Create the page "P c r" on this wiki! See also the search results found.

• 2015 IMO Problems
  a point <math>C</math> in <math>\mathcal{S}</math> such that <math>AC=BC</math>. We say tha ...tre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</math> in
  4 KB (692 words) - 22:33, 15 February 2021
• Wilson's Theorem
  ...math>(p-1)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathematician Lag ...<math>(p-1)! + 1</math>. Therefore <math>p</math> does not divide <math>(p-1)! + 1</math>.
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  ...(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> ...\frac{b^2}{4a}</math> if <math>a</math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (560 words) - 22:51, 13 January 2024
• Combination
  ...generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> The common forms of denoting the number of combinations of <math>{r}</math> objects from a set of <math>{n}</math> objects is:
  4 KB (615 words) - 11:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ...rrightarrow{v}</math> and <math>\overrightarrow{w}</math> in <math>\mathbb{R}^n</math>, where <math>\overrightarrow{v} \cdot \overrightarrow{w}</math> i ...ce]], and let <math> \langle \cdot, \cdot \rangle : V \times V \to \mathbb{R} </math> be an [[inner product]]. Then for any <math> \mathbf{a,b} \in V <
  13 KB (2,048 words) - 15:28, 22 February 2024
• Menelaus' theorem
  ...C</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R</math> on the intersection of <math>PQ</math> and <math>AB</math>, then pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R;
  5 KB (804 words) - 03:01, 12 June 2023
• Fermat's Little Theorem
  ...ibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>. ...to restrict ourselves to integers <math>{a}</math> not divisible by <math>{p}</math>.
  16 KB (2,675 words) - 10:57, 7 March 2024
• Geometric inequality
  ...ith sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>. The Inequality extends this to [[obtuse triangle| obtuse]] and [ ...an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.
  7 KB (1,296 words) - 14:22, 22 October 2023
• Ptolemy's theorem
  ...[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>: Given cyclic quadrilateral <math>ABCD,</math> extend <math>CD</math> to <math>P</math> such that <math>\angle BAC=\angle DAP.</math>
  7 KB (1,198 words) - 20:39, 9 March 2024
• Number theory/Advanced
  ...g <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line. <cmath>G_k(z)=\sum_{(c,d)\in Z^2\backslash 0} (cz+d)^{-k}</cmath>
  5 KB (849 words) - 16:14, 18 May 2021
• Real number
  ...<math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subsets of the real numbers are the [[rational numb ==The set <math>\mathbb{R}~</math>==
  3 KB (496 words) - 23:22, 5 January 2022
• Function
  * if <math>(a,b)\in f</math> and <math>(a,c)\in f</math> then <math>b=c</math>. (Here <math>(a,b)</math> is an [[ordered pair]].) ...ective (and therefore bijective) from <math>\mathbb{C} \rightarrow \mathbb{C}</math>.
  10 KB (1,761 words) - 03:16, 12 May 2023
• Circle
  ...the [[distance formula]], this gives <math>\sqrt{(x - h)^2 + (y - k)^2} = r</math> which is more commonly written as <cmath>(x - h)^2 + (y - k)^2 = r^2.</cmath>
  9 KB (1,555 words) - 20:05, 2 November 2023
• Muirhead's Inequality
  '''Muirhead's Inequality''' states that if a sequence <math>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a s Given <math>p=(p_1,p_2,\ldots,p_n)\in\mathbb{R}^n</math>, let's write
  8 KB (1,346 words) - 12:53, 8 October 2023
• Vector
  Every vector <math>\overrightarrow{PQ}</math> has a starting point <math>P\langle x_1, y_1\rangle</math> and an endpoint <math>Q\langle x_2, y_2\rangl pen p=linewidth(1);
  7 KB (1,265 words) - 13:22, 14 July 2021
• Exponential form
  ...te it as <math>z=r(\cos\theta+i\sin\theta)=r cis\theta</math>, where <math>r=|z| = \sqrt{a^2 + b^2}</math>. By [[Euler's identity|Euler's formula]], wh pair A,B,C,D,E;
  1 KB (238 words) - 22:51, 20 February 2022
• Rational approximation
  ...r and closer to <math>x</math>, we may have to use larger and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how w ...number <math>x</math> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not e
  7 KB (1,290 words) - 12:18, 30 May 2019
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  pair O=(0,0),A=(-1,0),B=(0,1),C=(1,0),P=(1/2,0),Q=(1/2,sqrt(3)/2),R=foot(P,Q,O); draw(B--O--C--arc(O,C,A)--O--R--P); rightanglemark(O,P,R);
  5 KB (912 words) - 20:06, 14 March 2023
• Goldbach Conjecture
  ...r of representations of a large integer <math>n</math> as the sum of <math>c</math> primes <math>n=p_1+\dotsb+p_c</math> with <math>p_1 \leq \dotsb \leq ...2 \leq x_1 \leq \dotsb \leq x_c: x_1+\ldots+x_c = n} \frac{dx_1 \ldots dx_{c-1}}{\ln x_1 \ldots \ln x_c} </math></center>
  7 KB (1,201 words) - 16:59, 19 February 2024
• Fundamental Theorem of Algebra
  ...s, every polynomial over <math>\mathbb{C}</math> splits over <math>\mathbb{C}</math>, or decomposes into linear factors. ...oss of generality, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. I
  5 KB (832 words) - 14:22, 11 January 2024

View (previous 20 | next 20) (20 | 50 | 100 | 250 | 500)