https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Arachnotron&feedformat=atomAoPS Wiki - User contributions [en]2021-05-09T11:50:31ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=International_Mathematical_Olympiad&diff=32424International Mathematical Olympiad2009-07-20T02:22:47Z<p>Arachnotron: /* Awards */</p>
<hr />
<div>The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest of all international scientific competitions. Each year, countries from around the world send a team of 6 students to compete in a grueling competition.<br />
<br />
== Format of the Competition ==<br />
The competition takes place over 2 consecutive days. Each day 3 problems are given to the students to work on for 4.5 hours. Following the general format of high school competitions, it does not require [[calculus]] or related topics.<br />
<br />
=== Scoring ===<br />
<br />
Scoring on each problem is done on a 0-7 scale (inclusive and integers only). Full credit is only given for complete, correct solutions. Each solution is intended to be in the form of a [[proof writing|mathematical proof]]. Since there are 6 problems, a perfect score is 42 points.<br />
<br />
=== Awards ===<br />
Medals, honorable mentions and sometimes, Special prize are given out.<br />
<br />
* Gold - the top 1/12 of individual scores.<br />
* Silver - the next 2/12 of individual scores.<br />
* Bronze - the next 3/12 of individual scores.<br />
* Honorable mention - any student who receives a score of 7 on any one problem but did not receive a medal.<br />
* Special Prize - Given to students who score 7 in one problem with an especially insightful solution.<br />
<br />
=== Team Competition ===<br />
There is no official team competition. Unofficially, however, the scores of each team are compared each year where a team's score is the sum of their individual scores.<br />
<br />
== History ==<br />
<br />
The IMO started in 1959 as a competition among Eastern European countries. Since then, it has evolved into the premier international competition in mathematics.<br />
<br />
== See also ==<br />
* [[IMO Problems and Solutions, with authors]]<br />
* [[Mathematics competition resources]]<br />
* [[Math books]]<br />
* [[Mathematics scholarships]]<br />
* [[Worldwide Online Olympiad Training]]<br />
<br />
== External Links ==<br />
* <url>index.php?f=87 AoPS-MathLinks IMO Forum</url><br />
* <url>resources.php AoPS-MathLinks Olympiad Resources</url><br />
* [http://www.imo-official.org Official IMO Site]<br />
* [http://www.imo2007.edu.vn/ IMO 2007 Vietnam]<br />
* [http://imo2006.dmfa.si/ IMO 2006 Slovenia]<br />
<br />
[[Category:Mathematics competitions]]</div>Arachnotronhttps://artofproblemsolving.com/wiki/index.php?title=International_Mathematical_Olympiad&diff=32423International Mathematical Olympiad2009-07-20T02:22:11Z<p>Arachnotron: /* Awards */</p>
<hr />
<div>The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest of all international scientific competitions. Each year, countries from around the world send a team of 6 students to compete in a grueling competition.<br />
<br />
== Format of the Competition ==<br />
The competition takes place over 2 consecutive days. Each day 3 problems are given to the students to work on for 4.5 hours. Following the general format of high school competitions, it does not require [[calculus]] or related topics.<br />
<br />
=== Scoring ===<br />
<br />
Scoring on each problem is done on a 0-7 scale (inclusive and integers only). Full credit is only given for complete, correct solutions. Each solution is intended to be in the form of a [[proof writing|mathematical proof]]. Since there are 6 problems, a perfect score is 42 points.<br />
<br />
=== Awards ===<br />
Medals, honorable mentions and sometimes, Special prize are given out.<br />
<br />
* Gold - the top 1/12 of individual scores.<br />
* Silver - the next 2/12 of individual scores.<br />
* Bronze - the next 3/12 of individual scores.<br />
* Honorable mention - any student who receives a score of 7 on any one problem but did not receive a medal.<br />
* Special PriZe - Given to students who score 7 in one problem with an especially insightful solution.<br />
<br />
=== Team Competition ===<br />
There is no official team competition. Unofficially, however, the scores of each team are compared each year where a team's score is the sum of their individual scores.<br />
<br />
== History ==<br />
<br />
The IMO started in 1959 as a competition among Eastern European countries. Since then, it has evolved into the premier international competition in mathematics.<br />
<br />
== See also ==<br />
* [[IMO Problems and Solutions, with authors]]<br />
* [[Mathematics competition resources]]<br />
* [[Math books]]<br />
* [[Mathematics scholarships]]<br />
* [[Worldwide Online Olympiad Training]]<br />
<br />
== External Links ==<br />
* <url>index.php?f=87 AoPS-MathLinks IMO Forum</url><br />
* <url>resources.php AoPS-MathLinks Olympiad Resources</url><br />
* [http://www.imo-official.org Official IMO Site]<br />
* [http://www.imo2007.edu.vn/ IMO 2007 Vietnam]<br />
* [http://imo2006.dmfa.si/ IMO 2006 Slovenia]<br />
<br />
[[Category:Mathematics competitions]]</div>Arachnotronhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10A_Problems/Problem_15&diff=284302007 AMC 10A Problems/Problem 152008-11-28T00:45:06Z<p>Arachnotron: /* Alternate Solution */</p>
<hr />
<div>==Problem==<br />
Four circles of radius <math>1</math> are each tangent to two sides of a square and externally tangent to a circle of radius <math>2</math>, as shown. What is the area of the square?<br />
<br />
[[Image:2007 AMC 10A -15 for wiki.png]]<br />
<br />
<math>\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}</math><br />
<br />
==Solution==<br />
The diagonal has length <math>\sqrt{2}+1+2+2+1+\sqrt{2}=6+2\sqrt{2}</math>. Therefore the sides have length <math>2+3\sqrt{2}</math>, and the area is<br />
<br />
<cmath>A=(2+3\sqrt{2})^2=4+6\sqrt{2}+6\sqrt{2}+18=22+12\sqrt{2} \Rightarrow \text{(B)}</cmath><br />
<br />
== Alternate Solution == <br />
<br />
Extend two radii from the larger circle to the centers of the two smaller circles above. This forms a right triangle of sides <math>3, 3, 3\sqrt{2}</math>. The length of the hypotenuse of the right triangle plus twice the radius of the smaller circle is equal to the side of the square. It follows, then <cmath> A = (2+3\sqrt{2})^2 = 22 + 12\sqrt{2} \Rightarrow \text{(B)}</cmath><br />
<br />
==See Also==<br />
{{AMC10 box|year=2007|ab=A|num-b=14|num-a=16}}<br />
<br />
[[Category:Introductory Geometry Problems]]</div>Arachnotronhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_10A_Problems/Problem_15&diff=284292007 AMC 10A Problems/Problem 152008-11-28T00:31:11Z<p>Arachnotron: /* Solution */</p>
<hr />
<div>==Problem==<br />
Four circles of radius <math>1</math> are each tangent to two sides of a square and externally tangent to a circle of radius <math>2</math>, as shown. What is the area of the square?<br />
<br />
[[Image:2007 AMC 10A -15 for wiki.png]]<br />
<br />
<math>\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}</math><br />
<br />
==Solution==<br />
The diagonal has length <math>\sqrt{2}+1+2+2+1+\sqrt{2}=6+2\sqrt{2}</math>. Therefore the sides have length <math>2+3\sqrt{2}</math>, and the area is<br />
<br />
<cmath>A=(2+3\sqrt{2})^2=4+6\sqrt{2}+6\sqrt{2}+18=22+12\sqrt{2} \Rightarrow \text{(B)}</cmath><br />
<br />
== Alternate Solution == <br />
<br />
Extend two radii from the larger circle to the centers of the two smaller circles above (or below -- it's irrelevant). This forms a right triangle of sides <math>3, 3, 3\sqrt{2}</math>. The length of the hypotenuse of the right triangle plus twice the radius of the smaller circle is equal to the side of the square. It follows, then <cmath> A = (2+3\sqrt{2})^2 = 22 + 12\sqrt{2} \Rightarrow \text{(B)}</cmath><br />
<br />
==See Also==<br />
{{AMC10 box|year=2007|ab=A|num-b=14|num-a=16}}<br />
<br />
[[Category:Introductory Geometry Problems]]</div>Arachnotron