Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1970 Canadian MO Problems/Problem 2"

(Created page with "== Problem == Given a triangle <math>ABC</math> with angle <math>A</math> obtuse and with altitudes of length <math>h</math> and <math>k</math> as shown in the diagram, prove th...")
 
(→‎Problem)
Line 4: Line 4:
  
  
−
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
 
  
 +
<asy>
 +
draw((0,0)--(5,0)--(16/5,12/5)--cycle,dot);
 +
draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black);
 +
MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE);
 +
MP("E",(2.5,0),NE);MP("A",(2.5,7.5/4),N);
 +
MP("h",(2.5,7.5/8),W);MP("k",(41/10,6/5),NE);
 +
draw((-.2,.2)--(2.5-.2,7.5/4+.2),arrow=ArcArrow(TeXHead));
 +
draw((2.5-.2,7.5/4+.2)--(-.2,.2),arrow=ArcArrow(TeXHead));
 +
MP("b",(2.3/2-.05,7.5/8+.25),N);
 +
draw((0,-.2)--(5,-.2),arrow=ArcArrow(TeXHead));
 +
draw((5,-.2)--(0,-.2),arrow=ArcArrow(TeXHead));
 +
MP("a",(2.5,-.2),S);
 +
draw((16/5,12/5)--(16/5-.2,12/5-.15)--(16/5-.2+.15,12/5-.15-.2)--(16/5+.15,12/5-.2)--cycle,black);
 +
</asy>
 +
 +
 +
== Solution ==
  
 
== Solution ==
 
== Solution ==

Revision as of 03:32, 8 October 2014

Problem

Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.


[asy] draw((0,0)--(5,0)--(16/5,12/5)--cycle,dot); draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black); MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE); MP("E",(2.5,0),NE);MP("A",(2.5,7.5/4),N); MP("h",(2.5,7.5/8),W);MP("k",(41/10,6/5),NE); draw((-.2,.2)--(2.5-.2,7.5/4+.2),arrow=ArcArrow(TeXHead)); draw((2.5-.2,7.5/4+.2)--(-.2,.2),arrow=ArcArrow(TeXHead)); MP("b",(2.3/2-.05,7.5/8+.25),N); draw((0,-.2)--(5,-.2),arrow=ArcArrow(TeXHead)); draw((5,-.2)--(0,-.2),arrow=ArcArrow(TeXHead)); MP("a",(2.5,-.2),S); draw((16/5,12/5)--(16/5-.2,12/5-.15)--(16/5-.2+.15,12/5-.15-.2)--(16/5+.15,12/5-.2)--cycle,black); [/asy]


Solution

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1970_Canadian_MO_Problems/Problem_2&oldid=64647"