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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
• Quaternion
  1,007 bytes (155 words) - 20:47, 14 October 2013
• 2020 AMC 10A Problems
  <cmath>\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}</cmath> What is the sum of all real numbers <math>x</math> for which <math>|x^2-12x+34|=2?</math>
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Logical Operators and Loops
  if (floor((i-j)/2)==((i-j)/2)) if (floor((i-j)/2)==((i-j)/2))
  2 KB (324 words) - 16:50, 2 October 2016
• Pigeonhole Principle/Solutions
  ...riends, then the remaining friends must have from <math>1</math> to <math>n-2</math> friends for the remaining friends not to also have no friends. By p ...same remainder when divided by 5. Since there are 5 possible remainders (0-4), by the pigeonhole principle, at least two of the integers must share the
  10 KB (1,617 words) - 01:34, 26 October 2021
• 2002 USAMO Problems/Problem 3
  ...it is necessary and sufficient that <math> P(x) = Q(x) + \prod_{i=1}^{n}(x-x_i) </math>. ...s. But a polynomial with real coefficients must have an even number of non-real roots, so <math>P(x) </math> must have <math>n </math> real roots. Sim
  4 KB (688 words) - 13:38, 4 July 2013
• 2007 AIME I Problems/Problem 10
  [[Image:AIME I 2007-10.png]] ...ath> (<math>j < k</math>) here. We can now use the [[Principle of Inclusion-Exclusion]] based on the stipulation that <math>j\ne k</math> to solve the p
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2004 USAMO Problems
  ...egers <math> x, y \in S </math>, if <math> x+y \in S </math>, then <math> x-y \in S </math>. * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2004-ua/04usamo-test.shtml 2004 USAMO Problems]
  3 KB (474 words) - 09:20, 14 May 2021
• 2007 AIME II Problems/Problem 3
  ...ince <math>\triangle ABE \cong \triangle CDF</math> and are both <math>5-12-13</math> [[right triangle]]s, we can come to the conclusion that the two ne A slightly more analytic/brute-force approach:
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 13
  {{AIME box|year=2007|n=II|num-b=12|num-a=14}}
  3 KB (600 words) - 11:10, 22 January 2023
• 2007 AIME II Problems
  <div style="text-align:center;">[[Image:2007 AIME II-3.png]]</div> ..._{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there?
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2003 USAMO Problems/Problem 1
  ...-1} + a </math> is divisible by 5, which is true when <math> k \equiv -3^{n-1}a \pmod{5} </math>. Since there is an odd digit in each of the residue cl ...s after it, and all multiples of 5 end in 5. Therefore, <math> a*10^x*5^{n-x}</math> always contains a 5 as its <math> (x+1)^{st}</math> digit, and we
  4 KB (736 words) - 22:17, 3 March 2023
• 2003 USAMO Problems/Problem 3
  ...{} c_j = \cdots = c_{j+i} = \cdots = c_k = j </math> (<math> 0 \le i \le k-j</math>). ''Proof.'' Since the <math>j </math> terms <math>c_0, \ldots, c_{j-1} </math> are all less than <math>j </math>, no other terms that precede <m
  3 KB (636 words) - 13:39, 4 July 2013
• Pentagon
  1 KB (191 words) - 09:59, 6 June 2022
• 2004 IMO Shortlist Problems/N2
  ...e-i}) = p^e + \sum_{i=0}^{e-1}p^i(p^{e-i} - p^{e-i-1}) = p^e + e(p^e - p^{e-1}) </math>. </center> ...^{e_i} - e_i \cdot p_i^{e_i}]}{\prod p_i^{e_i}} = \prod \left(e_i \frac{p_i-1}{p_i} + 1 \right) </math>.
  6 KB (1,007 words) - 09:10, 29 August 2011
• Parallelepiped
  2 KB (248 words) - 20:08, 17 August 2023
• Chicken McNugget Theorem
  ...am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. ...e proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.
  17 KB (2,748 words) - 19:22, 24 February 2024
• Dodecagon
  A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\circ}<
  1 KB (219 words) - 13:08, 15 June 2018
• Triangular number
  <math>1 +2+3 + 4............. +(n-1)+(n)</math>. ...math>T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n</math>.
  2 KB (275 words) - 08:39, 7 July 2021
• 2007 AMC 12A Problems/Problem 24
  ...mod{4}</math> is that <math>v_2(2j+1)=0,</math> so we must have <math>v_2(n-1)>v_2(n+1)</math> since <math>v_2(2k) \geq 1 >v_2(2j+1).</math> Therefore, {{AMC12 box|year=2007|num-b=23|num-a=25|ab=A}}
  4 KB (588 words) - 14:40, 23 August 2023
• User talk:1=2
  * [[1962 AHSME]] (Complete '''w/o solutions w/o problems 41-50''')
  13 KB (1,464 words) - 17:28, 6 January 2024
• 2006 AIME II Problems/Problem 13
  ...math>, <math>n\geq 0</math>. Then the final odd integer is <math>2n+1 + 2(j-1) = 2(n+j) - 1</math>. The odd integers form an [[arithmetic sequence]] wit ...ual <math>0</math>. We then perform casework based on the parity of <math>p-q</math>.
  4 KB (675 words) - 10:40, 14 July 2022
• 2006 AIME II Problems
  When rolling a certain unfair six-sided die with faces numbered <math>1, 2, 3, 4, 5</math>, and <math>6</math> ...h> and <math>t_2</math> intersect at <math>(x,y),</math> and that <math>x=p-q\sqrt{r},</math> where <math>p, q,</math> and <math>r</math> are positive i
  8 KB (1,350 words) - 12:00, 4 December 2022
• User:Temperal/The Problem Solver's Resource4
  ==<span style="font-size:20px; color: blue;">Algebra</span>== <cmath>f(x)=a_nx^n+a_{n-1}x^{n-1}\ldots+a_0</cmath>, where <math>a_n\ne 0</math>, and <math>a_i</math> are
  4 KB (828 words) - 21:45, 27 February 2020
• 1998 PMWC Problems
  for(int j = n-i; j > 0; --j){ ...ad a total of <math>30</math> balls, how many balls were there in the right-hand box?
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  F=O+(G-O)+(E-O); draw((xstart,A.y)--(xstart,A.y-len));
  4 KB (641 words) - 21:24, 21 April 2014
• 2002 PMWC Problems
  label("$" + string(i) + "^\textrm{" + ord[i-1] + "}$ Row:", (-5,(-i+1)*sqrt(3)/2+correction)); draw( (1-cos(pi/3)*correction,-sin(pi/3)*correction)--(0+cos(pi/3)*correction,-sqrt(3
  5 KB (725 words) - 16:07, 23 April 2014
• 2003 PMWC Problems
  filldraw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black); fill((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,white);
  7 KB (918 words) - 16:15, 22 April 2014
• 1989 USAMO Problems/Problem 3
  Let <math>P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n</math> be a [[polynomial]] in the complex variable <math>z <cmath> \lvert z_i-i \rvert \cdot \lvert z_j-i \rvert < 1. </cmath>
  2 KB (340 words) - 19:11, 18 July 2016
• Cramer's Rule
  2 KB (352 words) - 18:22, 11 October 2023
• Lucas' Theorem
  ...n of <math>n</math> in [[base]] <math>p</math> and <math>(\overline{i_mi_{m-1}\cdots i_0})_p</math> is the representation of <math>i</math> in base <mat For all <math>1\leq k \leq p-1</math>, <math>\binom{p}{k}\equiv 0 \pmod{p}</math>. Then we have
  1 KB (251 words) - 15:13, 11 August 2020
• Summation
  ...r algebraic structure), <math>\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}</math>. Here <math>i</math> refers to the index of summation, <math ...i= \frac{n(n+1)}{2}</math>, and in general <math>\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}</math>
  3 KB (482 words) - 16:39, 8 October 2023
• Zorn's Lemma
  ...intermediate results, viz., Bourbaki's Theorem (also known as the Bourbaki-Witt theorem). ...h>x \in A</math> does <math>x = f(x)</math>, which contradicts the Bourbaki-Witt Theorem. {{halmos}}
  9 KB (1,669 words) - 19:02, 1 August 2018
• Ideal
  ...ls in rings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. ...frak{a}</math> is both a left ideal and a right ideal, it is called a ''two-sided ideal''. In a [[commutative ring]], all three kinds of ideals are the
  8 KB (1,389 words) - 23:44, 17 February 2020
• Work
  1 KB (192 words) - 00:46, 16 July 2018
• Celsius
  4 KB (357 words) - 22:40, 17 April 2024
• 1999 USAMO Problems/Problem 4
  <cmath> \sum_{i=k+1}^n -a_i \le 2k-n . </cmath> <cmath> \sum_{i=k+1}^n a_i^2 \le \left( \sum_{i=k+1}^n-a_i \right)^2 \le (2k-n)^2 . </cmath>
  3 KB (499 words) - 09:47, 20 July 2016
• 2001 IMO Shortlist Problems/A3
  By the [[Cauchy-Schwarz Inequality]], By the [[Cauchy-Schwarz Inequality]]
  3 KB (441 words) - 00:54, 24 October 2023
• 1999 IMO Problems/Problem 2
  ...ath> and the right-hand side becomes <math>(x_a + x_b)^4/8</math>. By [[AM-GM]], ...ase the left-hand side of the desired inequality without changing the right-hand side; and then to use the inductive hypothesis.
  4 KB (838 words) - 01:04, 17 November 2023
• Proofs of AM-GM
  This pages lists some proofs of the weighted [[AM-GM Inequality]]. The inequality's statement is as follows: for all nonnegati ...and <math>\lambda_i \neq 0</math>, for some <math>i</math>, then the right-hand side of the inequality is zero and the left hand of the inequality is g
  12 KB (2,171 words) - 07:55, 11 May 2023
• Carleman's Inequality
  Define <math>c_n = n \left( 1 + \frac{1}{n} \right)^n = \frac{(n+1)^n}{n^{n-1}}</math>. Then for all positive integers <math>k</math>, Now, by [[AM-GM]],
  2 KB (278 words) - 16:39, 29 December 2021
• 2008 AMC 12B Problems/Problem 23
  ...square. When these are multiplied, they equal <math>2^{a+n-a} \times 5^{b+n-b} = 10^n</math>. <math>\log 10^n=n</math> so the number of factors divided .../math> satisfies <math>ab = 10^n</math> -- ex. <math>(1, 10^n), (2, 5*10^{n-1})</math>, etc. Then the sum of the base-<math>10</math> logarithms is <mat
  5 KB (814 words) - 18:02, 17 January 2023
• Vieta's formulas
  Let <math>P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0</math> be any polynomial with [[Complex number | c ....</cmath> This can be compactly summarized as <math>s_j = (-1)^j \frac{a_{n-j}}{a_n}</math> for some <math>j</math> such that <math>1 \leq j \leq n</mat
  3 KB (512 words) - 15:31, 22 February 2024
• 2000 AMC 12 Problems/Problem 16
  <math>f(i, j) = 17(i-1) + j = 17i + j - 17</math> {{AMC12 box|year=2000|num-b=15|num-a=17}}
  2 KB (310 words) - 11:28, 3 August 2021
• 2007 AMC 10A Problems
  draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle); ...at random from 0 to 2007, inclusive. What is the probability that <math>ad-bc</math> is even?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1990 USAMO Problems/Problem 4
  '''Lemma.''' Let <math>a</math> be an integer. Then there are <math>2^{k-1}</math> <math>k</math>-good sequences starting on <math>a</math>, and furt ...M</math>. Then the only possibilities for <math>a_{k+1}</math> are <math>m-1</math> and <math>M+1</math>; either way, <math>\{ a_i \}_{i=1}^{k+1}</math
  3 KB (529 words) - 19:15, 18 July 2016
• 2007 AMC 10A Problems/Problem 18
  ...\cdot 12 = \frac {36}5 </math>. (An alternate way is by seeing that the set-up AHGCM is similar to the 2 pole problem (http://www.artofproblemsolving.co {{AMC10 box|year=2007|ab=A|num-b=17|num-a=19}}
  6 KB (867 words) - 00:17, 20 May 2023
• 1974 USAMO Problems/Problem 1
  <cmath> a-b \mid P(a) - P(b) = b-c \mid P(b)-P(c) = c-a \mid P(c) - P(a) = a-b , </cmath> ...absolute value. In fact, two of them, say <math>(a-b)</math> and <math>(b-c)</math>, must be equal. Then
  7 KB (1,291 words) - 20:30, 27 April 2020
• 1994 USAMO Problems
  The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue ...sual, let <math>\, \binom{n}{k} \,</math> denote <math>\, n! \over k! \, (n-k)!</math>. Prove that <cmath> \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \s
  2 KB (391 words) - 07:58, 19 July 2016
• Isogonal conjugate
  Let <math>I'</math> and <math>J'</math> be the A-excenters of triangles <math>\triangle ABC</math> and <math>\triangle ADC,</ ...oordinates <cmath>(p' : q' : r'), p' = \frac {|B - C|^2}{p}, q' = \frac {|A-C|^2}{q}, r' = \frac {|A - B|^2}{r}.</cmath>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 1976 IMO Problems/Problem 2
  ...inct real solutions, where <math>2^{n-1}</math> are positive and <math>2^{n-1}</math> are negative. Also, for ever root <math>r</math>, <math>|r|<2</mat ...values less than 2, where <math>2^{k-1}</math> are positive and <math>2^{k-1}</math> are negative.
  3 KB (596 words) - 16:19, 28 July 2015
• 1976 IMO Problems/Problem 5
  {{IMO box|year=1976|num-b=4|num-a=6}}
  2 KB (377 words) - 16:28, 29 January 2021
• Partition (combinatorics)
  ...iven by <math>F(x)= \sum_{n \geq 0}P(n) x^n = \prod_{n=1}^\infty \frac{1}{1-x^n}</math>. Partitions can also be studied by using the [[Jacobi theta func ...epresents a different addend in the partition. The rows are ordered in non-increasing order so that that the row with the most dots is on the top and t
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2001 AMC 12 Problems/Problem 6
  A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents {{AMC12 box|year=2001|num-b=5|num-a=7}}
  2 KB (340 words) - 03:02, 28 June 2023
• 2008 AIME II Problems
  ...ch is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin bikin a^2 + y^2 = b^2 + x^2 = (a-x)^2 + (b-y)^2
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 3
  <math>(13,1)</math> and <math>(3,9)</math> give non-integral <math>b</math>, but <math>(8,5)</math> gives <math>b = 15</math>. T ...<math>2y+3z+4x=91</math>. Subtracting the two expressions we get <math>y+z-2x=-17</math>. Note that <math>-17</math> is odd, so one of <math>x,y,z</mat
  3 KB (564 words) - 22:15, 28 November 2023
• Mock AIME 5 Pre 2005 Problems
  Two 5-digit numbers are called "responsible" if they are:
  6 KB (909 words) - 07:27, 12 October 2022
• 2008 AIME II Problems/Problem 10
  ...our points using coordinates <math>0 \le x,y \le 3</math>, with the bottom-left point being <math>(0,0)</math>. By the [[Pythagorean Theorem]], the dis {{AIME box|year=2008|n=II|num-b=9|num-a=11}}
  4 KB (569 words) - 09:44, 25 November 2019
• 2005 USAMO Problems/Problem 2
  (1) <math>(x^3+y)(x^3+1) = (x^3+y)(x+1)(x^2-x+1) = 147^{157} = 7^{314}3^{152}</math>, <math>x^2-x+1 = (x+1)^2-3(x+1)+3 \rightarrow \gcd(x+1, x^2-x+1) \le 3</math>.
  7 KB (1,053 words) - 10:38, 12 August 2015
• 2005 USAMO Problems/Problem 1
  ...t with the circular arrangement of <math>n,p_{1}p_{2},p_{2}p_{3}\ldots,p_{k-1}p_{k}</math> as shown. ...by placing <math>p_k,p^{2}_{k},\ldots,p^{e_k}_{k}</math> between <math>p_{k-1}p_k</math> and <math>n</math>. It is easy to see that each element of <mat
  4 KB (650 words) - 13:40, 4 July 2013
• 2006 IMO Problems/Problem 5
  a_1 - a_0 &\mid P(a_1)- P(a_0) = a_2-a_1 \\ &\mid P(a_r)- P(a_{r-1}) = a_1 - a_0 .
  3 KB (704 words) - 14:42, 7 September 2021
• Schreier's Theorem
  ...iscovered it in 1928, and used it to give an improved proof of the [[Jordan-Hölder Theorem]]. Six years later, Hans Zassenhaus published his [[Zassenh ...ies in question. For integers <math>j \in [1,m-1]</math>, <math>i \in [0,n-1]</math>, let <math>H'_{im+j} = H_{i+1} \cdot (H_i \cap K_j)</math>, and fo
  2 KB (337 words) - 12:13, 9 April 2019
• 2008 Mock ARML 2 Problems
  ...th>f(x)</math> equal the number of zeroes to the right of the rightmost non-zero digit in the decimal form of <math>x!</math>, and let <math>n = \frac { ...sum_{i = 0}^{n}a_ia_{n - i} = 1</math> and <math>a_n > 0</math> for all non-negative integers <math>n</math>, evaluate <math>\sum_{j = 0}^{\infty}\frac
  4 KB (582 words) - 21:57, 8 May 2019
• 2008 Mock ARML 2 Problems/Problem 8
  ...sum_{i = 0}^{n}a_ia_{n - i} = 1</math> and <math>a_n > 0</math> for all non-negative integers <math>n</math>, evaluate <math>\sum_{j = 0}^{\infty}\frac By the given, the coefficients on the right-hand side are all equal to <math>1</math>, yielding the [[geometric series]]
  1 KB (190 words) - 00:57, 31 May 2016
• 2003 AMC 12A Problems/Problem 17
  <math>x^2 + (y-4)^2 = 4^2</math> <math>(x-2)^2 + y^2 = 2^2</math>
  6 KB (1,026 words) - 22:35, 29 March 2023
• 2003 AMC 12B Problems/Problem 18
  ...s 7 and 11 in order to keep <math>x</math> at a minimum. Moving all the non-y terms to the left hand side of the equation, we end up with: <cmath>7^{5b+1}11^{5d-1}=y^{13}</cmath>
  6 KB (914 words) - 11:07, 7 September 2023
• Jean-Victor Poncelet
  '''Jean-Victor Poncelet''' (July 1, 1788 – December 22, 1867) was a French mathema ...son of Claude de Poncelet, a lawyer in the local Parliament, and a well-to-do landowner. Poncelet attended local schools in his home town before going
  2 KB (253 words) - 11:41, 19 December 2018
• KGS math club/solution 6 1
  ...know that <math>\forall i: 2 | (W-w_i)</math>, that is, each remaining ball-mass is divisible by two. Combining these, we get <math>\forall i,j: 2 | (w_i-w_j)</math> by subtracting the case i from the case j.
  4 KB (759 words) - 06:27, 18 July 2009
• Legendre's Formula
  ...i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}</cmath> <cmath>e_2(27!)=\frac{27-S_2(27)}{2-1}=27-S_2(27)</cmath>
  4 KB (699 words) - 17:55, 5 August 2023
• 2001 IMO Shortlist Problems/C2
  ...that <math>S(x)\equiv y\bmod{n}</math> for all integers <math>0\leq y\leq n-1</math>. This means that the sum of every <math>S(a)</math> is congruent to ...(c_i\sum_{j=1}^{n} (n-1)!*a_j\right)=\sum_{i=1}^{n} c_i\cdot \frac{n(n+1)(n-1)!}{2}=\frac{c_i(n+1)!}{2}\bmod{n!}</cmath>
  2 KB (364 words) - 08:55, 31 August 2011
• 2001 IMO Shortlist Problems/C5
  519 bytes (73 words) - 20:07, 29 May 2020
• 2001 IMO Shortlist Problems/N6
  ...ath>(c+d-b)^2+b^2\equiv c^2+d^2\pmod p</math>. Factorizing, <cmath>2(b^2-bc-bd+cd)\equiv0\pmod p</cmath> <cmath>\implies(b-c)(b-d)\equiv0\pmod p</cmath>
  2 KB (443 words) - 13:08, 17 August 2011
• Millennium Problems
  ...], the [[Poincaré Conjecture]], the [[Riemann Hypothesis]], and the [[Yang-Mills Theory]]. In 2003, the Poincaré Conjecture was proven by Russian math ...ent. He gave three lectures, titled "Ricci Flow and Geometrization of Three-Manifolds", on April 7, 9, and 11. These were his first public presentation
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1989 AHSME Problems/Problem 29
  {{AHSME box|year=1989|num-b=28|num-a=30}}
  1 KB (173 words) - 19:52, 30 December 2020
• 2008 IMO Problems/Problem 5
  a_i-n, & \mbox{ if } a_i \in B .../math> elements from <math>\cal{N}</math>, which would imply <math>N = 2^{k-n}M</math> and solve the problem.
  4 KB (677 words) - 01:10, 19 November 2023
• 2000 USAMO Problems
  Call a real-valued function <math>f</math> very convex if
  3 KB (495 words) - 19:02, 18 April 2014
• 2000 USAMO Problems/Problem 4
  if((i == 0 || j == 9) && !(j-i == 9)) fill(shift(i,j)*unitsquare,rgb(0.3,0.3,0.3)); ...r of the rows containing the colored squares. Hence, each of the <math>1999-m</math> colored squares must be placed in different rows, but as there are
  2 KB (382 words) - 13:37, 4 July 2013
• 2000 USAMO Problems/Problem 5
  ...e externally tangent at the point <math>A_k</math>, so <math>O_k, A_k, O_{k-1}</math> are [[collinear]]. ...ath>\theta_k = 180 - \theta_{k-1}</math>, and so <math>\theta_k = \theta_{k-2}</math>. Thus, <math>\theta_{1} = \theta_{7}</math>.
  3 KB (609 words) - 09:52, 20 July 2016
• 2001 USAMO Problems/Problem 1
  ...olors in 3 boxes and <math>3n-48</math> colors in 2. Thus <math>n\geq 20+48-2n,</math> so <math>3n\geq 68</math>, and <math>n\geq23</math> and we are do {{USAMO newbox|year=2001|before=First question|num-a=2}}
  5 KB (841 words) - 17:19, 5 May 2022
• Karamata's Inequality
  ...a\leq x_1\leq x_2 \leq b</math> and <math>\Gamma(x, y):=\frac{f(y)-f(x)}{y-x}</math>, <math>\Gamma(x_1, x)\leq \Gamma (x_2, x)</math> <cmath>\Gamma(x_1, x)=\frac{f(x_1)-f(x)}{x_1-x}\leq \frac{f(x_2)-f(x)}{x_2-x}=\Gamma(x_2, x)</cmath>
  2 KB (370 words) - 03:39, 28 March 2024
• 1973 USAMO Problems/Problem 2
  <center> <math>X_0=1</math>, <math>X_1=1</math>, <math>X_{n+1}=X_n+2X_{n-1}</math> <math>(n=1,2,3,\dots),</math></center> <center><math>Y_0=1</math>, <math>Y_1=7</math>, <math>Y_{n+1}=2Y_n+3Y_{n-1}</math> <math>(n=1,2,3,\dots)</math>.</center>
  2 KB (342 words) - 18:54, 3 July 2013
• 2004 IMO Problems/Problem 4
  ...math>t_3 = t_2 + t_1 + k</math> for <math>k \ge 0</math>; then (by [[Cauchy-Schwarz Inequality]]) ...} \right) + t_n \sum_{i=1}^{n-1} \frac 1{t_i} + \frac{1}{t_n} \sum_{i=1}^{n-1} t_i + 1\end{align*}</cmath>
  2 KB (322 words) - 00:54, 19 November 2023
• 2002 AMC 10A Problems
  Given that <math>a, b,</math> and <math>c</math> are non-zero real numbers, define <math>(a, b, c) = \frac{a}{b} + \frac{b}{c} + \fra ..., and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
  11 KB (1,733 words) - 11:04, 12 October 2021
• 2002 AMC 10A Problems/Problem 20
  {{AMC10 box|year=2002|ab=A|num-b=19|num-a=21}}
  3 KB (439 words) - 22:15, 9 June 2023
• 1973 USAMO Problems/Problem 1
  http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1973Problem1 {{USAMO box|year=1973|before=First Question|num-a=2}}
  2 KB (276 words) - 08:10, 4 January 2022
• 1975 USAMO Problems/Problem 3
  ...], we can write <math>Q(x)</math> as <cmath>Q(x) = c(x)(x-1)(x-2) \cdots (x-n)</cmath> where <math>c</math> is a constant. Thus, <cmath>(x+1)P(x) - x = c(x)(x-1)(x-2) \cdots (x-n).</cmath>
  3 KB (465 words) - 17:29, 8 January 2024
• 1962 IMO Problems/Problem 7
  <math>M_{12}=\frac{V_1-V_2}{2}=\left( \frac{\sqrt{2}}{6}r_c ,\;\frac{\sqrt{6}}{6}r_c,\;-\frac{r_c}{ <math>M_{13}=\frac{V_1-V_3}{2}=\left( \frac{\sqrt{2}}{6}r_c ,\;-\frac{\sqrt{6}}{6}r_c,\;-\frac{r_c}
  11 KB (1,928 words) - 20:52, 21 November 2023
• 1963 IMO Problems/Problem 3
  ...textrm{-coordinate} = \sum_{k = 1}^{\frac{n}{2}}a_k \cdot \sin \frac{2\pi(k-1)}{n}.\textbf{ (1)}</cmath> ...xtrm{-coordinate} = \sum_{k = \frac{n}{2}+1}^{n}a_k \cdot \sin \frac{2\pi(k-1)}{n}</cmath>
  4 KB (836 words) - 17:58, 7 December 2022
• 2001 AMC 10 Problems
  two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit o ...t), and the rest sells for half price. How much money is raised by the full-price tickets?
  14 KB (1,983 words) - 16:25, 2 June 2022
• 2000 AMC 10 Problems/Problem 16
  ..., hence we can compute the slope, <math>m_1</math> to be <math>\frac{0-3}{6-0}=-\frac{1}{2}</math>, so <math>l_1</math> is the line <math>y=\frac{-1}{2} ...th> gives us <math>b_2=-2</math>, so <math>l_2</math> is the line <math>y=x-2</math>.
  7 KB (966 words) - 23:22, 31 July 2023
• 2007 IMO Shortlist Problems/A1
  <center><math>\max\{|x_i-a_i|:1\leq i\leq n\}\geq \frac{d}{2}</math>.</center> ..._j:i\le j\le n\}</math>, all <math>d_i</math> can be expressed as <math>a_p-a_q</math>, where <math>1\le p\le i\le q \le n</math>.
  3 KB (734 words) - 05:11, 26 March 2024
• 2004 AMC 12B Problems/Problem 14
  pair J=intersectionpoint(A--B, M--(M+rotate(90)*(B-A)) ); pair K=intersectionpoint(A--B, N--(N+rotate(90)*(B-A)) );
  7 KB (1,083 words) - 22:41, 23 November 2020
• 1976 IMO Problems
  For all positive integral <math>n</math>, <math>u_{n+1}=u_n(u_{n-1}^2-2)-u_1</math>, <math>u_0=2</math>, and <math>u_1=2\frac12</math>. Prove that
  2 KB (409 words) - 16:25, 29 January 2021
• 1971 IMO Problems/Problem 6
  ...})(i, j = 1, 2, \cdots, n)</math> be a square matrix whose elements are non-negative integers. Suppose that whenever an element <math>a_{ij} = 0</math>, ...h>. We thus infer that<cmath>s(\ell_j)+s(c_{\sigma(j)})\ge (n-t)+(n-k+t)=2n-k\ge n</cmath>so <math>s(\ell_j)+s(c_{\sigma(j)})\ge n,\ \forall j\ge k+1</m
  6 KB (1,192 words) - 14:14, 29 January 2021
• 2001 AMC 12 Problems/Problem 10
  {{AMC12 box|year=2001|num-b=9|num-a=11}} {{AMC10 box|year=2001|num-b=17|num-a=19}}
  2 KB (252 words) - 00:11, 15 August 2022
• 2001 AMC 12 Problems/Problem 22
  Therefore <math>[EHJ]=[ACD]-[AHD]-[DEH]-[EJC]=35-14-\frac {21}2-\frac{15}2 = \boxed{3}</math>. {{AMC12 box|year=2001|num-b=21|num-a=23}}
  7 KB (1,120 words) - 02:27, 29 August 2023
• 2009 AMC 12B Problems
  Each morning of her five-day workweek, Jane bought either a <math>50</math>-cent muffin or a <math>75 Twenty percent less than <math>60</math> is one-third more than what number?
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 20
  ...rtices, implying each vertex owns 4 edge ends. There are twice as many edge-ends as there are edges, and <math>2 \cdot 100 = 200</math>. {{AMC12 box|year=2009|ab=B|num-b=19|num-a=21}}
  9 KB (1,567 words) - 13:43, 19 August 2023
• 2009 AMC 12B Problems/Problem 25
  Given any square, we can circumscribe another axes-parallel square around it. In the picture below, the original square is red ...n a clever bijection given in [http://web2.slc.qc.ca/sh/Contest/AMC12_2009B-S.pdf this page].
  15 KB (2,229 words) - 03:36, 4 September 2021
• Maximal ideal
  968 bytes (183 words) - 19:50, 23 August 2009
• Polynomial ring
  ...biggr)\cdot \biggl(\sum_j b_jx^j\biggr) = \sum_k\biggl(\sum_{i=0}^k a_ib_{k-i}\biggr)x^k ...nto <math>R[x]/(x-a)</math> and the canonical homomorphism of <math>R[x]/(x-a)</math> into <math>R</math>.)
  12 KB (2,010 words) - 00:10, 3 August 2020
• User talk:Moplam
  ...etitions</td><td>2006-2007</td><td>2007-2008</td><td>2008-2009</td><td>2009-2010</td></tr>
  2 KB (383 words) - 22:32, 21 April 2010
• 1989 AJHSME Problems
  Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed ...per is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one o
  13 KB (1,765 words) - 11:55, 22 November 2023
• Hilbert's Basis Theorem
  ascending chain of (two-sided) [[ideal]]s of <math>A[x]</math>. <math>\mathfrak{c}_{i,j}</math> is a two-sided ideal of <math>A</math>; furthermore,
  4 KB (617 words) - 19:59, 23 April 2023
• Determinant
  ...n\times n</math> determinant can be written in terms of <math>(n-1)\times(n-1)</math> determinants: where <math>m_i</math> is the <math>(n-1)\times(n-1)</math> matrix formed by removing the <math>1</math>st row and <math>i</ma
  8 KB (1,345 words) - 00:31, 9 May 2020
• 2004 USAMO Problems/Problem 2
  By Schur's theorem (a generalization of the more well-known Chicken McNugget theorem), every integer greater than some integer <ma {{USAMO newbox|year=2004|num-b=1|num-a=3}}
  3 KB (508 words) - 14:23, 17 July 2014
• 1968 AHSME Problems/Problem 35
  {{AHSME 35p box|year=1968|num-b=34|after=Last Problem}}
  3 KB (597 words) - 01:52, 16 August 2023
• 1989 AJHSME Problems/Problem 19
  <cmath>4.8-2.2=2.6\approx 2.5 \rightarrow \boxed{\text{B}}</cmath> {{AJHSME box|year=1989|num-b=18|num-a=20}}
  2 KB (255 words) - 03:28, 28 November 2019
• 1989 AJHSME Problems/Problem 22
  {{AJHSME box|year=1989|num-b=21|num-a=23}}
  1 KB (177 words) - 16:03, 19 April 2021
• User:PowerOfPi
  **2010 - 65-67, qualified for CMOQR **2011 - 40-50, qualified for CMO
  6 KB (796 words) - 17:48, 26 December 2017
• 1965 IMO Problems/Problem 4
  ...= 1-P \Longleftrightarrow x_i = 1 \pm t</math> where <math>t = \pm \sqrt{1-P} \in \mathbb{R}</math>. Which has no non-zero solutions for <math>t</math>.
  2 KB (382 words) - 12:51, 29 January 2021
• AoPS Wiki:Competition ratings
  ...Test.] When considering problem difficulty, '''put more emphasis on problem-solving aspects and less so on technical skill requirements'''. ...iple choice tests like AMC are rated as though they are free-response. Test-takers can use the answer choices as hints, and so correctly answer more AMC
  41 KB (6,742 words) - 16:27, 15 April 2024
• 2009 USAMO Problems
  ...ile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling
  4 KB (718 words) - 18:16, 17 September 2012
• 2009 USAMO Problems/Problem 6
  {{USAMO newbox|year=2009|num-b=5|after=Last question}}
  2 KB (463 words) - 12:19, 21 August 2020
• 2009 IMO Problems/Problem 6
  ...h> be distinct positive integers and let <math>M</math> be a set of <math>n-1</math> positive integers not containing <math>s=a_1+a_2+\ldots+a_n</math>. ...ts are identical, we can treat them as one and use induction on <math>n = k-1</math>. We now consider four cases:
  5 KB (1,055 words) - 01:18, 19 November 2023
• 1998 USAMO Problems/Problem 3
  By AM-GM, {{USAMO newbox|year=1998|num-b=2|num-a=4}}
  2 KB (322 words) - 13:31, 23 August 2023
• Semisimple module
  Let us take a well-ordering on <math>I</math> (such an ordering exists by the [[well-ordering theorem]], a consequence of the [[axiom of choice]]),
  6 KB (1,183 words) - 15:02, 18 August 2009
• Tower law
  3 KB (567 words) - 08:42, 21 August 2009
• 1994 IMO Problems/Problem 3
  For space-time saving we say that a positive integer is a T-number or simply is T if it has exactly three 1s in his base two representat * <math>f(2^n) = \tbinom n2 = \sum_{i=1}^{n-1} i</math> whence
  7 KB (1,298 words) - 19:59, 9 February 2024
• 1989 IMO Problems/Problem 6
  ...\sum_{x \in \mathbb{U}} (1-I_{(1,3)}(x))(1-I_{(3,1)}(x))(1-I_{(2,4)}(x))(1-I_{(4,2)}(x))</math> So there are <math>2 \times 2 = 4</math> non-zero pairs. Each pair however has 2 ways to rearrange. So the third sum equa
  11 KB (1,928 words) - 12:26, 26 July 2023
• Mock AIME 3 2006-2007/Problem 3
  *[[Mock AIME 3 2006-2007/Problem 2 | Next Problem]] *[[Mock AIME 3 2006-2007]]
  862 bytes (151 words) - 01:35, 30 December 2009
• Characteristic polynomial
  In the context of problem-solving, the characteristic polynomial is often used to find closed forms fo ...{2n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{n1} & -a_{n2} & \cdots & t-a_{nn}\end{vmatrix}</cmath></center>
  19 KB (3,412 words) - 14:57, 21 September 2022
• KGS math club/solution 9 1
  629 bytes (125 words) - 06:42, 4 March 2010
• Proofs without words
  http://mathoverflow.net/questions/8846/proofs-without-words http://gurmeet.net/computer-science/mathematical-recreations-proofs-without-words/
  55 KB (7,986 words) - 17:04, 20 December 2018
• Pell equation
  ...al number]] <math>D</math>. Generally, <math>D</math> is taken to be square-free, since otherwise we can "absorb" the largest square factor <math>d^2 | ...using [[difference of squares]]. We would have <math>x^2 - Dy^2 = (x+dy)(x-dy) = 1</math>, from which we can use casework to quickly determine the solu
  6 KB (1,076 words) - 10:20, 29 October 2023
• 2010 AIME I Problems/Problem 11
  pair A = (8,10), B = (4.5,6.5), C= (9.75,8.25), F=foot(A,B,C), G=2*F-A; fill(A--B--C--cycle,rgb(0.9,0.9,0.9)); label("$y \le 10 - |x-8|$",(max,Q(max)),E,fontsize(10));
  4 KB (636 words) - 16:46, 25 November 2023
• Mock AIME 2 2010 Answer Key
  ...to 5 less. We can repeat this step again, and we will end up at <math>2010-15=1995</math>. This process can be repeated every for every 15 lemmings, so ...0})) = (b_1, b_2, \ldots, b_{10})</math>, with <math>0 \le b_i \le 2, 3|a_i-b_i</math> for <math>1 \le i \le 10</math>. Given that <math>f</math> can ta
  36 KB (6,214 words) - 20:22, 13 July 2023
• User talk:Zero.destroyer
  Thanks - <font style="font-family:Georgia,sans-serif">[[User:Azjps|azjps]] ([[User talk:Azjps|<font color="green">talk</fon
  956 bytes (163 words) - 19:33, 19 March 2010
• 2007 IMO Problems
  <cmath>\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} \qquad (*)</cmath> ...be positive integers. Show that if <math>4ab-1</math> divides <math>(4a^2-1)^2</math>, then <math>a=b</math>.
  3 KB (505 words) - 09:24, 10 September 2020
• 2007 IMO Problems/Problem 1
  <cmath>\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} (*)</cmath> ..._j:i\le j\le n\}</math>, all <math>d_i</math> can be expressed as <math>a_p-a_q</math>, where <math>1\le p\le i\le q \le n</math>.
  4 KB (768 words) - 23:15, 31 March 2010
• 2010 AIME II Problems
  Let <math>K</math> be the product of all factors <math>(b-a)</math> (not necessarily distinct) where <math>a</math> and <math>b</math> ...of <math>P(z)</math> are <math>w+3i</math>, <math>w+9i</math>, and <math>2w-4</math>, where <math>i^2=-1</math>. Find <math>|a+b+c|</math>.
  8 KB (1,246 words) - 21:58, 10 August 2020
• 2010 AIME II Problems/Problem 9
  <cmath>MN=BI-NI-BM=BI-(BM+MH).</cmath> ...he [[Law of cosines]] on <math>\triangle BCI</math>, <math>BI=\sqrt{2^2+1^2-2(2)(1)(\cos(120^\circ))}=\sqrt{7}</math>
  5 KB (857 words) - 22:22, 27 August 2023
• LaTeX:LaTeX on AoPS
  ==== In-line Math Mode ==== ...e>$</code> signs to enclose the math we want to display, and it displays in-line with our text. For example, typing <code>$\sqrt{x} = 5$</code> gives us
  8 KB (1,356 words) - 22:35, 26 June 2020
• 2010 USAMO Problems
  ...ath>h_k</math> is standing directly behind a student with height <math>h_{k-2}</math> or less, the two students are permitted to switch places. Prove th Let <math>q = \dfrac{3p-5}{2}</math> where <math>p</math> is an odd prime, and let
  3 KB (525 words) - 13:44, 4 July 2013
• 2010 USAMO Problems/Problem 3
  <cmath> \prod_{i=1}^{1005}(4i-1) = 3\times 7 \times \ldots \times 4019. </cmath> <math>(a_{2i-1},a_{2i}), 1\le i \le 1005</math>, the product is at most
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2010 USAJMO Problems
  ...be an integer. Find, with proof, all sequences <math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following three properties: <ol style="list-style-type:lower-alpha">
  3 KB (538 words) - 13:55, 16 June 2020
• 2010 USAJMO Problems/Problem 1
  ...closed under multiplication and a non-square times a square is always a non-square. ...lement <math>g_m</math> of the equivalence class <math>C_m</math> is square-free, since if it were divisible by the square of a prime, the quotient woul
  12 KB (2,338 words) - 20:30, 13 February 2024
• 2010 USAJMO Problems/Problem 2
  <math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following <ol style="list-style-type:lower-alpha">
  6 KB (1,177 words) - 23:09, 20 November 2022
• 2010 USAMO Problems/Problem 1
  // Semi-circle: centre O, radius r, diameter A--B. pair P = foot(Y, A, X); dot(P); label("$P$", P, unit(P-Y)); dot(P);
  13 KB (2,178 words) - 14:14, 11 September 2021
• 2010 USAMO Problems/Problem 2
  with height <math>h_{k-2}</math> or less, the two students are permitted to \binom{0}{0} = 1, \quad \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}
  5 KB (823 words) - 19:20, 3 October 2017
• Scholarships by State
  ...s primarily for students in the U.S. Some of these scholarships are subject-specific. ...fornia [http://www.cagovernorsscholars.org/ website] (NOTE: granted in 2000-2002)
  9 KB (1,107 words) - 15:32, 16 July 2010
• User:Azjps/geogebra
  real labelscalefactor = 0.5; /* changes label-to-point distance */ ...pair hyperbolaRight1 (real t) {return (0.49*(-1-t^2)/(1-t^2),1.04*(-2)*t/(1-t^2));}
  47 KB (6,507 words) - 18:49, 17 October 2019
• 2008 iTest Problems/Problem 77
  ...<math>4</math> triangles of size <math>2^{n-1}</math> which include the all-zero "inner" triangle with vertices <math>\Bigg\{\binom{2^{n-1}}{1},\binom{2^{n-1}}{2^{n-1}-1},\binom{2^n-2}{2^{n-1}-1}\Bigg\}</math>
  5 KB (788 words) - 05:15, 28 January 2019
• 2010 IMO Problems/Problem 2
  Let <math>A'</math> be A-excenter <math>\triangle ABC \implies</math> {{IMO box|year=2010|num-b=1|num-a=3}}
  3 KB (525 words) - 14:52, 16 July 2023
• 2010 IMO Problems/Problem 5
  Type 1) Choose a non-empty box <math>B_j</math>, <math>1\leq j \leq 5</math>, remove one coin fro Type 2) Choose a non-empty box <math>B_k</math>, <math>1\leq k \leq 4</math>, remove one coin fro
  3 KB (411 words) - 06:25, 25 June 2018
• 2010 IMO Problems/Problem 6
  <cmath>a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.</cmath> :<math>\text{To this end, let’s define an n-type to be a vector T = ⟨t1,...,ts⟩ of nonnegative integers such that}</
  4 KB (786 words) - 08:46, 12 March 2024
• 1971 IMO Problems
  All the faces of tetrahedron <math>ABCD</math> are acute-angled triangles. We consider all closed polygonal paths of the form <math>X ...})(i, j = 1, 2, \cdots, n)</math> be a square matrix whose elements are non-negative integers. Suppose that whenever an element <math>a_{ij} = 0</math>,
  3 KB (495 words) - 13:52, 29 January 2021
• 2010 IMO Problems
  Type 1) Choose a non-empty box <math>B_j</math>, <math>1\leq j \leq 5</math>, remove one coin fro Type 2) Choose a non-empty box <math>B_k</math>, <math>1\leq k \leq 4</math>, remove one coin fro
  4 KB (603 words) - 09:22, 10 September 2020
• 2004 AMC 8 Problems
  How many different four-digit numbers can be formed by rearranging the four digits in <math>2004</ma ...lton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the t
  13 KB (1,860 words) - 19:58, 8 May 2023
• Fractional ideal
  2 KB (288 words) - 20:05, 23 January 2017
• 1997 AJHSME Problems
  Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the l ...speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If
  12 KB (1,702 words) - 12:35, 6 November 2022

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