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• Mill's Constant
  ...we are not sure) and Mill's Constant is the smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>
  794 bytes (105 words) - 01:59, 15 January 2022
• 2020 CIME II Problems/Problem 10
  ...ath>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c\}</math>, for if <math>a+b\geq 6c</math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Right
  2 KB (332 words) - 09:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ..._i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Inequality]], Cauchy-Schwarz forms the found ...math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulation comes in handy in linear algeb
  13 KB (2,048 words) - 15:28, 22 February 2024
• Discriminant
  ...the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. *If <math>D=0</math>, then at least two of the roots are equal;
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  If line <math>PQ</math> intersecting <math>AB</math> on <math>\triangle ABC</m The converse of Menelaus' theorem is also true. If <math>\frac{BP}{PC} \cdot \frac{CQ}{QA} \cdot \frac{AR}{RB} = 1</math> in t
  5 KB (804 words) - 03:01, 12 June 2023
• Number theory
  Some branches of number theory may only deal with a certain subset of the real numbers, such as [[integer]]s, [[pos ...p://www.amazon.com/exec/obidos/ASIN/0262680289 (details)] Out of print but if you can find it in a library or used, you might love it and learn a lot. Wr
  3 KB (399 words) - 23:08, 8 January 2024
• Fermat's Little Theorem
  ...r arithmetic]] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is If <math>{a}</math> is an [[integer]], <math>{p}</math> is a [[prime number]]
  16 KB (2,658 words) - 16:02, 8 May 2024
• Parabola
  ...p) upwards/right if <math>a</math> is positive, and will be downwards/left if <math>a</math> is negative.
  3 KB (551 words) - 16:22, 13 September 2023
• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if On the other hand, if <math> a_1\geq a_2\geq ... \geq a_n </math> and
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  ...h>\{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 integer [[r .... All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the sa
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  The [[Isoperimetric Inequality]] states that if a figure in the plane has [[area]] <math>A</math> and [[perimeter]] <math>P ...e the total value of the expression. To prove this, all we need to show is if <math>0<A,B<180</math>, then <math>\sin(A+B)+\sin(A-B)<2\sin A</math>. This
  7 KB (1,296 words) - 14:22, 22 October 2023
• Brahmagupta's Formula
  ...formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side [[length]]s, given as follows: <cmath>[ABCD] = \sqrt{(s-a)(s- If we draw <math>AC</math>, we find that <math>[ABCD]=\frac{ab\sin B}{2}+\frac
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...> be the regular heptagon. Consider the quadrilateral <math> ABCE </math>. If <math> a </math>, <math> b </math>, and <math> c </math> represent the leng
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  ...th>k</math> of 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>,
  2 KB (275 words) - 12:51, 26 July 2023
• Cyclic quadrilateral
  ...rty is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
  1 KB (162 words) - 20:39, 9 March 2024
• Algebra
  As if to add to the confusion, "[[algebra (structure) |algebra]]" is the name for
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...mmarize the group law by saying <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line. ...ssociated to <math>z</math> and <math>z'</math> are isomorphic if and only if there is a relation
  5 KB (849 words) - 16:14, 18 May 2021
• 1959 IMO Problems/Problem 2
  ...<math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? '''Case I:''' If <math>x \le 1</math>, then <math>|x-1| = 1 - x</math>, and the equation red
  3 KB (466 words) - 12:04, 12 April 2024
• Harmonic mean
  ...to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we
  1 KB (196 words) - 00:49, 6 January 2021
• Euler's totient function
  ...an or equal to <math> n </math> that are divisible by <math> p_1 </math>. If we do the same for each <math> p_i </math> and add these up, we get (d) If <math>p</math> is prime and <math>n\ge{1},</math> then <math>\phi(p^n)=p^n-
  5 KB (898 words) - 19:12, 28 January 2024
• Probability
  ...slate a very basic situation into measure-theoretic language. Furthermore, if one wanted to determine whether the coin was fair or weighted, it would be
  4 KB (588 words) - 12:47, 2 October 2022
• Contest strategies
  ...imation can narrow down the possible values. It can also be used to check if an answer is reasonable. ...ompletely solving the problem, especially when the problem stumps you, and if you can narrow the problem down to a couple of choices, you may be able to
  3 KB (538 words) - 13:13, 16 January 2021
• Divisibility
  ...— or using [[modular arithmetic]], that <math>b \equiv 0 \pmod a</math>. If <math>b</math> does ''not'' divide <math>a</math>, we write that <math>b \n
  2 KB (277 words) - 16:21, 29 April 2023
• Rearrangement Inequality
  ...zed when <math>A</math> and <math>B</math> are oppositely sorted (that is, if <math>a_k</math> is less than or equal to exactly <math>{i}</math> of the o ...number of tens, etc. Meanwhile, you would find yourself very disappointed if there were more pennies than nickels, more nickels than dimes, and so on.
  5 KB (804 words) - 13:54, 26 January 2023
• Real number
  We say that <math>\alpha</math> is a '''cut''' if and only if (2) If <math>a\in\alpha</math>, <math>p\in\alpha\forall p<a</math>
  3 KB (496 words) - 23:22, 5 January 2022
• Divisibility rules
  ...her [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other bases have their own, different versions of these rules. ...r is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
  8 KB (1,315 words) - 18:18, 2 March 2024
• Induction
  ...t shown to hold for a certain value (the Base Case); it is then shown that if the desired result holds for a certain value, it then holds for another, cl ...<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
• Orthic triangle
  The orthic triangle does not exist if <math>\triangle ABC</math> is right. The two cases of when <math>\triangle If <math>\triangle ABC</math> is acute, then the incenter of the orthic triang
  8 KB (1,408 words) - 11:54, 8 December 2021
• Vertex
  ...y</math>-value of the graph is minimal (if <math>a > 0</math>) or maximal (if <math>a < 0</math>).
  1,000 bytes (162 words) - 13:08, 20 February 2024
• Overcounting
  ...order to count the number of handshakes. It's not a matter of a great deal if there are less than 10 persons. But assume a party from the President, ther ...ople. According to the formula we get 2 handshakes, but wait, we will have only 1 handshake between two persons. That means we have overcounted somewhere.
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  ...idea, but there are multiple ways one might proceed with the construction. If we were to go like before and break the problem down by each box, we'd get ...ir position is actually predetermined by choosing the <math>A</math>s; the only place the three <math>B</math>s can go is in the three empty boxes, so we d
  12 KB (1,896 words) - 23:55, 27 December 2023
• Jensen's Inequality
  If <math>{F}</math> is a concave function, we have: We only prove the case where <math>F</math> is concave. The proof for the other cas
  3 KB (623 words) - 13:10, 20 February 2024
• Casework
  ...xists, casework may be considered [[brute force]]. This is especially true if that alternative solution uses [[complementary counting]]. ''How many words are less than four letters long and contain only the letters A, B, C, D, and E? Here, 'word' refers to any string of letters
  5 KB (709 words) - 10:28, 19 February 2024
• Euclidean algorithm
  ...<math>mb</math> and, thereby, their difference <math>r</math>. Similarly, if <math>{d}</math> divides both <math>b</math> and <math>r</math>, it should * If <math>b=0</math>, then <math>\gcd(a,b)=a</math>.
  6 KB (924 words) - 21:50, 8 May 2022
• Conic section
  * An [[ellipse]] is formed if the cutting plane makes an [[angle]] with an [[axis]] that is larger than t If the cutting plane passes through the vertex of the cone, the result is a de
  5 KB (891 words) - 01:14, 9 January 2023
• Generating function
  If we let <math>A(k)={n \choose k}</math>, then we have <math>{n \choose 0}+{n ...0</math>) or <math>x \in \left( \frac{1}{a}, -\frac{1}{a} \right)</math> (if <math>a < 0</math>), generating the powers of <math>a</math> for any real <
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  Therefore, if the theorem holds under <math>n+1</math>, it must be valid. ...e complicated [[polynomial]]s with [[binomial coefficient]]s. For example, if a contest problem involved the polynomial <math>x^5+4x^4+6x^3+4x^2+x</math>
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  ...' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. ...s being neither prime nor [[composite number|composite]] because it is its only factor among the [[natural number|natural numbers]].
  6 KB (985 words) - 12:38, 25 February 2024
• Recursion
  ...example of a recursive definition is that of the [[Fibonacci sequence]]. If we let <math>F_n</math> be the <math>n</math>th Fibonacci number, the seque
  2 KB (316 words) - 16:03, 1 January 2024
• Function
  ...A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if * if <math>(a,b)\in f</math> and <math>(a,c)\in f</math> then <math>b=c</math>.
  10 KB (1,761 words) - 03:16, 12 May 2023
• Complementary counting
  More formally, if <math>B</math> is a [[subset]] of <math>A</math>, complementary counting ex ...any house a single color. In how many ways can she color the seven houses if at least one pair of consecutive houses must have the same color?
  8 KB (1,192 words) - 17:20, 16 June 2023
• Combinatorial identity
  If <math>k > n</math> then <math>\binom{n}{k} = 0 = \binom{n - 1}{k - 1}sfds + Also, we can look at Pascal's Triangle to see why this is. If we were to extend Pascal's Triangle to row n, we would see the term <math>\
  12 KB (1,996 words) - 12:01, 18 May 2024
• Integer
  ...), or the negatives of the natural numbers (<math>-1,-2,-3,\ldots</math>). If <math>a</math> and <math>b</math> are integers, then their sum <math>a+b</m ...egers). Integers in Python can be of arbitrary size, theoretically limited only by computer memory.
  2 KB (296 words) - 15:04, 5 August 2022
• Prime factorization
  ...s (or powers of primes) are a factor and then move up to a different prime if said prime is not a factor. When a prime is a factor, we factor out the p
  3 KB (496 words) - 22:14, 5 January 2024
• Circle
  ...ometry)|tangent]] of that circle. Note that any point on a circle can have only one tangent. ...nt that has endpoints on the circle is called the [[chord]] of the circle. If the chord is extended to a line, that line is called a secant of the circle
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  ...on <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>. In particular, if the center of the ellipse is the origin this simplifies to <math>\frac{x^2}
  5 KB (892 words) - 21:52, 1 May 2021
• Divisor
  ...al number <math>{k}</math> such that <math>n=kd</math> or, in other words, if <math>\frac nd</math> is also a natural number (i.e <math>d</math> divides See the main article on [[counting divisors]]. If <math>n=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}}\cdot\dots\cdot p_m^{\al
  1 KB (274 words) - 19:50, 29 August 2023
• Base numbers/Common bases
  ...11103_8</math> only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).
  2 KB (351 words) - 10:39, 1 October 2015
• Henstock-Kurzweil integral
  ...h> is ''Generalized Riemann Integrable'' on <math>[a,b]</math> if and only if, <math>\forall\epsilon>0</math>, there exists a [[gauge]] <math>\delta:[a,b if <math>\mathcal{\dot{P}}</math> is a <math>\delta</math>-fine [[Partition of
  2 KB (401 words) - 09:46, 31 January 2018
• Multiple
  ...me other integer. Thus <math>k</math> is a multiple of <math>m</math> only if <math>k</math> can be written in the form <math>mn</math>, where <math>m</m
  860 bytes (142 words) - 22:51, 26 January 2021

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