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  • ...ften possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elementary means. One can find the area of even ...y. It makes sense that the area of a rectangle with length <math>l</math> and width <math>w</math> is simply <math> l\cdot w</math>.
    7 KB (1,185 words) - 19:48, 25 November 2024
  • Two is <math>10 \%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>? The equations <math>2x + 7 = 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <ma
    13 KB (1,965 words) - 21:18, 7 September 2024
  • ...ath>6</math> points, each incorrect answer is worth <math>0</math> points, and each problem left unanswered is worth <math>2.5</math> points. If Charlyn l ...randdaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
    13 KB (1,953 words) - 23:31, 25 January 2023
  • ...certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer ...that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
    12 KB (1,792 words) - 12:06, 19 February 2020
  • ...lar 2 </math> each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
    13 KB (2,028 words) - 15:32, 22 March 2022
  • ...tangent to the circles as shown, and the sides <math>\overline{AB}</math> and <math>\overline{AC}</math> are congruent. What is the area of <math>\triang Let the centers of the smaller and larger circles be <math>O_1</math> and <math>O_2</math> , respectively.
    5 KB (811 words) - 10:44, 30 November 2024
  • ...integers, <math> m </math> and <math> p </math> are [[relatively prime]], and <math> n </math> is not divisible by the square of any [[prime]], find <mat ...math>H</math>. Let the endpoints of the chord/tangent be <math>A,B</math>, and the foot of the perpendicular from <math>O_3</math> to <math>\overline{AB}<
    4 KB (693 words) - 12:03, 28 December 2021
  • ...e area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \rfloor</math> (the [[floor function]]). .... Thus, the radius of <math>C</math> has a length of <math>3r = 30</math>, and so <math>r = 10</math>. <math>K = 30^2\pi - 6(10^2\pi) = 300\pi</math>, so
    1 KB (213 words) - 12:17, 22 July 2017
  • ..._1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> Rewrite the given equations as <math>(x+5)^2 + (y-12)^2 = 256</math> and <math>(x-5)^2 + (y-12)^2 = 16</math>.
    12 KB (2,001 words) - 19:26, 23 July 2024
  • ...th>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ldots+a_{98}=137</math>. ...ath>t_{3}</math> in the figure, have areas <math>4</math>, <math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>.
    6 KB (933 words) - 00:15, 19 June 2022
  • How many even integers between 4000 and 7000 have four different digits? ...math>0 < a < b < c < d < 500\,</math> satisfy <math>a + d = b + c\,</math> and <math>bc - ad = 93\,</math>?
    8 KB (1,275 words) - 05:55, 2 September 2021
  • where <math> k </math> and <math> n </math> are positive integers and <math> n </math> is as large as possible, find <math> k + n. </math> ...radius 100 can be expressed as <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m + n.
    6 KB (965 words) - 15:36, 8 September 2019
  • The product <math>N</math> of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possib ...>B</math>, <math>B</math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many se
    7 KB (1,127 words) - 08:02, 11 July 2023
  • ...ntersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. Find the square of the length of <math>Q ...th>P</math> to get another circle (centered at a new point <math>C</math>, and with radius <math>6</math>) that intersects circle <math>A</math> at <math>
    14 KB (2,351 words) - 20:06, 8 December 2024
  • ...th>PB^{}_{}=15</math>, <math>BQ^{}_{}=20</math>, <math>PR^{}_{}=30</math>, and <math>QS^{}_{}=40</math>. Let <math>\frac{m}{n}</math>, in lowest terms, de Let <math>O</math> be the center of the rhombus. Via [[parallel]] sides and alternate interior angles, we see that the opposite [[triangle]]s are [[con
    8 KB (1,270 words) - 22:36, 27 August 2023
  • ...(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <mat ...lly have a regular dodecagon whose vertices are the centers of the smaller triangles circumscribed about a circle of radius <math>1</math>.
    4 KB (740 words) - 16:46, 24 May 2024
  • ...at <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>? == Solution 1 (Similar Triangles) ==
    3 KB (541 words) - 13:41, 4 November 2024
  • ...RS\,</math> can be expressed as <math>m/n\,</math>, where <math>m\,</math> and <math>n\,</math> are relatively prime integers. Find <math>m + n\,</math>. From the [[Pythagorean Theorem]], <math>AH^2+CH^2=1994^2</math>, and <math>(1995-AH)^2+CH^2=1993^2</math>.
    3 KB (449 words) - 20:39, 21 September 2023
  • ...ath> where <math>q, r,\,</math> and <math>s\,</math> are positive integers and <math>s\,</math> is not divisible by the square of any prime. What is <mat ...he respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overline{AB}, \overline{BC}, \overline{CA}</math>.
    4 KB (717 words) - 21:20, 3 June 2021
  • ...th>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. ...9 = \frac{2 \cdot O_6A_6 + 1 \cdot O_3A_3}{3} = 5</math> (consider similar triangles). Applying the [[Pythagorean Theorem]] to <math>\triangle O_9A_9P</math>, w
    3 KB (605 words) - 10:30, 5 May 2024

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