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  • ...aving these point as vertices. Prove that no more than <math>70 \%</math> of these triangles are acute-angled. ...</math> replaced by <math>4</math>. Hence at most <math>\frac{3}{4}</math> of the <math>10</math>, or <math>7.5</math>, can be acute, and hence at most <
    6 KB (1,054 words) - 17:09, 11 December 2024
  • A '''triangle''' is a type of [[polygon]]. ...ree [[edge | sides]], with the smaller angle measures of the intersections of the sides summing to 180 degrees. Triangles exist in Euclidean [[geometry]]
    4 KB (631 words) - 20:16, 8 October 2024
  • ...nar figure]] consisting of straight [[line segment]]s. There are two types of polygons: [[convex polygon|convex]] and [[concave polygon|concave]]. In their most general form, polygons are an ordered [[set]] of [[vertex|vertices]], <math>\{A_1, A_2, \ldots, A_n\}</math>, <math>n \geq 3
    2 KB (372 words) - 18:04, 30 May 2015
  • ...ng with figures in a given [[space]]. It is one of the two oldest branches of mathematics, along with [[arithmetic]] (which eventually branched into [[nu ...collegiate]] [[mathematics competitions]] is Euclidean geometry. This type of geometry was first formally outlined by the Greek [[mathematician]] [[Eucli
    3 KB (478 words) - 00:52, 25 August 2024
  • ...nter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]] ...section of its (interior) [[angle bisector]]s. The incenter is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter.
    2 KB (381 words) - 18:38, 24 November 2011
  • ...>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math>. Let <math>x</math> denote the common side length of the rhombi.
    5 KB (730 words) - 14:05, 15 January 2024
  • The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each numb each of the resulting numbers is doubled. What is the sum of the final two
    13 KB (1,957 words) - 11:53, 24 January 2024
  • ...s, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the d What is the value of
    10 KB (1,547 words) - 03:20, 9 October 2022
  • ...8}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ldo ...is the smallest positive multiple of <math>15</math> such that every digit of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\f
    6 KB (933 words) - 00:15, 19 June 2022
  • What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x} ...math>x_3</math>, <math>x_4</math>, and <math>x_5</math> satisfy the system of equations below.
    5 KB (847 words) - 23:35, 19 December 2024
  • ...> as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations .../math>, let <math>f_1(k)</math> denote the square of the sum of the digits of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}
    6 KB (902 words) - 07:57, 19 June 2021
  • ...nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>.
    6 KB (870 words) - 09:14, 19 June 2021
  • ...s of square <math>S_{i+2}.</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form < Find the last three digits of the product of the positive roots of
    6 KB (1,000 words) - 23:25, 26 March 2024
  • ...The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as <math> m/n, </math> where <math> m </math> a ...she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
    6 KB (965 words) - 15:36, 8 September 2019
  • ...math>PD=6</math>, <math>PE=3</math>, and <math>CF=20</math>, find the area of <math>\triangle ABC</math>. ...ke use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math> intersects line <math>Y
    14 KB (2,234 words) - 15:31, 22 December 2024
  • ...h interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>? The formula for the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.
    3 KB (516 words) - 18:18, 16 April 2024
  • ...>. Let <math>\frac{m}{n}</math>, in lowest terms, denote the [[perimeter]] of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. ...gle CRQ</math>). Quickly we realize that <math>O</math> is also the center of the rectangle.
    8 KB (1,270 words) - 22:36, 27 August 2023
  • ...are positive integers and <math>d_{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> .../math> be the midpoint of <math>\overline{AB}</math>. Then <math>\overline{OF} \perp \overline{AB}</math>.
    3 KB (484 words) - 12:11, 14 January 2023
  • ...> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...x of the triangle coincides with that of the rectangle. Set one [[vertex]] of the triangle at <math>A</math>, and the other two points <math>E</math> and
    5 KB (811 words) - 20:39, 20 July 2024
  • ...>A_1</math>, <math>A_n</math>, and <math>B</math> are consecutive vertices of a regular polygon? ...0}{n}</math>, <math>\angle A_nA_1B = \frac{(m-2)180}{m}</math>, and <math>\angle A_2A_1B = 60^{\circ}</math>. Since those three angles add up to <math>360^{
    3 KB (497 words) - 23:39, 21 December 2018

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