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• 2006 AMC 12B Problems/Problem 13
  ...math> ABCD</math> is 24, and <math> \angle BAD = 60^\circ</math>. What is the area of rhombus <math> BFDE</math>? ...h>. One-third the area of <math>ABCD</math> is equal to <math>8</math>. So the answer is <math>\boxed{\text{C}}</math>.
  3 KB (447 words) - 03:49, 16 January 2021
• 2005 AIME II Problems/Problem 12
  ...th> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> Let <math>G</math> be the foot of the [[perpendicular]] from <math>O</math> to <math>AB</math>. Denote <math>x =
  13 KB (2,080 words) - 13:14, 23 July 2024
• 2005 AIME II Problems/Problem 14
  By the [[Law of Sines]] and since <math>\angle BAE = \angle CAD, \angle BAD = \ang ...= \frac{15}{BE} - 1 \Longrightarrow BE = \frac{13^2 \cdot 15}{463}</math>. The answer is <math>q = \boxed{463}</math>.
  14 KB (2,340 words) - 16:38, 21 August 2024
• 2004 AIME II Problems/Problem 7
  ...dimensions <math> AE=8, BE=17, </math> and <math> CF=3 </math> are given. The perimeter of rectangle <math> ABCD </math> is <math> m/n, </math> where <ma ...>\overline{BB'}</math>, it follows that <math>BE = B'E</math> (by SAS). By the [[Pythagorean Theorem]], we have <math>AB' = 15</math>. Similarly, from <ma
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2002 AIME I Problems
  ...ndrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is <math>m/n</math>, where <ma ...hown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as <math>\frac{1}{2}\left(\sqrt{p}-q\right
  8 KB (1,374 words) - 21:09, 27 July 2023
• 1985 AIME Problems/Problem 6
  ...ior point. The [[area]]s of four of these triangles are as indicated. Find the area of triangle <math>ABC</math>. ...ases. Moreover, the two pairs of bases are actually the same, and thus in the same ratio. As a result, we have:
  5 KB (789 words) - 03:09, 23 January 2023
• 1986 AIME Problems/Problem 15
  ...ng the lines <math>y=x+3</math> and <math>y=2x+4</math> respectively, find the area of triangle <math>ABC</math>. ...e <math>x</math>-axis. The tangents of these two angles are the slopes of the respective medians; in other words, <math>\tan \theta_1 = 1</math>, and <ma
  11 KB (1,722 words) - 09:49, 13 September 2023
• 2006 AMC 10B Problems/Problem 19
  ...n in the figure, which is bounded by <math>BD</math>, <math>BE</math>, and the minor arc connecting <math>D</math> and <math>E</math>? ...f sector <math>DOE</math> minus the area of triangle <math>DOE</math> plus the area of triangle <math>DBE</math>.
  5 KB (873 words) - 15:39, 29 May 2023
• 1997 AIME Problems/Problem 15
  ...ath>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...f the rectangle. Set one [[vertex]] of the triangle at <math>A</math>, and the other two points <math>E</math> and <math>F</math> on <math>BC</math> and <
  5 KB (811 words) - 21:39, 20 July 2024
• 2002 AIME I Problems/Problem 10
  ...that <math>EB=9</math> and <math>FC=27</math>, find the integer closest to the area of quadrilateral <math>DCFG</math>. ...hagorean Theorem, <math>BC=35</math>. Letting <math>BD=x</math> we can use the Angle Bisector Theorem on triangle <math>ABC</math> to get <math>x/12=(35-x
  4 KB (643 words) - 22:44, 8 August 2023
• 2003 AIME II Problems/Problem 11
  ...math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> We use the [[Pythagorean Theorem]] on <math>ABC</math> to determine that <math>AB=25.<
  5 KB (772 words) - 19:47, 1 August 2023
• 2002 AIME II Problems/Problem 13
  ...th> It is given that the ratio of the area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n,</math> where <math>m</math> Let <math>X</math> be the intersection of <math>\overline{CP}</math> and <math>\overline{AB}</math>.
  6 KB (937 words) - 20:06, 24 August 2024
• 1971 Canadian MO Problems/Problem 1
  ...the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle. First, extend <math>CO</math> to meet the circle at <math>P.</math> Let the radius be <math>r.</math> Applying [[power of a point]],
  680 bytes (114 words) - 16:33, 4 September 2024
• Euler line
  ...iangle ABC</math>, the '''Euler line''' is a [[line]] which passes through the [[orthocenter]] <math>H</math>, [[centroid]] <math>G</math>, [[circumcenter Euler line is the central line <math>L_{647}</math>.
  59 KB (10,203 words) - 04:47, 30 August 2023
• 1985 IMO Problems/Problem 1
  ...adrilateral]] <math>ABCD</math>. The other three sides are [[tangent]] to the circle. Prove that <math>AD + BC = AB</math>. ...entioned in the problem. Let <math>T</math> be the second intersection of the circumcircle of <math>CDO </math> with <math>AB </math>. By measures of ar
  4 KB (684 words) - 07:28, 3 October 2021
• 2007 AIME II Problems/Problem 9
  ...>BEF</math> is [[tangent]] to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> ...math>BE = DF = \sqrt{63^2 + 84^2} = 21\sqrt{3^2 + 4^2} = 105</math>. Also, the length of <math>EF = \sqrt{63^2 + (448 - 2\cdot84)^2} = 7\sqrt{9^2 + 40^2}
  5 KB (818 words) - 11:05, 7 June 2022
• 2007 AIME I Problems/Problem 9
  ...ath>, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as <math>p/q</math>, where <math>p</m ...\cong \triangle ADO_1</math>. Call <math>x = AD = AF</math> and <math>y = EB = BG</math>. We know that <math>x + y + 2r = 34</math>.
  11 KB (1,853 words) - 20:10, 21 July 2024
• 2007 AIME I Problems/Problem 13
  ...ath>, <math>BC</math>, and <math>CD</math>. The plane's intersection with the pyramid has an area that can be expressed as <math>\sqrt{p}</math>. Find < Note first that the intersection is a [[pentagon]].
  7 KB (1,034 words) - 23:30, 18 June 2024
• 2007 BMO Problems/Problem 1
  ...ath>AB=BC=CD </math>, <math>AC \neq BD </math>, and let <math>E </math> be the intersection point of its diagonals. Prove that <math>AE=DE </math> if and Now, by the [[Law of Sines]],
  3 KB (566 words) - 23:59, 14 September 2014
• 1971 Canadian MO Problems
  ...the circle at <math>C.</math> Given <math>EC=1,</math> find the radius of the circle Determine all real numbers <math>a</math> such that the two polynomials <math>x^2+ax+1</math> and <math>x^2+x+a</math> have at leas
  3 KB (519 words) - 08:58, 13 September 2012

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