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2016 USAMO Problems/Problem 3
...ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and cir 1. Let <math>I_A</math> be the <math>A</math>-excenter, then <math>I_A,O,</math> and <math>P</math> are colinear. This can be proved by the Trigonom

6 KB (998 words) - 21:36, 17 October 2022
2016 USAMO Problems/Problem 5
...t <math>\overline{OI}</math> is parallel to <math>\ell,</math> where <math>O</math> is the circumcenter of triangle <math>ABC,</math> and <math>I</math> ...ath>\overline{JK}</math> bisects <math>\overline{JK}</math>. Hence, <cmath>O = \left(\frac{b}{2},\frac{b}{2} \cot B\right) \qquad (5)</cmath> if triangl

8 KB (1,449 words) - 00:09, 12 October 2023
1986 IMO Problems
...h> under a rotation center <math>A_{k+1}</math> through an angle <math>120^o</math> clockwise for <math>k=0,1,2,\ldots</math>. Prove that if <math>P_{19 ...ces of a regular <math>n</math>-gon (<math>n\ge5</math>) with center <math>O</math>. A triangle <math>XYZ</math>, which is congruent to and initially co

3 KB (482 words) - 00:04, 30 January 2021
1960 AHSME Problems/Problem 2
...ormly spaced, how long, in seconds, does it take to strike <math>12</math> o'clock?

706 bytes (104 words) - 12:47, 20 December 2018
1976 AHSME Problems/Problem 18
O=(0,0), c1=circle(O, r1);

2 KB (361 words) - 08:05, 9 April 2023
Mock AIME 4 Pre 2005/Problems
...> Call the points of tangency <math>M</math> and <math>S.</math> Let <math>O</math> and <math>P</math> be the points of intersection between <math>\omeg

7 KB (1,094 words) - 15:39, 24 March 2019
1979 AHSME Problems/Problem 12
pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0); draw(arc(O,1, 0, 180));

3 KB (392 words) - 13:34, 17 June 2021
1979 AHSME Problems/Problem 21
...s where the incircle meets the triangle as <math>X,Y,Z</math>, where <math>O</math> is the incenter, and denote <math>AX = AY = z, BX = BZ = y, CY = CZ

1 KB (278 words) - 13:06, 9 January 2017
Mock AIME 3 2010 Problems
. Let ω, with center I, be the circumcircle of 4abc, and let Γ, with center O, be the incircle of 4abc. Let -7. In 4ABC, let AB = 9, BC = 13, and CA = 14. Let G, H, I, and O be the centroid, orthocenter, incenter, and

14 KB (2,904 words) - 18:24, 16 May 2017
2017 AMC 10A Problems/Problem 22
label("O", (4, 3.4)); ...line{AB}</math> and <math>\overline{AC}</math> are tangent to circle <math>O</math>, then <math>\angle OBA = \angle OCA = 90^{\circ}</math>, so <math>\a

6 KB (993 words) - 20:28, 23 February 2024
2017 AMC 12A Problems/Problem 24
Quadrilateral <math>ABCD</math> is inscribed in circle <math>O</math> and has side lengths <math>AB=3, BC=2, CD=6</math>, and <math>DA=8</ ...math>\overline{AC}</math>. Let <math>G</math> be the point on circle <math>O</math> other than <math>C</math> that lies on line <math>CX</math>. What is

8 KB (1,322 words) - 03:24, 7 October 2023
2017 AMC 12A Problems/Problem 16
...dius of <math>6</math> and centered at the left corner of the semi-circle (O) with radius <math>3</math>. Extend the three semicircles to full circles. label("$O$", origin, W);

13 KB (1,982 words) - 17:12, 20 December 2022
2017 AMC 12A Problems
...math>\overline{AC}</math>. Let <math>G</math> be the point on circle <math>O</math> other than <math>C</math> that lies on line <math>CX</math>. What is

15 KB (2,418 words) - 16:58, 7 November 2022
2017 AMC 10B Problems/Problem 22
pair A, B, C, D, E, O; O = (A+B)/2;

6 KB (980 words) - 17:16, 13 May 2024
2017 AMC 10B Problems/Problem 15
But, note that <math>\sin(m\angle EAD)=\sin(m\angle CAD)=\frac{O}{H}=\frac{3}{5}</math>. Thus, we see that

4 KB (656 words) - 01:20, 4 December 2023
2017 AMC 12B Problems/Problem 18
label("$O$", (-0.10330578512396349,-0.39365890308038826), NE * labelscalefactor); Let <math>O</math> be the center of the circle. Note that <math>EC + CA = EA = \sqrt{AD

7 KB (886 words) - 04:01, 23 January 2023
2017 AIME I Problems/Problem 4
...h>\triangle ABC</math>, which we will call <math>O</math>. Note that <math>O</math> is equidistant from each of <math>A</math>, <math>B</math>, and <mat

7 KB (1,177 words) - 15:55, 5 January 2024
2017 AIME I Problems/Problem 6
...Converting to degrees, since the angles of a triangle add up to <math>180^o</math>, we find that <math>\frac{4}{15} \cdot 180 =\boxed{048}</math>, whic

4 KB (726 words) - 16:18, 5 January 2024
2017 AIME I Problems/Problem 8
...and uniformly at random from the interval <math>(0, 75)</math>. Let <math>O</math> and <math>P</math> be two points on the plane with <math>OP = 200</m Put <math>\triangle POQ</math> and <math>\triangle POR</math> with <math>O</math> on the origin and the triangles on the <math>1^{st}</math> quadrant.

9 KB (1,539 words) - 15:47, 17 February 2024
2017 AIME I Problems/Problem 15
Let the right triangle's lower-left point be at <math>O(0,0)</math>. Notice the 2 other points will determine a unique equilateral

22 KB (3,622 words) - 17:11, 6 January 2024

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