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• The positive three-digit integer N has a ones digit of 0. What is the probability that N is divisible by 4? Express your answer as a common fraction.
  ...a <math>\boxed{1/2}</math> probability that <math>N</math> is divisible by 4.
  427 bytes (71 words) - 17:35, 12 June 2022

Page text matches

• Pythagorean Theorem
  B = (4, 3); C = (4, 0);
  5 KB (886 words) - 21:12, 22 January 2024
• Physics
  Isaac Newton was born on January 4, 1643, in Lincolnshire, England. Newton was born very shortly after the dea ...places a force on the matter with the same mass <math>n</math>, then <math>n</math> will put an equivalent force in the opposite direction.
  9 KB (1,355 words) - 07:29, 29 September 2021
• 2005 Alabama ARML TST Problems/Problem 7
  ...e series: <center><math>3+\frac{11}4+\frac 94 + \cdots + \frac{n^2+2n+3}{2^n}+\cdots</math>.</center> ...^{\infty} \left(\frac{2n}{2^n}\right)+\sum_{n=1}^{\infty} \left(\frac{3}{2^n}\right)</math>
  1 KB (193 words) - 21:13, 18 May 2021
• 2015 IMO Problems
  ...gers <math>n\geq 3</math>, there exists a balanced set consisting of <math>n</math> points. </li> ...ath> for which there exists a balanced centre-free set consisting of <math>n</math> points. </li>
  4 KB (692 words) - 22:33, 15 February 2021
• Wilson's Theorem
  ...emainder 1. Show that there is an integer <math>{n}</math> such that <math>n^2 + 1</math> is divisible by <math>{p}</math>.
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  *Show that <math>\sum_{k=1}^{n}a_k^2 \geq a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1</math>. [[Inequality_Introductory_Problem_2|Solution]] *Show that <math>x^2+y^4\geq 2x+4y^2-5</math> for all real <math>x</math> and <math>y</math>.
  3 KB (560 words) - 22:51, 13 January 2024
• Arithmetic mean
  <center><math>AM=\frac{x_1+x_2+\cdots+x_n}{n}</math></center> is the arithmetic mean of the <math>{n}</math> numbers <math>x_1,x_2,\ldots,x_n</math>.
  699 bytes (110 words) - 12:44, 20 September 2015
• Polynomial
  * <math>5x^4 - 2x^2 + 9</math>, in the variable <math>x</math> A polynomial in one variable is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. Here, <math>a_i</math> is the <m
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  ...integers <math>(x,y)</math> that are solutions to the equation <math>\frac{4}{x}+\frac{5}{y}=1</math>. (2021 CEMC Galois #4b) ...ice how I artificially grouped up the <math>y</math> terms by adding <math>4*5=20</math>).
  7 KB (1,107 words) - 07:35, 26 March 2024
• Factoring
  <cmath>a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})</cmath> If <math>n=2</math>, this creates the difference of squares factorization, <cmath>a^2-
  3 KB (532 words) - 22:00, 13 January 2024
• Mathematical problem solving
  ...um of the [[series]] <math>\frac11 + \frac14 + \frac19 + \cdots + \frac{1}{n^2} + \cdots</math><br> ...}+\frac{x^4}{5!}-\cdots=\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{4\pi^2}\right)\left(1-\frac{x^2}{9\pi^2}\right)\cdots</math><br>
  2 KB (314 words) - 06:45, 1 May 2014
• Pigeonhole Principle
  ...e''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one hole must contain two or more pigeons. This seemingly tri ...if <math>n</math> balls are to be placed in <math>k</math> boxes and <math>n>k</math>, then at least one box must contain more than one ball.
  11 KB (1,985 words) - 21:03, 5 August 2023
• 1965 AHSME Problems/Problem 40
  ...^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: <math>\textbf{(A)}\ 4 \qquad
  3 KB (571 words) - 00:42, 22 October 2021
• Geometric mean
  ...numbers. Note that if <math>n</math> is even, we take the positive <math>n</math>th root. It is analogous to the [[arithmetic mean]] (with addition r ...1 and 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>.
  2 KB (282 words) - 22:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...four guys in order. By the same logic as above, this is <math>2!\binom{6}{4}=30</math>. Again, <math>|A\cap C|</math> would be putting five guys in ord If <math>(A_i)_{1\leq i\leq n}</math> are finite sets, then:
  9 KB (1,703 words) - 07:25, 24 March 2024
• Cauchy-Schwarz Inequality
  ...th>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Ineq ...ghtarrow{v}</math> and <math>\overrightarrow{w}</math> in <math>\mathbb{R}^n</math>, where <math>\overrightarrow{v} \cdot \overrightarrow{w}</math> is t
  13 KB (2,048 words) - 15:28, 22 February 2024
• Factorial
  ...[[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>. * <math>4! = 24</math>
  10 KB (809 words) - 16:40, 17 March 2024
• Discriminant
  ==Discriminant of polynomials of degree n== .../math> with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it.
  4 KB (734 words) - 19:19, 10 October 2023
• 2006 AIME II
  * [[2006 AIME II Problems/Problem 4]] {{AIME box|year=2006|n=II|before=[[2006 AIME I]]|after=[[2007 AIME I]], [[2007 AIME II|II]]}}
  1 KB (133 words) - 12:32, 22 March 2011
• Fermat's Little Theorem
  ...s, then <math>a^{\varphi(n)} \equiv 1 \pmod{n}</math>, where <math>\varphi(n)</math> denotes [[Euler's totient function]]. In particular, <math>\varphi( ...let <math>S = \{\text{natural numbers relatively prime to and less than}\ n\}</math> <math>\square</math>
  16 KB (2,675 words) - 10:57, 7 March 2024
• Euler's Totient Theorem
  ...range <math>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive ...{k}\equiv 1\pmod{n}</math>, and by [[Lagrange's Theorem]] <math>bk|\varphi(n)</math> which means
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  ...[[area]] <math>A</math> and [[perimeter]] <math>P</math>, then <math>\frac{4\pi A}{P^2} \le 1</math>. This means that given a perimeter <math>P</math> f <b>Proof of Lemma: </b> Let <math>M</math> and <math>N</math> be the projections of <math>E</math> and <math>F</math> onto line <m
  7 KB (1,296 words) - 14:22, 22 October 2023
• Elementary symmetric sum
  ...f those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then: ...um_{sym}f(x)</math>. The <math>n</math>th can be written <math>\sum_{sym}^{n}f(x)</math>
  2 KB (275 words) - 12:51, 26 July 2023
• Number theory/Advanced
  <cmath>f(z)=\sum_{n\ge 0}a_nq^n.</cmath> ...n series <math>G_4</math> and <math>G_6</math> are modular forms of weight 4 and 6 respectively.
  5 KB (849 words) - 16:14, 18 May 2021
• Divisibility rules
  ...nly if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. A number is divisible by <math>5^n</math> if and only if the last <math>n</math> digits are divisible by that power of 5.
  8 KB (1,315 words) - 18:18, 2 March 2024
• Induction
  ...it works for <math>n=1+1=2</math>, which in turn means it works for <math>n=2+1=3</math>, and so on. ...nd show that if <math>{n=k}</math> gives the desired result, so does <math>n=k+2</math>. If you wish, you can similarly induct over the powers of 2.
  5 KB (768 words) - 20:45, 1 September 2022
• Greatest common divisor
  <math>270=2\cdot3^3\cdot5</math> and <math>144=2^4\cdot3^2</math>. The common factors are 2 and <math>3^2</math>, so <math>GCD ...an use the recursive formula <math>GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)</math>.
  2 KB (288 words) - 22:40, 26 January 2021
• Overcounting
  "How many numbers less than or equal to 100 are divisible by 2 or 3 but not 4?". ...Suppose we know the total number of people invited to the party, say <math>n</math>.
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  ...uare \square \square \square,</cmath> which we have to populate with <math>4</math> <math>A</math>s and <math>3</math> <math>B</math>s. Using constructi ...ath>B</math>s. Starting with the <math>A</math>s, we must choose the <math>4</math> boxes of their placement; because all the <math>A</math>s are indist
  12 KB (1,896 words) - 23:55, 27 December 2023
• Euclidean algorithm
  <math>r_{n-1} \pmod {r_n} \equiv 0</math><br> <math>r_{n-1} = r_n \cdot q_{n+1} +0</math><br>
  6 KB (924 words) - 21:50, 8 May 2022
• Generating function
  ...se 0}+{n \choose 1}x + {n \choose 2}x^2+\cdots+</math><math>{n \choose n}x^n</math>. ...s the number of ways we can get <math>{k}</math> heads when flipping <math>n</math> different coins.
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  ...x]] <math>a</math>, <math>b</math>, and [[non-negative]] [[integer]] <math>n</math>, <center><math>(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k</math></center>
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  ...N = p_1p_2\cdots p_n + 1</math> is not divisible by any of them, but <math>N</math> must [[#Importance of Primes|have]] a prime factor, which leads to a ...ividing larger numbers would result in a quotient smaller than <math>\sqrt{n}</math>.
  6 KB (985 words) - 12:38, 25 February 2024
• Function
  ...in the second. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value ...ve from <math>\mathbb{R} \rightarrow \mathbb{R}</math> (since <math>f(2) = 4 = f(-2)</math>) nor surjective from <math>\mathbb{R} \rightarrow \mathbb{R}
  10 KB (1,761 words) - 03:16, 12 May 2023
• Complementary counting
  ...ructive]] approach, the first digit can be one of seven integers; <math>1, 4, 5, 6, 7, 8,</math> and <math>9</math>. Note that the first digit cannot be ...use can be, four options for the second, and so on. Hence, there are <math>4^7 = 16384</math> ways she can color the four houses.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Prime factorization
  ...the '''prime factorization''' of <math>n</math> is an expression for <math>n</math> as a product of powers of [[prime number]]s. An important theorem o <math>n = {p_1}^{e_1} \cdot {p_2}^{e_2}\cdot{p_3}^{e_3}\cdots{p_k}^{e_k}</math>
  3 KB (496 words) - 22:14, 5 January 2024
• Composite number
  ...me composite numbers are <math>4=2^2</math> and <math>12=2\times 6=3\times 4</math>. Composite numbers '''atleast have 2 distinct [[prime]] [[divisors]] 4 6 8 9 10 12 14 15 16 18 20 21 22 24 25 26 27 28 30 32 33 34 35 36 38 39 40
  6 KB (350 words) - 12:58, 26 September 2023
• Natural number
  ...tegers (sometimes called [[whole number]]s). In particular, <math>\mathbb{N}</math> usually includes zero in the contexts of [[set theory]] and [[abstr
  1 KB (162 words) - 21:44, 13 March 2022
• Ellipse
  ...ts are perpendicular. Drawing all four semi-axes divides the ellipse into 4 [[congruent (geometry)|congruent]] quarters. pair P=(3,12/5), F1=(-4,0), F2=(4,0);
  5 KB (892 words) - 21:52, 1 May 2021
• Base numbers
  ...46. This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...</math>, spits out <math>P(n)</math>, the value of the polynomial at <math>n</math>. However, the oracle charges a fee for each such computation, so you
  4 KB (547 words) - 17:23, 30 December 2020
• Common factorizations
  \text{\textbullet}&&x^{n}-y^{n}&=(x-y)(x^{n-1}+x^{n-2}y+\cdots +xy^{n-1}+y^n) ...^2 \\\phantom{\text{\textbullet}}&&- b^4 + 2 b^2 d^2 - 4 b c^2 d + c^4 - d^4&=\det\begin{bmatrix}a&b&c&d\\d&a&b&c\\c&d&a&b\\b&c&d&a\end{bmatrix}\\&&&=(a
  2 KB (327 words) - 13:13, 6 July 2023
• Pi
  ...ost surprising places, such as in the sum <math>\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for p ...approximates <math>\frac{\pi}{4}</math>. This can simply be multiplied by 4 to approximate <math>\pi</math>.
  8 KB (1,469 words) - 21:11, 16 September 2022
• Fibonacci sequence
  ...<math>F_1 = F_2 = 1</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n \geq 3</math>. This is the simplest nontrivial example of a [[linear recur ...>n \geq 2</math>. In general, one can show that <math>F_n = (-1)^{n+1}F_{-n}</math>.
  6 KB (957 words) - 23:49, 7 March 2024
• Muirhead's Inequality
  ...equence <math>(5,1)</math> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</ma x^5y^1+y^5x^1&=\frac{3}{4}\left(x^5y^1+y^5x^1\right)+\frac{1}{4}\left(x^5y^1+y^5x^1\right)\\
  8 KB (1,346 words) - 12:53, 8 October 2023
• Cardinality
  ...t]]s in that set, i.e. the size of the set. The cardinality of <math>\{3, 4\}</math> is 2, the cardinality of <math>\{1, \{2, 3\}, \{1, 2, 3\}\}</math> ...4\}</math> is <math>|\{3, 4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (A)</math> are used.
  2 KB (263 words) - 00:54, 17 November 2019
• Fundamental Theorem of Calculus
  * Evaluate: <math>\int_2^5 x^3 dx</math> and <math>\int_{.2}^{.4} \cos(x) dx</math>. (The next few questions are meant as hints for how to ...ctly how far the object moved between times <math>t=.2</math> and <math>t=.4</math>. Interpret the distance that the object moved geometrically, as an
  11 KB (2,082 words) - 15:23, 2 January 2022
• Polygon
  ...et]] of [[vertex|vertices]], <math>\{A_1, A_2, \ldots, A_n\}</math>, <math>n \geq 3</math>, with [[edge]]s <math>\{\overline{A_1A_2}, \overline{A_2A_3} ...ewer -- it will have "degenerated" from an <math>n</math>-gon to an <math>(n - 1)</math>-gon. (In the case of triangles, this will result in either a l
  2 KB (372 words) - 19:04, 30 May 2015
• 2011 AMC 10B Problems/Problem 22
  <math> \textbf{(A)}\ 5\sqrt{2} - 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad\textbf{(D)}\ \frac{ ...s$",(W--Z),E,red); label("$s$",(X--Y),W,red); label("$s\sqrt{2}$",(W--X),N,red);
  4 KB (691 words) - 18:38, 19 September 2021
• Vector
  ...rected 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 abs ...sional vector can be described in this coordinate form as an ordered <math>n</math>-tuple of numbers within angle brackets or parentheses, <math>(x\,\,y
  7 KB (1,265 words) - 13:22, 14 July 2021
• Logarithm
  ...2.285669651531203956336043826\ldots=x</cmath> such that:<cmath>(^24)^x=4^{4^x}\approx(3^5)^6</cmath> # Evaluate <math>(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)</math>.
  4 KB (680 words) - 12:54, 16 October 2023
• Law of Cosines
  draw(D--(30,4)--(34,4)--(34,0)--D); 1. If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
  6 KB (1,003 words) - 09:11, 7 June 2023
• Angle bisector
  A=(4,2); D=(3,4);
  3 KB (575 words) - 15:27, 19 March 2023
• Law of Sines
  pair A=(-1,5), B=(-4,-1), C=(4,-1), D, O; *If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
  4 KB (658 words) - 00:37, 8 September 2018
• Sequence
  ...ath>\mathbb{N}</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>. ...> is called the [[limit]] of <math>(x_n)</math> and is written <math>\lim_{n \to \infty} x_n</math>. The statement that <math>(x_n)</math> converges to
  2 KB (413 words) - 21:18, 13 November 2022
• Geometric sequence
  For example, <math>1, 2, 4, 8</math> is a geometric sequence with common ratio <math>2</math> and <mat ...progression if and only if <math>a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}</math>. A similar definition holds for infinite geometric sequences. It
  4 KB (644 words) - 12:55, 7 March 2022
• Arithmetic sequence
  ...on difference <math>-8</math>; however, <math>7, 0, 7, 14</math> and <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference ...progression if and only if <math>a_2 - a_1 = a_3 - a_2 = \cdots = a_n - a_{n-1}</math>. A similar definition holds for infinite arithmetic sequences. It
  4 KB (736 words) - 02:00, 7 March 2024
• Fermat's Last Theorem
  ...\geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...he never published it, though he did publish a proof for the case <math>n=4</math>. It seems unlikely that he would have circulated a proof for the sp
  3 KB (453 words) - 11:13, 9 June 2023
• 2005 USAMO Problems/Problem 4
  ...ece of length <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? For <math>0 \le x \le n</math>, it is easy to see that the number of stable tables is <math>(x+1)^2
  7 KB (1,276 words) - 20:51, 6 January 2024
• Diophantine equation
  If <math>n>1</math>, <math>2n, n^2 - 1, n^2 + 1</math> is a Pythagorean triple. ...ny <math>m,n</math>(<math>m>n</math>), we have <math>m^2 - n^2, 2mn, m^2 + n^2</math> is a Pythagorean triple.
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...s to the [[circumcenter]]. This creates a triangle that is <math>\frac{1}{n},</math> of the total area (consider the regular [[octagon]] below as an ex ...ound using [[trigonometry]] to be of length <math>\frac s2 \cot \frac{180}{n}^{\circ}</math>.
  6 KB (1,181 words) - 22:37, 22 January 2023
• Floor function
  ...r+\left\lfloor a+\frac{1}{n}\right\rfloor+\ldots+\left\lfloor a+\frac{n-1}{n}\right\rfloor</cmath> *<math>\lfloor -3.2 \rfloor = -4</math>
  3 KB (508 words) - 21:05, 26 February 2024
• Pascal's triangle
  ...he sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>. ...ved from the combinatorics identity <math>{n \choose k}+{n \choose k+1} = {n+1 \choose k+1}</math>. Thus, any number in the interior of Pascal's Triang
  5 KB (838 words) - 17:20, 3 January 2023
• Newton's Sums
  Consider a polynomial <math>P(x)</math> of degree <math>n</math>, <center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center>
  4 KB (690 words) - 13:11, 20 February 2024
• 1985 IMO Problems/Problem 5
  ...ath> and <math>BC</math> again at distinct points <math>K</math> and <math>N</math> respectively. Let <math>M</math> be the point of intersection of the {{IMO box|year=1985|num-b=4|num-a=6}}
  3 KB (496 words) - 13:35, 18 January 2023
• Trigonometry
  ...=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \cdots</cmath>
  8 KB (1,217 words) - 20:15, 7 September 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power. ...1}{a}}</math> and the harmonic mean's root mean power is -1 as <math>\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}=\sqrt[-1]{\frac{x_1^{-1}+\cdots+x_a^{-
  5 KB (912 words) - 20:06, 14 March 2023
• Goldbach Conjecture
  <math>4 = 2 + 2</math> ...might expect the total number of ways to write a large even integer <math>n</math> as the sum of two odd primes to be roughly
  7 KB (1,201 words) - 16:59, 19 February 2024
• Riemann Hypothesis
  ...function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldots</math>. These are called the trivial zeros. This hypothesis is o ...uld hold. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>.
  2 KB (425 words) - 12:01, 20 October 2016
• Quadratic residues
  ...ulo]] <math>m</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...p-1}{2}}</math>, so <math>\left(\frac{-1}{p}\right)=1 \iff p \equiv 1 \mod 4</math>
  5 KB (778 words) - 13:10, 29 November 2017
• Power of a Point Theorem
  Let <math>P</math> be a point, and let <math>S</math> be an <math>n</math>-sphere. Let two arbitrary lines passing through <math>P</math> inter ...th> intersect at <math>R</math>. If <math>AR:BR=1:4</math> and <math>CR:DR=4:9</math>, find the ratio <math>AB:CD</math>.
  5 KB (827 words) - 17:30, 21 February 2024
• 2004 AIME I
  * [[2004 AIME I Problems/Problem 4]] {{AIME box|year=2004|n=I|before=[[2003 AIME I]], [[2003 AIME II|II]]|after=[[2004 AIME II]]}}
  1 KB (135 words) - 18:15, 19 April 2021
• Collatz Problem
  ...sitive integer]] <math>n</math>, the sequence <math>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people h ==Properties of <math>f(n)</math> ==
  1 KB (231 words) - 19:45, 24 February 2020
• 2004 AIME II
  * [[2004 AIME II Problems/Problem 4]] {{AIME box|year=2004|n=II|before=[[2004 AIME I]]|after=[[2005 AIME I]], [[2005 AIME II|II]]}}
  1 KB (135 words) - 12:24, 22 March 2011
• 2005 AIME I
  * [[2005 AIME I Problems/Problem 4 | Problem 4]] {{AIME box|year=2005|n=I|before=[[2004 AIME I|2004 AIME I]], [[2004 AIME II|II]]|after=[[2005 AIME
  1 KB (154 words) - 12:30, 22 March 2011
• 2006 AIME I
  * [[2006 AIME I Problems/Problem 4]] {{AIME box|year=2006|n=I|before=[[2005 AIME I]], [[2005 AIME II|II]]|after=[[2006 AIME II]]}}
  1 KB (135 words) - 12:31, 22 March 2011
• 2005 AIME II
  * [[2005 AIME II Problems/Problem 4]] {{AIME box|year=2005|n=II|before=[[2005 AIME I]]|after=[[2006 AIME I]], [[2006 AIME II|II]]}}
  1 KB (135 words) - 12:30, 22 March 2011
• Mock AMC
  | n/a | n/a
  51 KB (6,175 words) - 20:58, 6 December 2023
• Bertrand's Postulate
  ...em. A more widely known version states that there is a prime between <math>n</math> and <math>2n</math>. ...closer look at the [[combinations|binomial coefficient]] <math>\binom{2n}{n}</math>. Assuming that the reader is familiar with that proof, the Bertrand
  2 KB (309 words) - 21:43, 11 January 2010
• Riemann zeta function
  <cmath>\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}= 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots</cmath>
  9 KB (1,547 words) - 03:04, 13 January 2021
• Modular arithmetic/Introduction
  draw((0,0), linewidth(4)); <math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>
  15 KB (2,396 words) - 20:24, 21 February 2024
• Perfect square
  ...so that <math>m^2=n</math>. The first few perfect squares are <math>0, 1, 4, 9, 16, 25, 36</math>. ...th>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>
  954 bytes (155 words) - 01:14, 29 November 2023
• Modular arithmetic/Intermediate
  ...th>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...ntegers modulo <math>n</math>''' (usually known as "the integers mod <math>n</math>," or <math>\mathbb{Z}_n</math> for short). This structure gives us
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2006 AIME I Problems
  {{AIME Problems|year=2006|n=I}} == Problem 4 ==
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  === Solution 4 === {{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 14
  ...> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the triple M=(B+C)/2,S=(4*A+T)/5;
  6 KB (980 words) - 21:45, 31 March 2020
• 2006 AIME I Problems/Problem 13
  ...h> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]] .../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
• 2006 AIME I Problems/Problem 10
  ...ough the tangent point of the other two circles. This clearly will cut the 4 circles into two regions of equal area. Using super advanced linear algebra {{AIME box|year=2006|n=I|num-b=9|num-a=11}}
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 8
  pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1, label("$\mathcal{Q}$",(4.2,-1),NW);
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AIME I Problems/Problem 7
  for(int i=0; i<4; i=i+1) { pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);
  4 KB (709 words) - 01:50, 10 January 2022
• 2006 AIME I Problems/Problem 5
  <math> b = 4 </math> <math>b=4</math>
  3 KB (439 words) - 18:24, 10 March 2015
• 2006 AIME I Problems/Problem 4
  ...oduct <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. ...<math>5</math> dividing it, for <math>76</math> extra; every n! for <math>n\geq 50</math> has one more in addition to that, for a total of <math>51</ma
  2 KB (278 words) - 08:33, 4 November 2022
• 2006 AIME I Problems/Problem 3
  ...math> so we take <math>N_0 = 25</math> and <math>n = 2.</math> Then <cmath>N = 7 \cdot 10^2 + 25 = \boxed{725},</cmath> and indeed, <math>725 = 29 \cdot ...ow that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1,</math> <math>2</math> or <math>3</math> digits. Checkin
  4 KB (622 words) - 03:53, 10 December 2022
• 2022 AIME I Problems/Problem 1
  Q(20) &= &\hspace{1mm}-800 + 20c + d &= 53, &&(4) ...]</math> from <math>5\cdot[(1)+(2)]</math> to get <cmath>b+d=5\cdot(54+54)-4\cdot(53+53)=\boxed{116}.</cmath>
  4 KB (670 words) - 13:03, 13 November 2023
• 2021 JMC 10
  ...bf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 7</math> ==Problem 4==
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  ...ne <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>? == Problem 4 ==
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  == Problem 4 == [[2006 AMC 12A Problems/Problem 4|Solution]]
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  ...8 \qquad (\mathrm {B}) \ -4 \qquad (\mathrm {C})\ 2 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 8 == Problem 4 ==
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  == Problem 4 == [[2004 AMC 12A Problems/Problem 4|Solution]]
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  Members of the Rockham Soccer League buy socks and T-shirts. Socks cost &#36;4 per pair and each T-shirt costs &#36;5 more than a pair of socks. Each memb <math> \mathrm{(A) \ } 4.5\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 18
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  <math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </
  12 KB (1,792 words) - 13:06, 19 February 2020

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