late April by ACS Loc ...www.amazon.com/gp/product/0618221565/ref=pd_lpo_k2_dp_k2a_1_img/102-5655201-2084940?%5Fencoding=UTF8 ''Chemistry''] by Steven S. Zumdahl, Susan A. Zumda
  2 KB (258 words) - 19:31, 8 March 2023
• Inequality
  ...b</math> if <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive. ...b</math> if <math>a</math> is smaller than <math>b</math>, that is, <math>a-b</math> is negative.
  12 KB (1,798 words) - 16:20, 14 March 2023
• 2015 IMO Problems
  <math>\mathcal{S}</math> is <i>centre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</ma <ol style="list-style-type: lower-latin;">
  4 KB (692 words) - 22:33, 15 February 2021
• Proof writing
  ...wider population. Students who spend time studying maths can develop proof-writing skills over time. ...proof|two-column format]], as favored by many geometry teachers. In higher-level mathematics (taken as meaning an advanced undergraduate level of mathe
  3 KB (502 words) - 18:16, 18 January 2016
• Simon's Favorite Factoring Trick
  ...bles in a linear term on the other side. An example would be: <cmath>xy+66x-88y=23333</cmath>where <math>23333</math> is the constant term, <math>xy</ma ...e previous example, <math>xy+66x-88y=23333</math> is the same as: <cmath>(x-88)(y+66)=(23333)+(-88)(66)</cmath>
  7 KB (1,107 words) - 07:35, 26 March 2024
• Pigeonhole Principle
  ...th> where <math>i>j</math>, we may multiply <math>2^jb</math> by <math>2^{i-j}</math> to get <math>2^ib</math>, as desired. <math>\square</math> ...math> and <math>n</math> possible remainders (namely, <math>0, 1, \ldots, n-1</math>) modulo <math>n</math>. Then by the pigeonhole principle, there exi
  11 KB (1,985 words) - 21:03, 5 August 2023
• Cauchy-Schwarz Inequality
  ...ormulations in abstract algebra, calculus, and contest mathematics. In high-school competitions, its applications are limited to elementary and linear a ...intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
  13 KB (2,048 words) - 15:28, 22 February 2024
• Fermat's Little Theorem
  ...ath> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>. ...> denotes [[Euler's totient function]]. In particular, <math>\varphi(p) = p-1</math> for prime numbers <math>p</math>. In turn, this is a special case o
  16 KB (2,675 words) - 10:57, 7 March 2024
• Brute forcing
  1 KB (190 words) - 13:22, 5 May 2023
• Real number
  Let <math>\alpha\subset\mathbb{Q}</math> be non-empty
  3 KB (496 words) - 23:22, 5 January 2022
• Pi
  ...imes(0,1)}\frac {x^ky^\ell}{1-xy}\,dx\,dy</math>. Expanding <math>\frac 1{1-xy}=\sum_{j=0}^\infty x^jy^j</math> and integrating term by term, we get <ma <math>I(k,\ell)=\frac 1{\ell-k}\left[\frac 1{k+1}+\frac 1{k+2}+\dots+\frac 1\ell\right]</math>
  8 KB (1,469 words) - 21:11, 16 September 2022
• Chain Rule
  Here are some more precise statements for the single-variable and multi-variable cases. ===Multi-dimensional Chain Rule===
  12 KB (2,377 words) - 11:48, 22 July 2009
• Muirhead's Inequality
  ...an inequality ''can'' be proved with AM-GM before demonstrating the full AM-GM proof. ...</math> from the [[AM-GM Inequality#Weighted AM-GM Inequality|(weighted) AM-GM inequality]].
  8 KB (1,346 words) - 12:53, 8 October 2023
• Vector
  ...ected line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abst ...h (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at
  7 KB (1,265 words) - 13:22, 14 July 2021
• Area
  ...math>c</math> are the lengths of the sides and <math>s</math> is the [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>. ...>, where <math>r</math> is the radius of the [[incircle]] and s is the semi-perimeter.
  6 KB (1,181 words) - 22:37, 22 January 2023
• Improper fractional base
  787 bytes (118 words) - 19:20, 23 October 2010
• Newton's Sums
  <center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> <cmath>a_nP_1 + a_{n-1} = 0</cmath>
  4 KB (690 words) - 13:11, 20 February 2024
• Fundamental Theorem of Algebra
  <cmath> \lvert P(z) \rvert \ge \lvert z^n \rvert - (R-1) \lvert z^{n-1} \rvert = \lvert z^{n-1} \rvert \cdot \bigl[ \lvert z \rvert - (R-1) \bigr]
  5 KB (832 words) - 14:22, 11 January 2024
• ...bda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...he left-hand side of the desired inequality but may contribute to the right-hand side.
  4 KB (774 words) - 12:12, 29 October 2016
• Mock AMC
  Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to {| class="wikitable" style="text-align:center;width:100%"
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2006 AIME I Problems
  Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </ma A collection of 8 cubes consists of one cube with edge-length <math> k </math> for each integer <math> k, 1 \le k \le 8. </math> A
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possi {{AIME box|year=2006|n=I|num-b=14|after=Last Question}}
  6 KB (910 words) - 19:31, 24 October 2023
• 2006 AIME I Problems/Problem 13
  ...For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 ...math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math>
  10 KB (1,702 words) - 00:45, 16 November 2023
• LaTeX:Symbols
  ....)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list of supported commands]. [http://mirrors.ctan.org/info/symbols/comprehensive/symbols-a4.pdf The Comprehensive LaTeX Symbol List].
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AIME I Problems/Problem 10
  ...will not work in general). Denote the intersection of the line and the [[x-axis]] as <math>(x, 0)</math>. ...ezoid]] and a [[rectangle]], and the areas are <math>\frac{1}{2}((1-x) + (2-x))(3)</math> and <math>2 \cdot 1 = 2</math>, totaling <math>\frac{13}{2} -
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 8
  ...ath>44.5^2</math> is also less than <math>2006</math>, so we have numbers 1-44, times two because 0.5 can be added to each of time, plus 1, because 0.5 ...have <math>[GIHJ]=0,</math> implying the minimum possible positive-integer-valued area is <math>1.</math>
  5 KB (730 words) - 15:05, 15 January 2024
• 2021 JMC 10
  <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</m ...h>74</math> and <math>83</math> are pretentious. How many pretentious three-digit numbers are odd?
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  ...1.99</math>, and <math>\textdollar0.99</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the <ma How many even three-digit integers have the property that their digits, read left to right, are
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2001 AMC 12 Problems
  two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit o A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2006 AMC 12B Problems/Problem 14
  {{AMC12 box|year=2006|ab=B|num-b=13|num-a=15}}
  1 KB (227 words) - 17:21, 8 December 2013
• 2006 AMC 12B Problems/Problem 25
  A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> is a 1-1 correspondence between the odd and even numbers less than and relatively p
  5 KB (924 words) - 12:02, 15 June 2022
• 2005 AMC 12B Problems/Problem 24
  ...ive x- axis, the answer is <math>\dfrac{\tan(BOJ) + \tan(BOJ+60) + \tan(BOJ-60)}{2}</math>. {{AMC12 box|year=2005|ab=B|num-b=23|num-a=25}}
  4 KB (761 words) - 09:10, 1 August 2023
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. Let <cmath>P(x) = c_nx^n + c_{n-1}x^{n-1} + \dots + c_1x + c_0 = \sum_{j=0}^{n} c_jx^j</cmath> with all <math>c_j
  8 KB (1,427 words) - 21:37, 13 March 2022
• Prime ideal
  967 bytes (176 words) - 18:08, 7 April 2012
• 2005 AIME II Problems/Problem 12
  ...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);dr ...*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), O=(4.5,4.5), G=O+(E-O)*dir(-90), J=O+(F-O)*dir(-90); draw(A--B--C--D--A);draw(E--O--F);draw(G--O--J);draw(F--G,linet
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME I Problems/Problem 13
  ...n range from <math>l = 5</math> to <math>9</math>; notice that <math>d = 10-l</math> gives the number of diagonals. Let <math>R</math> represent a move ...jacent. The <math>D</math>'s split the string into three sections (<math>-D-D-</math>): by the [[Pigeonhole principle]] all of at least one of the two l
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME II Problems/Problem 14
  ...from Pythagorean theorem, <math>SC = \frac{18}{5}</math> and <math>AS = 14-SC = \frac{52}{5}</math>. We can also use the Pythagorean theorem on triangl ...and <math>RE</math> as <math>x</math>. <math>RB</math> then equals <math>13-y</math>. Then, we have two similar triangles.
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 7
  where the left-hand sum can be computed from: <center><math>\sum_{i=1}^{15} S_i = S_{15} + \left(\sum_{i=1}^{7} S_{2i-1} + S_{2i}\right) = -15 + 7 = -8</math></center>
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME II Problems/Problem 10
  ...math> is clearly never divisible by 9, so we can just focus on <math>1+2^{y-x}</math>. ...1+2^{y-x} \equiv 0 \pmod 9 \implies 2^{y-x} \equiv -1 \pmod 9 \implies 2^{y-x} \equiv 8 \pmod 9</cmath>
  7 KB (1,091 words) - 18:41, 4 January 2024
• 2019 AIME I Problems/Problem 2
  ...\ne 0</math> because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry. The probability that <math>B-J = 1</math> is <math>\frac{19}{20 \times 19} = \frac{1}{20}</math> because
  5 KB (830 words) - 22:15, 28 December 2023
• 1989 AIME Problems
  <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  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
• 1998 AIME Problems
  7 KB (1,084 words) - 02:01, 28 November 2023
• 2000 AIME II Problems
  ...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{n
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...the largest number of students who could study both languages. Find <math>M-m</math>. x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2003 AIME II Problems
  ...th>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? ...ne{EF,}</math> <math>\overline{CD}\parallel \overline{FA},</math> and the y-coordinates of its vertices are distinct elements of the set <math>\{0,2,4,6
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1985 AIME Problems/Problem 10
  ...changes when <math>\{x\}=\frac{m}{n}</math>, where <math>m\in\{1,2,3,...,n-1\}</math>, <math>n\in\{2,4,6,8\}</math>. Using [[Euler's Totient Function]] ...6x\rfloor+\lfloor 8x\rfloor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 5
  {{AIME box|year=1985|num-b=4|num-a=6}}
  2 KB (410 words) - 13:37, 1 May 2022
• 1986 AIME Problems/Problem 11
  The [[polynomial]] <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_ ...<math>1 - x + x^2 + \cdots - x^{17} = \frac {1 - x^{18}}{1 + x} = \frac {1-x^{18}}{y}</math>. Since <math>x = y - 1</math>, this becomes <math>\frac {1
  6 KB (872 words) - 16:51, 9 June 2023
• 1987 AIME Problems/Problem 7
  {{AIME box|year=1987|num-b=6|num-a=8}}
  3 KB (547 words) - 22:54, 4 April 2016
• 1989 AIME Problems/Problem 5
  ...rac{1}{3}</math>. The answer we are looking for is <math>{5\choose3}(h)^3(1-h)^2 = 10\left(\frac{1}{3}\right)^3\left(\frac{2}{3}\right)^2 = \frac{40}{24 {{AIME box|year=1989|num-b=4|num-a=6}}
  2 KB (258 words) - 00:07, 25 June 2023
• 1990 AIME Problems/Problem 9
  ...ing this pattern, we find that there are <math>\sum_{i=6}^{11} {i\choose{11-i}} = {6\choose5} + {7\choose4} + {8\choose3} + {9\choose2} + {{10}\choose1} ...th>n-1</math> flips must fall under one of the configurations of <math>S_{n-1}</math>.
  3 KB (425 words) - 19:31, 30 July 2021
• 1991 AIME Problems/Problem 3
  ...>. Taking logarithms in both sides of this last equation and using the well-known fact <math>\log(a_{}^{}b)=\log a + \log b</math> (valid if <math>a_{}^ ...t[\prod_{j=1}^{k}\frac{(N-j+1)x}{j}\right]=\sum_{j=1}^{k}\log\left[\frac{(N-j+1)x}{j}\right]\, .
  5 KB (865 words) - 12:13, 21 May 2020
• 2006 AMC 10B Problems/Problem 22
  {{AMC10 box|year=2006|ab=B|num-b=21|num-a=23}}
  2 KB (394 words) - 00:51, 25 November 2023
• 1993 AIME Problems/Problem 13
  ...h>, <math>CX=AC\cdot\left(\frac{CD}{AB-CD}\right)=200\cdot\left(\frac{t}{3t-t}\right)=100</math>. ...f this line with <math>AB</math> as <math>S</math>. Then <math>SB=AB-AS=3t-t=2t</math>.
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1994 AIME Problems/Problem 4
  ...th> such that <math>\lfloor\log_2{a}\rfloor=j</math>, and there are <math>n-2^k+1</math> such integers such that <math>\lfloor\log_2{a}\rfloor=k</math>. ...3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor= \sum_{j=0}^{k-1}(j\cdot2^j) + k(n-2^k+1) = 1994</math>.
  2 KB (264 words) - 13:33, 11 August 2018
• 1995 AIME Problems/Problem 12
  ...>AP = 1.</math> It follows that <math>\triangle OPA</math> is a <math>45-45-90</math> [[right triangle]], so <math>OP = AP = 1,</math> <math>OB = OA = \ Without loss of generality, place the pyramid in a 3-dimensional coordinate system such that <math>A = (1,0,0),</math> <math>B =
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1996 AIME Problems/Problem 12
  <cmath>|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.</cmath> ...obtained from these paired sequences are also obtained in another <math>2^5-1</math> ways by permuting the adjacent terms <math>\{a_1,a_2\},\{a_3,a_4\},
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 7
  ...l be <br /> three other equivalent boards.</font></td><td><font style="font-size:85%">For those symmetric about the center, <br /> there is only one oth ...re rotationally symmetric about the center square; there are <math>\frac{49-1}{2}=24</math> such pairs. There are then <math>{49 \choose 2}-24</math> pa
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 3
  ...itive [[integer]] <math>n</math> for which the expansion of <math>(xy-3x+7y-21)^n</math>, after like terms have been collected, has at least 1996 terms. ...3)^n</math>. Both [[binomial expansion]]s will contain <math>n+1</math> non-like terms; their product will contain <math>(n+1)^2</math> terms, as each t
  3 KB (515 words) - 04:29, 27 November 2023
• 1998 AIME Problems/Problem 15
  ...s a path that retraces no segment. Each time that such a path reaches a non-terminal vertex, it must leave it. ...it can be arranged that <math>n-2</math> segments will emanate from <math>n-2</math> of the vertices and that an odd number of segments will emanate fro
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1999 AIME Problems/Problem 7
  ...math> take on the values <math>0, 1, \ldots, 9</math>. At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are ...}{d_{i}}= 2^{9-x_{i}}3^{9-y_{i}}5^{9-z_{i}}</math>. In general, the divisor-count of <math>\frac{N}{d}</math> must be a multiple of 4 to ensure that a s
  3 KB (475 words) - 13:33, 4 July 2016
• 2001 AIME I Problems/Problem 15
  .../math>, which can be done in <math>4! = 24</math> ways. Then choose a three-edge path along tetrahedron <math>DBEG</math> which, because it must start a ...faces, and one face adjacent to the three B-faces, which we will call the C-face.
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 12
  ...ween a plane and a point <math>I</math> can be calculated as <math>\frac{(I-G) \cdot P}{|P|}</math>, where G is any point on the plane, and P is a vecto ...rpendicular to plane <math>ABC</math> can be found as <math>V=(A-C)\times(B-C)=\langle 8, 12, 24 \rangle</math>
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2003 AIME I Problems/Problem 3
  ...1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math> she writes on her list the gre Each [[element]] of the [[set]] will appear in <math>7</math> two-element [[subset]]s, once with each other number.
  2 KB (317 words) - 00:09, 9 January 2024
• 2003 AIME II Problems/Problem 15
  ...that <math> \left( x^{23} + x^{22} + \cdots + x^2 + x + 1 \right) \cdot (x-1) = x^{24} - 1 </math>. The five-element sum is just <math>\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \s
  4 KB (675 words) - 17:23, 30 July 2022
• 2003 AIME II Problems/Problem 13
  ...ing the starting vertex in the next move. Thus <math>P_n=\frac{1}{2}(1-P_{n-1})</math>. ...-1</math> steps plus the number of ways to get to <math>C</math> in <math>n-1</math> steps.
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2001 AIME II Problems/Problem 15
  {{AIME box|year=2001|n=II|num-b=14|after=Last Question}}
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 10
  ...plies that <math>10^{6} - 1 | 10^{6k} - 1</math>, and so any <math>\boxed{j-i \equiv 0 \pmod{6}}</math> will work. ...le 5</math>, and let <math>j-i-a = 6k</math>. Then we can write <math>10^{j-i} - 1 = 10^{a} (10^{6k} - 1) + (10^{a} - 1)</math>, and we can easily verif
  4 KB (549 words) - 23:16, 19 January 2024
• 2001 AIME II Problems/Problem 6
  Let point <math>A</math> be the top-left corner of square <math>ABCD</math> and the rest of the vertices be arra ...factor. We get <math>b = 2/5a</math>, meaning the ratio of areas <math>((a-2b)/a)^2</math> = <math>(1/5)^2</math> = <math>1/25</math> = <math>m/n.</mat
  4 KB (772 words) - 19:31, 6 December 2023
• 2000 AIME II Problems/Problem 14
  ...8!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>. ...{k=1}^{n-1} {((k+1)!- k!)} = 1 + ((2! - 1!) + (3! - 2!) + \cdots + (n! - (n-1)!)) = n!</math>.
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2006 USAMO Problems/Problem 1
  if and only if <math>s </math> is not a divisor of <math>p-1 </math>. ...>s|(p-1)</math>. Then for some positive integer <math>k</math>, <math>sk=p-1</math>. The conditions given are equivalent to stating that <math>sm \bmo
  3 KB (506 words) - 17:54, 22 June 2023
• 2006 USAMO Problems/Problem 3
  Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with any non-negative integer roots, so <math>a_i > 1</math> and thus <math>b_i > 0</math
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  ...is from <math>2^m(2^{i_0-m} - 1) = 2^{i_0} - 2^m</math> to <math>2^m(2^{i_0-m}+1) = 2^{i_0} + 2^m</math>. ...- 2^{i_0} = 2^p(2t - 2^{i_0-p} + 1)</math> and the number <math>2t + 2^{i_0-p} + 1</math> is odd, a jump of size <math>2^{p+1}</math> can be made from <
  7 KB (1,280 words) - 17:23, 26 March 2016
• Multinomial Theorem
  For a positive integer <math>k</math> and a non-negative integer <math>n</math>, ..._1 + b_2 + b_3 + ... + b_{k-1} + b_k= n}{\binom{n}{b_1, b_2, b_3, ..., b_{k-1}, b_k} \prod_{j=1}^{k}{x_j^{b_j}}}</cmath>
  3 KB (476 words) - 19:37, 4 January 2023
• Dedekind domain
  ...\cdots P_{n-1})\not\subseteq (a)</math>. Take any <math>b\in P_1\cdots P_{n-1}\setminus (a)</math> let <math>\gamma = b/a\in K</math>. We claim that thi ...ot\in R</math>. Now for any <math>x\in J</math>, <math>bx\in P_1\cdots P_{n-1}J\subseteq P_1\cdots P_n\subseteq (a)</math>, and so <math>bx = ar</math>
  9 KB (1,648 words) - 16:36, 14 October 2017
• 2006 AMC 10B Problems
  ...</math> and <math>y</math>, define <math> x \mathop{\spadesuit} y = (x+y)(x-y) </math>. What is <math> 3 \mathop{\spadesuit} (4 \mathop{\spadesuit} 5) < ...rcle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2000 AMC 12 Problems/Problem 8
  <cmath>S_{100}=\frac{100[2\cdot 4+(100-1)4]}{2}</cmath> ...se Figure 0 exists) <math>\dbinom{101-1}{0}+4\dbinom{101-1}{1}+4\dbinom{101-1}{2}=20201</math> or <math>\textbf{(C)}</math>
  7 KB (988 words) - 15:14, 10 April 2024
• User:Rowechen
  <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m
  22 KB (3,778 words) - 19:29, 2 June 2020
• 2005 AMC 10A Problems/Problem 1
  So the desired difference is <math>m-j=20-10=10 \Rightarrow \boxed{\textbf{(D) }10}</math> {{AMC10 box|year=2005|ab=A|before=First Problem|num-a=2}}
  909 bytes (134 words) - 19:05, 25 December 2022
• Navier-Stokes Equation
  ...interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting a ...cal terms these rates correspond to their [[derivative]]s. Thus, the Navier-Stokes equations for the most simple case of an ideal fluid with zero viscos
  3 KB (553 words) - 22:08, 2 May 2022
• 2015 IMO Problems/Problem 6
  ...th>b</math> and <math>N</math> for which<cmath>\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2</cmath>for all integers <math>m</math> and <math>n</m <u>'''Theorem'''</u> ''Let <math>T</math> be a non-negative integer parameter. If given a sequence <math>a_1,a_2,\dots</math> t
  4 KB (833 words) - 01:33, 31 December 2019
• Volume
  ...t defined by an upper bound of <math>f(x,y,z)</math> in the Cartesian three-space can be found using a triple [[integral]]: <math>\int_{a_z}^{b_z}\int_{
  3 KB (523 words) - 20:24, 17 August 2023
• Divisibility rules/Rule for 3 and 9 proof
  Suppose that the [[base numbers | base-ten]] representation of <math>N</math> is <center><math>N = a_k a_{k-1} \cdots a_2 a_1 a_0</math>,</center>
  2 KB (316 words) - 20:29, 6 March 2014
• AwesomeMath
  ...or bright middle and high school students who wish to sharpen their problem-solving skills and further their mathematics education. Many of our particip ...ve, instructor-led, virtual classes are taught once per weekend for each 12-week semester. Students choose their classes and there is no application req
  1 KB (195 words) - 11:19, 30 November 2023
• Stirling number
  <cmath> S(n,k) = \frac{1}{k!}\sum_{j=0}^{k}(-1)^j{k \choose j} (k-j)^n. </cmath>
  2 KB (253 words) - 00:04, 5 December 2020
• 2007 iTest Problems
  Compute the sum of all twenty-one terms of the geometric series <cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath>
  33 KB (5,177 words) - 21:05, 4 February 2023
• Euler line
  ...th>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ ...riangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.
  59 KB (10,203 words) - 04:47, 30 August 2023
• Pick's Theorem
  2 KB (301 words) - 13:08, 20 February 2024
• Fermat point
  We shall present a standard triangle inequality proof as well as a less-known vector proof: ...\vec{0}</math> and adding, we see that <math>|a|+|b|+|c|\leq |a-x|+|b-x|+|c-x|</math>, or <math>AP+BP+CP\leq AX+BX+CX</math>. Thus, the origin or point
  4 KB (769 words) - 16:07, 29 December 2019
• Chinese Remainder Theorem
  ...or instance, [[Fermat's Little Theorem]] may be generalized to the [[Fermat-Euler Theorem]] in this manner. ...: the proof of the general case follows by induction to the above result (k-1) times.
  6 KB (1,022 words) - 14:57, 6 May 2023
• Matrix
  4 KB (856 words) - 15:29, 30 March 2013
• 1985 IMO Problems/Problem 3
  Now assume that the problem holds for <math>k-1</math>. We now have two cases: <math> i_1 \ge k</math>, and <math>i_1 < k ..._{j=0}^{k-1}a_j x^j + (1+x)^k \sum_{j=0}^{k-1} b_j x^j \equiv \sum_{j=0}^{k-1}\left[ (a_j + b_j)x^j + b_j x^{j+k} \right] \pmod{2}
  2 KB (354 words) - 04:56, 11 March 2023
• 2001 AMC 8 Problems
  Granny Smith has \$63. Elberta has \$2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? ...3, 4 and 9 are each used once to form the smallest possible '''even''' five-digit number. The digit in the tens place is
  13 KB (1,994 words) - 13:04, 18 February 2024
• 2003 AMC 12A Problems/Problem 12
  {{AMC10 box|year=2003|ab=A|num-b=23|num-a=25}} {{AMC12 box|year=2003|ab=A|num-b=11|num-a=13}}
  2 KB (368 words) - 18:04, 28 December 2020
• Minkowski Inequality
  1 KB (240 words) - 16:49, 29 December 2021
• 2001 IMO Shortlist Problems/A2
  ...>b_0, b_1, \ldots</math> of positive integers for which <math>1+b_n\le b_{n-1}\sqrt[n]{2}</math>, it is clear that there will not exist any sequence <ma ...exist such a sequence. Then, define <math>x_0=b_0</math> and <math>x_n=x_{n-1}\sqrt[n]{2} -1</math>. It is clear that <math>x_n\ge b_n</math> for all <m
  2 KB (411 words) - 04:38, 17 June 2019

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