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 "1959 IMO Problems/Problem 5"

m
(→‎Part c))
 
(12 intermediate revisions by 5 users not shown)
Line 9: Line 9:
 
(c) Find the locus of the midpoints of the segments <math>PQ </math> as <math>M </math> varies between <math>A </math> and <math>B </math>.
 
(c) Find the locus of the midpoints of the segments <math>PQ </math> as <math>M </math> varies between <math>A </math> and <math>B </math>.
  
−
== Solutions ==
+
== Solution ==
  
 
=== Part A ===
 
=== Part A ===
Line 19: Line 19:
 
=== Part B ===
 
=== Part B ===
  
−
We observe that <math> \frac{AM}{MB} = \frac{CM}{MB} = \frac{AN}{NB} </math> since the triangles <math>ABN, BCN </math> are similar.  Then <math>NM </math> bisects <math>ANB </math>.
+
We observe that <math> \frac{AM}{MB} = \frac{CM}{MB} = \frac{AN}{NB} </math> since the triangles <math>ABN, BCM</math> are similar.  Then <math>NM </math> bisects <math>ANB </math>.
  
 
We now consider the circle with diameter <math>AB </math>.  Since <math>ANB </math> is a right angle, <math>N </math> lies on the circle, and since <math>MN </math> bisects <math>ANB </math>, the arcs it intercepts are congruent, i.e., it passes through the bisector of arc <math>AB </math> (going counterclockwise), which is a constant point.
 
We now consider the circle with diameter <math>AB </math>.  Since <math>ANB </math> is a right angle, <math>N </math> lies on the circle, and since <math>MN </math> bisects <math>ANB </math>, the arcs it intercepts are congruent, i.e., it passes through the bisector of arc <math>AB </math> (going counterclockwise), which is a constant point.
Line 27: Line 27:
 
Denote the midpoint of <math>PQ </math> as <math>R </math>.  It is clear that <math>R </math>'s distance from <math>AB </math> is the average of the distances of <math>P </math> and <math>Q </math> from <math>AB </math>, i.e., half the length of <math>AB</math>, which is a constant.  Therefore the locus in question is a line segment.
 
Denote the midpoint of <math>PQ </math> as <math>R </math>.  It is clear that <math>R </math>'s distance from <math>AB </math> is the average of the distances of <math>P </math> and <math>Q </math> from <math>AB </math>, i.e., half the length of <math>AB</math>, which is a constant.  Therefore the locus in question is a line segment.
  
 +
== Solution 2 ==
 +
[[Image:IMO19595A.png]]
 +
=== Part a) ===
 +
Notice that <math>\angle BAF = \arctan (\frac{MF}{AM}) = \frac{MB}{AM} </math> and <math>\angle ABC = \arctan (\frac{MC}{MB}) = \frac{AM}{MB} </math>.
  
−
{{alternate solutions}}
+
<math>\implies </math> <math>\angle BAF + \angle ABC = 90^{\circ} </math>.
 +
 
 +
Now notice that <math>\angle FNB = 90^{\circ} </math>. Considering <math>\angle BAF </math> as <math>\theta </math>, this gives <math>\angle MBN = 90^{\circ} - \theta </math> and thus <math>\angle MFN = 90^{\circ} + \theta </math>. But notice that <math>\angle MFA = 90^{\circ} - \theta </math>, which means that <math>\angle AFN = 180^{\circ} </math>. Therefore points <math>A, F, N </math> are collinear. Now <math>\angle BNF = 90^{\circ} </math> and <math>\angle ANC = 90^{\circ} </math>. Therefore, <math>\angle BNC = 180^{\circ} </math> and thus points <math>B, N, C </math> are collinear. Therefore, AF and BC intersect at N.
 +
 
 +
=== Part b) ===
 +
Construct the bisector of arc AB above AB. Call it X. <math>\angle ANM = \angle ACM = 45^{\circ} </math>. Now <math>\angle ANB = 90^{\circ}</math> which means N lies on the circle with AB as diameter.
 +
 
 +
<math>\implies </math> <math>\angle ANX = \angle ABX = 45^{\circ} = \angle ANM </math>. Therefore since M and X are on the same side of <math>N </math>, <math>NM </math> passes through <math>X </math> wherever we choose <math>M </math> on <math>AB </math>.
 +
 
 +
=== Part c) ===
 +
Let the midpoint of <math>PQ </math> be <math>R </math>. Let <math>G </math> be the midpoint of <math>AM </math>. Let <math>F </math> be the midpoint of <math>MB </math>. Let <math>I </math> be the foot of the perpendicular from <math>R </math> onto <math>AB </math>. Therefore by Midpoint Theorem, <math>RI = \frac{PG + HQ}{2} = \frac{AG + HB}{2} = \frac{AB}{4} </math>. Therefore the distance <math>RI </math> is a constant and thus the locus is a straight line parallel to <math>AB </math> at a distance (to <math>AB </math>, of course) of <math>\frac{AB}{4} </math>.
  
 
== See Also ==
 
== See Also ==
  
  
 +
Quadrados e Circulos circunscritos / IMO 1959-#5
 +
Link do vídeo: https://youtu.be/UNcHD5JI6wU
 
{{IMO box|year=1959|num-b=4|num-a=6}}
 
{{IMO box|year=1959|num-b=4|num-a=6}}
  
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]
 +
[[Category:Geometric Construction Problems]]

Latest revision as of 07:23, 23 May 2024

Problem

An arbitrary point $M$ is selected in the interior of the segment $AB$. The squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with the segments $AM$ and $MB$ as their respective bases. The circles about these squares, with respective centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$.

(a) Prove that the points $N$ and $N'$ coincide.

(b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$.

(c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

Solution

Part A

Since the triangles $AFM, CBM$ are congruent, the angles $AFM, CBM$ are congruent; hence $AN'B$ is a right angle. Therefore $N'$ must lie on the circumcircles of both quadrilaterals; hence it is the same point as $N$.

1IMO5A.JPG

Part B

We observe that $\frac{AM}{MB} = \frac{CM}{MB} = \frac{AN}{NB}$ since the triangles $ABN, BCM$ are similar. Then $NM$ bisects $ANB$.

We now consider the circle with diameter $AB$. Since $ANB$ is a right angle, $N$ lies on the circle, and since $MN$ bisects $ANB$, the arcs it intercepts are congruent, i.e., it passes through the bisector of arc $AB$ (going counterclockwise), which is a constant point.

Part C

Denote the midpoint of $PQ$ as $R$. It is clear that $R$'s distance from $AB$ is the average of the distances of $P$ and $Q$ from $AB$, i.e., half the length of $AB$, which is a constant. Therefore the locus in question is a line segment.

Solution 2

IMO19595A.png

Part a)

Notice that $\angle BAF = \arctan (\frac{MF}{AM}) = \frac{MB}{AM}$ and $\angle ABC = \arctan (\frac{MC}{MB}) = \frac{AM}{MB}$.

$\implies$ $\angle BAF + \angle ABC = 90^{\circ}$.

Now notice that $\angle FNB = 90^{\circ}$. Considering $\angle BAF$ as $\theta$, this gives $\angle MBN = 90^{\circ} - \theta$ and thus $\angle MFN = 90^{\circ} + \theta$. But notice that $\angle MFA = 90^{\circ} - \theta$, which means that $\angle AFN = 180^{\circ}$. Therefore points $A, F, N$ are collinear. Now $\angle BNF = 90^{\circ}$ and $\angle ANC = 90^{\circ}$. Therefore, $\angle BNC = 180^{\circ}$ and thus points $B, N, C$ are collinear. Therefore, AF and BC intersect at N.

Part b)

Construct the bisector of arc AB above AB. Call it X. $\angle ANM = \angle ACM = 45^{\circ}$. Now $\angle ANB = 90^{\circ}$ which means N lies on the circle with AB as diameter.

$\implies$ $\angle ANX = \angle ABX = 45^{\circ} = \angle ANM$. Therefore since M and X are on the same side of $N$, $NM$ passes through $X$ wherever we choose $M$ on $AB$.

Part c)

Let the midpoint of $PQ$ be $R$. Let $G$ be the midpoint of $AM$. Let $F$ be the midpoint of $MB$. Let $I$ be the foot of the perpendicular from $R$ onto $AB$. Therefore by Midpoint Theorem, $RI = \frac{PG + HQ}{2} = \frac{AG + HB}{2} = \frac{AB}{4}$. Therefore the distance $RI$ is a constant and thus the locus is a straight line parallel to $AB$ at a distance (to $AB$, of course) of $\frac{AB}{4}$.

See Also

Quadrados e Circulos circunscritos / IMO 1959-#5 Link do vídeo: https://youtu.be/UNcHD5JI6wU

1959 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=1959_IMO_Problems/Problem_5&oldid=219593"