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2001 PMWC Problems
pair A,B,C,D,E,F,G,O; O=(A+B+C+D)/4;

4 KB (641 words) - 21:24, 21 April 2014
1971 Canadian MO Problems
...of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>O

3 KB (519 words) - 08:58, 13 September 2012
1999 USAMO Problems/Problem 5
...s 2000</math> grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes t ...quare surrounded by two empty squares. In both cases, the player places an O in that square. In both situations, it is easy to verify that the first pla

2 KB (433 words) - 13:35, 4 July 2013
1989 USAMO Problems/Problem 4
...enter of the inscribed circle of triangle <math>ABC</math> and point <math>O</math> is the center of the circumscribed circle, prove that line <math>IO< Consider the lines that pass through the circumcenter <math>O</math>. Extend <math>AO</math>, <math> BO</math>, <math>CO</math> to <math>

2 KB (290 words) - 19:11, 18 July 2016
2007 AMC 12B Problems
The point <math>O</math> is the center of the circle circumscribed about triangle <math>ABC</ ...math>AB=BC=CD=DE=EA=2</math> and <math>\angle ABC=\angle CDE=\angle DEA=90^o</math>. The plane of <math>\triangle ABC</math> is parallel to <math>\overl

12 KB (1,814 words) - 12:58, 19 February 2020
1960 IMO Problems/Problem 6
...be the angle with arms <math>AO</math> and <math>AA_1</math>, where <math>O</math> is the center of the circle; <math>0 < \theta < \frac{\pi}{4}</math> Knowing <math>\angle A_1 A H_1 = 2 \times \angle A_1 A O = 2 \theta</math>, and using the formula for a double-angle tangent, we can

7 KB (1,214 words) - 18:49, 29 January 2018
Asymptote: Useful functions
...ons, you can replace <code>line l1, line l2</code> with <code>pair A, pair O, pair B</code>, which would mark the angle <math>\angle AOB</math>. The fun pair O = (0,0);

4 KB (646 words) - 21:18, 26 March 2024
Excircle
...ertices <math>B,C</math> of triangle <math>\triangle ABC</math>. Let <math>O</math> be the circumcenter <math>\triangle ABC</math>. Prove that <cmath>OA

5 KB (843 words) - 03:02, 1 July 2020
2001 USAMO Problems/Problem 2
...ion for OA */ pair Ca = 2C-A, Cb = bisectorpoint(Ca,C,B), OA = IP(A--A+10*(O-A),C--C+50*(Cb-C)), D2 = D(MP("D_2",foot(OA,B,C))), Fa=2B-A, Ga=2C-A, F=MP(

11 KB (2,091 words) - 08:35, 16 November 2017
1992 USAMO Problems/Problem 2
...h>M_k = (1, \tan k^\circ)</math> in the coordinate plane with origin <math>O=(0,0)</math>, for integers <math>0 \le k \le 89</math>. pair O=(0,0), a=expi(0), b=expi(1/6), c=expi(2/6), d=expi(3/6), y=expi(32/30), z=

4 KB (628 words) - 07:41, 19 July 2016
De Longchamps point
Let <math>H, O,</math> and <math>L</math> be orthocenter, circumcenter, and De Longchamps ...point of intersection of the heights <math>H</math> with respect to <math>O.</math>

10 KB (1,780 words) - 09:23, 17 November 2022
2000 AMC 12 Problems/Problem 17
A circle centered at <math>O</math> has radius <math>1</math> and contains the point <math>A</math>. The pair O=(0,0);

6 KB (979 words) - 12:50, 17 July 2022
2000 AMC 12 Problems/Problem 24
...eral, it follows that <math>MB=18/{\pi}</math>, and <math>OA</math> (where O is the center of the circle) is <math>36/{\pi}-r</math>. By the Pythagorea

2 KB (263 words) - 19:59, 18 April 2024
2007 AMC 10A Problems
...and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the midpoint of <math>\overline{AB}</math>, and <math>OA = 2\sqrt pair O=(0,0);

13 KB (2,058 words) - 17:54, 29 March 2024
1995 AHSME Problems
Two rays with common endpoint <math>O</math> form a <math>30^\circ</math> angle. Point <math>A</math> lies on one ...d <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\ov

17 KB (2,387 words) - 22:44, 26 May 2021
1995 AHSME Problems/Problem 28
We let <math>O</math> be the center, <math>\overline{A_1AA_2}</math>, <math>\overline{B_1B Let <math>O</math> be the center of the circle, <math>H</math> be the midpoint of <math

3 KB (509 words) - 22:56, 5 December 2023
1951 AHSME Problems
At <math> 2: 15</math> o'clock, the hour and minute hands of a clock form an angle of: <math>AB</math> is a fixed diameter of a circle whose center is <math>O</math>. From <math>C</math>, any point on the circle, a chord <math>CD</mat

23 KB (3,646 words) - 21:53, 21 June 2024
1966 AHSME Problems
<math>AB</math> is the diameter of a circle centered at <math>O</math>. <math>C</math> is a point on the circle such that angle <math>BOC</ Five points <math>O,A,B,C,D</math> are taken in order on a straight line with distances <math>O

19 KB (3,159 words) - 22:10, 11 March 2024
2003 AMC 12B Problems/Problem 22
pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18 pair A = 2*O-C, B = 2*O-D;

4 KB (551 words) - 14:17, 23 June 2022
2003 AMC 12B Problems/Problem 21
<math>O</math> is the point that satisfies the system of equations: <math>\begin{ca ..., <math>y = - \frac52</math>, <math>x = \frac{5 \sqrt{3}}{2}</math>, <math>O = (\frac{5 \sqrt{3}}{2}, - \frac52)</math>

2 KB (372 words) - 09:38, 1 September 2022

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