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 "1965 IMO Problems/Problem 5"
 
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
 
Consider <math>\triangle OAB</math> with acute angle <math>AOB</math>. Through a point <math>M \neq O</math> perpendiculars are drawn to <math>OA</math> and <math>OB</math>, the feet of which are <math>P</math> and <math>Q</math> respectively. The point of intersection of the altitudes of <math>\triangle OPQ</math> is <math>H</math>. What is the locus of <math>H</math> if <math>M</math> is permitted to range over (a) the side <math>AB</math>, (b) the interior of <math>\triangle OAB</math>?
 
Consider <math>\triangle OAB</math> with acute angle <math>AOB</math>. Through a point <math>M \neq O</math> perpendiculars are drawn to <math>OA</math> and <math>OB</math>, the feet of which are <math>P</math> and <math>Q</math> respectively. The point of intersection of the altitudes of <math>\triangle OPQ</math> is <math>H</math>. What is the locus of <math>H</math> if <math>M</math> is permitted to range over (a) the side <math>AB</math>, (b) the interior of <math>\triangle OAB</math>?
 +
  
 
== Solution ==
 
== Solution ==
−
{{solution}}
 
 
Let <math>O(0,0),A(a,0),B(b,c)</math>.
 
Let <math>O(0,0),A(a,0),B(b,c)</math>.
 
Equation of the line <math>AB: y=\frac{c}{b-a}(x-a)</math>.
 
Equation of the line <math>AB: y=\frac{c}{b-a}(x-a)</math>.
Line 17: Line 18:
 
a line segment <math>MN , M \in OA , N \in OB</math>.
 
a line segment <math>MN , M \in OA , N \in OB</math>.
 
Second question: the locus consists in the <math>\triangle OMN</math>.
 
Second question: the locus consists in the <math>\triangle OMN</math>.
 +
 +
 +
== Solution 2 ==
 +
 +
This solution is a simplified version of the previous solution,
 +
it fills in some gaps. and it provides more information.  The
 +
idea is to just follow the degrees of the expressions and
 +
equations in <math>\lambda, x, y</math> involved.  If we manage to conclude
 +
that the equation for <math>H</math> is an equation of degree <math>1</math>, then we
 +
will know that it is a line.  We don't need to know the equation
 +
explicitly.
 +
 +
Just like in the previous solution, we use analytic (coordinate)
 +
geometry, but we don't care how the axes are chosen.
 +
 +
The coordinates of <math>M</math> are expressions of degree <math>1</math> in <math>\lambda</math>.
 +
 +
The equation for <math>MP</math> is an equation of degree <math>1</math> in <math>x, y</math>
 +
with constant coefficients for <math>x, y</math>, and whose constant term
 +
is an expression of degree <math>1</math> in <math>\lambda</math>.
 +
 +
The coordinates of <math>P</math> (the intersection of <math>MP</math> and <math>OA</math>) are
 +
expressions of degree <math>1</math> in <math>\lambda</math>.
 +
 +
The equation of the perpendicular from <math>P</math> to <math>OB</math> is of degree
 +
<math>1</math> in <math>x, y</math>, with constant coefficients for <math>x, y</math>, and whose
 +
constant term is an expression of degree <math>1</math> in <math>\lambda</math>.  This
 +
corresponds to equation (2) in the above solution.
 +
 +
Similarly, the equation of the perpendicular from <math>Q</math> to <math>OA</math>
 +
is of degree <math>1</math> in <math>x, y</math>, with constant coefficients for
 +
<math>x, y</math>, and whose constant term is an expression of degree
 +
<math>1</math> in <math>\lambda</math>.  This corresponds to equation (1) in the
 +
above solution.
 +
 +
Now, in principle, we would have to solve the system of two
 +
equations to obtain the coordinates of <math>H</math> as expressions
 +
of <math>\lambda</math>, and then eliminate <math>\lambda</math> to obtain an
 +
equation in <math>x, y</math>.  Or, as a shortcut, we can eliminate
 +
<math>\lambda</math> directly from the two equations.  Either way,
 +
the result is an equation of degree <math>1</math> in <math>x, y</math>.
 +
 +
This tells us that the locus is on this line.  We just need to
 +
specify which set of points on this line is the locus.
 +
 +
The previous solution, with a good amount of hand waving, tells
 +
us that the solution is "a line segment
 +
<math>B_1A_1, B_1 \in OA, A_1 \in OB</math>".  (On top of the hand waving
 +
the solution uses the unhappy notation <math>M</math> for <math>B_1</math> and <math>N</math>
 +
for <math>A_1</math>, which is bad because <math>M</math> has already been used!)
 +
We will do better than that.
 +
 +
Let <math>A_1</math> be the foot of the perpendicular from <math>A</math> to <math>OB</math>, and
 +
<math>B_1</math> be the foot of the perpendicular from <math>B</math> to <math>OA</math>.
 +
(For this paragraph see the picture shown in Solution 3.)
 +
Consider the limit situation when <math>M = A</math>.  Then <math>Q = A_1</math>, and
 +
<math>P = A</math>.  It follows that the intersection <math>H</math> of the
 +
perpendiculars from <math>P</math> to <math>OB</math> and <math>Q</math> to <math>OA</math> is <math>A_1</math>.
 +
Similarly, the limit situation when <math>M = B</math> yields <math>H = B_1</math>.
 +
Now it is reasonable to say that when <math>M</math> moves from <math>A</math> to <math>B</math>,
 +
<math>H</math> moves from <math>A_1</math> to <math>B_1</math>.  So, the locus is the line segment
 +
joining the feet <math>A_1, B_1</math> of the perpendiculars in
 +
<math>\triangle OAB</math> from <math>A, B</math>.  This answers question (a).
 +
 +
Again with a good amount of hand waving, the previous solution
 +
says "the locus consists in the <math>\triangle OB_1A_1</math>".  We justify
 +
this by pointing out that if <math>M</math> is inside <math>\triangle OAB</math>, then
 +
we can take the triangle <math>\triangle OA'B'</math>, such that <math>A' \in OA</math>,
 +
<math>B' \in OB</math>, <math>A'B'</math> going through <math>M</math> and parallel to <math>AB</math>.  Then
 +
<math>H</math> will be on the corresponding segment <math>A_1'B_1'</math> determined by
 +
the feet of the perpendiculars in <math>\triangle OA'B'</math>.  Conversely,
 +
it is easy to see that any point <math>H \in \triangle OA_1B_1</math> is on
 +
a segment <math>A_1'B_1'</math> obtained from a triangle <math>\triangle OA'B'</math>,
 +
and <math>H</math> is obtained from a point <math>M \in A'B'</math>.  This answers
 +
question (b).
 +
 +
(Solution by pf02, October 2024)
 +
 +
 +
== Solution 3 ==
 +
 +
 +
 +
 +
 +
 +
 +
TO BE CONTINUED.  SAVING MID WAY, SO I DON'T LOSE WORK DONE SO FAR.
 +
 +
 +
== See Also ==
 +
{{IMO box|year=1965|num-b=4|num-a=6}}

Latest revision as of 20:01, 30 October 2024

Problem

Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$, (b) the interior of $\triangle OAB$?


Solution

Let $O(0,0),A(a,0),B(b,c)$. Equation of the line $AB: y=\frac{c}{b-a}(x-a)$. Point $M \in AB : M(\lambda,\frac{c}{b-a}(\lambda-a))$. Easy, point $P(\lambda,0)$. Point $Q = OB \cap MQ$, $MQ \bot OB$. Equation of $OB : y=\frac{c}{b}x$, equation of $MQ : y=-\frac{b}{c}(x-\lambda)+\frac{c}{b-a}(\lambda-a)$. Solving: $x_{Q}=\frac{1}{b^{2}+c^{2}}\left[b^{2}\lambda+\frac{c^{2}(\lambda-a)b}{b-a}\right]$. Equation of the first altitude: $x=\frac{1}{b^{2}+c^{2}}\left[b^{2}\lambda+\frac{c^{2}(\lambda-a)b}{b-a}\right] \quad (1)$. Equation of the second altitude: $y=-\frac{b}{c}(x-\lambda)\quad\quad (2)$. Eliminating $\lambda$ from (1) and (2): \[ac \cdot x + (b^{2}+c^{2}-ab)y=abc\] a line segment $MN , M \in OA , N \in OB$. Second question: the locus consists in the $\triangle OMN$.


Solution 2

This solution is a simplified version of the previous solution, it fills in some gaps. and it provides more information. The idea is to just follow the degrees of the expressions and equations in $\lambda, x, y$ involved. If we manage to conclude that the equation for $H$ is an equation of degree $1$, then we will know that it is a line. We don't need to know the equation explicitly.

Just like in the previous solution, we use analytic (coordinate) geometry, but we don't care how the axes are chosen.

The coordinates of $M$ are expressions of degree $1$ in $\lambda$.

The equation for $MP$ is an equation of degree $1$ in $x, y$ with constant coefficients for $x, y$, and whose constant term is an expression of degree $1$ in $\lambda$.

The coordinates of $P$ (the intersection of $MP$ and $OA$) are expressions of degree $1$ in $\lambda$.

The equation of the perpendicular from $P$ to $OB$ is of degree $1$ in $x, y$, with constant coefficients for $x, y$, and whose constant term is an expression of degree $1$ in $\lambda$. This corresponds to equation (2) in the above solution.

Similarly, the equation of the perpendicular from $Q$ to $OA$ is of degree $1$ in $x, y$, with constant coefficients for $x, y$, and whose constant term is an expression of degree $1$ in $\lambda$. This corresponds to equation (1) in the above solution.

Now, in principle, we would have to solve the system of two equations to obtain the coordinates of $H$ as expressions of $\lambda$, and then eliminate $\lambda$ to obtain an equation in $x, y$. Or, as a shortcut, we can eliminate $\lambda$ directly from the two equations. Either way, the result is an equation of degree $1$ in $x, y$.

This tells us that the locus is on this line. We just need to specify which set of points on this line is the locus.

The previous solution, with a good amount of hand waving, tells us that the solution is "a line segment $B_1A_1, B_1 \in OA, A_1 \in OB$". (On top of the hand waving the solution uses the unhappy notation $M$ for $B_1$ and $N$ for $A_1$, which is bad because $M$ has already been used!) We will do better than that.

Let $A_1$ be the foot of the perpendicular from $A$ to $OB$, and $B_1$ be the foot of the perpendicular from $B$ to $OA$. (For this paragraph see the picture shown in Solution 3.) Consider the limit situation when $M = A$. Then $Q = A_1$, and $P = A$. It follows that the intersection $H$ of the perpendiculars from $P$ to $OB$ and $Q$ to $OA$ is $A_1$. Similarly, the limit situation when $M = B$ yields $H = B_1$. Now it is reasonable to say that when $M$ moves from $A$ to $B$, $H$ moves from $A_1$ to $B_1$. So, the locus is the line segment joining the feet $A_1, B_1$ of the perpendiculars in $\triangle OAB$ from $A, B$. This answers question (a).

Again with a good amount of hand waving, the previous solution says "the locus consists in the $\triangle OB_1A_1$". We justify this by pointing out that if $M$ is inside $\triangle OAB$, then we can take the triangle $\triangle OA'B'$, such that $A' \in OA$, $B' \in OB$, $A'B'$ going through $M$ and parallel to $AB$. Then $H$ will be on the corresponding segment $A_1'B_1'$ determined by the feet of the perpendiculars in $\triangle OA'B'$. Conversely, it is easy to see that any point $H \in \triangle OA_1B_1$ is on a segment $A_1'B_1'$ obtained from a triangle $\triangle OA'B'$, and $H$ is obtained from a point $M \in A'B'$. This answers question (b).

(Solution by pf02, October 2024)


Solution 3

TO BE CONTINUED. SAVING MID WAY, SO I DON'T LOSE WORK DONE SO FAR.


See Also

1965 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1965_IMO_Problems/Problem_5&oldid=231037"