176. Another nice problems for middle school.
by Virgil Nicula, Nov 23, 2010, 9:51 PM
Proposed problem 1. Prove that 
is 
-isosceles or -right angle 
.
Proof. I"ll use the well-known relation
a.s.o. Therefore, 
is -isosceles or -right angled.
Otherwise,
or 
or 
or 
.
Proposed problem 2. Let
be an -right triangle with incenter 
. Denote 
and the midpoint 
of ![$[EF]$](//latex.artofproblemsolving.com/7/6/3/763239c0ce4fccc63411d3d6cb0011f7f6cc3a31.png)
. Prove that 
.
Proof 1 (synthetic). Denote the projections
, 
on 
of 
, 
respectively. Observe that 
and 
. Therefore, 
and
, i.e. is the circumcenter of 
. In conclusion, 
is middleline in the right trapezoid 
.
Remark. Prove easily that the point is the 
-exincenter of 
and is the 
-exincenter of 
. Thus 
and 
.
Proof 2 (metric). I"ll show that
what is equivalently with . Indeed, 
,
i.e.
. Observe that 
. Thus, 
.
Therefore,
![$\left[2\cdot \left(BE^2+BF^2\right)-EF^2\right]-$](//latex.artofproblemsolving.com/c/8/0/c80ba1323ecc2480ce2ad3d52e3baabf1c0bc212.png)
![$\left[2\cdot \left(CF^2+CE^2\right)-EF^2\right]$](//latex.artofproblemsolving.com/0/a/e/0aec074f7e260750437a8602e344b7651e11564a.png)
. In conclusion, 
, i.e. .
Proposed problem 3. In the convex cyclical quadrilateral
denote the projections , 
, 
of the vertex on lines 
, , 
respectively. Prove that there is the relation 
.
Proof. I"ll use the remarkable relation
in any triangle (standard notations). Denote 
, 
, 
,
, 
, 
. Therefore, the relation 
. So the proposed relation becomes
which is equivalently with the Ptolemy's relation 
. Observe that 
- the Simson's line of w.r.t. 
.
Proposed problem 4. Let with 
. Denote the midpoint of ![$[AH]$](//latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
and 
for which 
. Prove that 
.
Proof. Denote the midpoints
of ![$[AC] , [AB]$](//latex.artofproblemsolving.com/6/7/0/67097a379cb4ca1b13259c0d563ebe25abf4bd18.png)
respectively. Observe that the quadrilaterals 
and 
are cyclically
,
. Since 
, 
obtain that the quadrilateral 
is a parallelogram. In conclusion,
.
Proposed problem 5. Let be a convex quadrilateral for which 
and 
. Prove that 
.
Proof 1 (synthetic). Observe that is -right and isosceles triangle. Denote 
for which 
.
Observe that
, i.e. the quadrilateral 
is cyclically. Since 
obtain that
. In conclusion, 
.
Proof 2 (trigonometric). Denote
.Apply the trigonometric form of Ceva's theorem in the convex quadrilateral 
.
Proposed problem 6 (IMO 1985). A circle
has center on the side ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
of the cyclical .
The other three sides are tangent to the circle. Prove that
.
Proof 1 (trigonometric). Denote the tangent points
of 
with 
resperctively and 
.
Suppose w.l.o.g. that
. Prove easily that 
. Therefore, 
, which is truly because
(prove easily). From the remarks 
can find a synthetical proof of this problem.
Proof 2 (synthetic). ?!
Proposed problem 7.
and 
.
Proof.![$\sum_{i=1}^{n}\left(a_i+\frac{1}{a_i}\right)^2 \ge \frac{1}{n} \cdot \left[ \sum_{i=1}^{n}\left(a_i+\frac{1}{a_i}\right) \right]^2=$](//latex.artofproblemsolving.com/e/8/9/e89a376551130df8caac190aecbe23629df17557.png)
.
Proposed problem 8. Let be an -isosceles such that the orthocentre lies on its incircle 
. Ascertain 
.
Proof. Denote the midpoint
of the side ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Show easily that 
.
Denote
. Thus, 
.
Proposed problem 9. Let be an arbitrary point in . Through construct 
, 
and 
, where 
,
and 
. Prove that ![$[AIPF] + [BGPD] + [CEPJ] \le \frac{2}{3}\cdot [ABC] $](//latex.artofproblemsolving.com/d/c/2/dc2faf50b70e613c1bf0db1cc57eb5441b2c5b56.png)
, where through ![$[T]$](//latex.artofproblemsolving.com/f/4/2/f42aef7deb65ea67b2d33762ac74fbe5b3b83bfb.png)
denote area of 
.
Proof. Denote
, 
and 
.Observe that 
and ![$\frac {[PDI]}{[ABC]}+\frac {[PGJ]}{[ABC]}+\frac {[PFE]}{[ABC]}=$](//latex.artofproblemsolving.com/5/e/8/5e88e294df2e67366927e832f384147d0c5b8e65.png)
![$[PDI]+[PGJ]+[PFE]\ge $](//latex.artofproblemsolving.com/a/f/6/af6cd6f949f463dc8b695ebba1b45892ec9e0f71.png)
![$\frac 13\cdot [ABC]$](//latex.artofproblemsolving.com/7/e/0/7e0988eec58f2d19c57ba9b386a1f662929fe9a7.png)
. In conclusion,
![$[AIPF] + [BGPD] + [CEPJ]=$](//latex.artofproblemsolving.com/e/e/9/ee981669ce82df941912d44d6b0f14fb50012ec5.png)
![$[ABC]-\left([PDI]+[PGJ]+[PFE]\right)\ \le\ [ABC]-\frac 13\cdot [ABC]=\frac{2}{3}\cdot [ABC] $](//latex.artofproblemsolving.com/6/7/8/678d87d7de0baccb939216051448d0cde92c1f97.png)
.
Proposed problem 10. Prove that
.
Proof. I"ll use the well-known relation
a.s.o. Therefore, 
.
Proposed problem 11. Let
and 
be two positive real numbers. Prove the inequality 
.
Proof.
.
PP11. Let be a trapezoid with 
. Denote the midpoints , of , ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
and 
, 
. Prove that 
.
Proof[/u].[/b] Denote
and 
. Observe that 
and 
. Since 
obtain that
, i.e. 
. Analogously denote 
and 
.
Observe that
and 
. Since 
obtain that 
, i.e. 
. In conclusion, and 
and/or 
.
PP12. Let be a triangle. Prove that 
.
Proof 1.
.
Proof 2.
so that 
and 
.
Proof 3.
so that 
and 
and 
. Prove easily that in the isosceles trapezoid have
, i.e. . Otherwise, note 
and from 
obtain that 
. Since 
obtain 
, i.e. 
.
PP13. Let an equilateral and for a point 
construct the parallelogram 
, where 
. Denote the
projections
of the points 
on the line 
respectively and 
. Prove that 
and find 
.
Proof. Let
and 
, i.e. 
. Apply theorem of Sines to ![$[NP]$](//latex.artofproblemsolving.com/4/b/f/4bf4a2a9ab593d9ee3bd6f3316f38ad187f4cbb9.png)
in 
and 
exists 
so that 
Since ![$2\cdot [NAP]=MN\cdot MP\cdot\sin\widehat{NMP}=lh$](//latex.artofproblemsolving.com/d/3/6/d368e29b077b8caf85e5335cf97b08939087eb54.png)
get 
and 
.
is 
-isosceles or -right angle 
. Proof. I"ll use the well-known relation
a.s.o. Therefore, 
is -isosceles or -right angled.Otherwise,
or 
or 
or 
.Proposed problem 2. Let
be an -right triangle with incenter 
. Denote 
and the midpoint 
of ![$[EF]$](http://latex.artofproblemsolving.com/7/6/3/763239c0ce4fccc63411d3d6cb0011f7f6cc3a31.png)
. Prove that 
. Proof 1 (synthetic). Denote the projections
, 
on 
of 
, 
respectively. Observe that 
and 
. Therefore, 
and
, i.e. is the circumcenter of 
. In conclusion, 
is middleline in the right trapezoid 
.Remark. Prove easily that the point
is the 
-exincenter of 
and is the 
-exincenter of 
. Thus 
and 
.Proof 2 (metric). I"ll show that
what is equivalently with . Indeed, 
,i.e.
. Observe that 
. Thus, 
.Therefore,
![$\left[2\cdot \left(BE^2+BF^2\right)-EF^2\right]-$](http://latex.artofproblemsolving.com/c/8/0/c80ba1323ecc2480ce2ad3d52e3baabf1c0bc212.png)
![$\left[2\cdot \left(CF^2+CE^2\right)-EF^2\right]$](http://latex.artofproblemsolving.com/0/a/e/0aec074f7e260750437a8602e344b7651e11564a.png)
. In conclusion, 
, i.e. . Proposed problem 3. In the convex cyclical quadrilateral
denote the projections , 
, 
of the vertex
on lines 
, , 
respectively. Prove that there is the relation 
.Proof. I"ll use the remarkable relation
in any triangle (standard notations). Denote 
, 
, 
,
, 
, 
. Therefore, the relation 
. So the proposed relation becomes
which is equivalently with the Ptolemy's relation 
. Observe that 
- the Simson's line of w.r.t. 
.Proposed problem 4. Let
with 
. Denote the midpoint of ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
and 
for which 
. Prove that 
.Proof. Denote the midpoints
of ![$[AC] , [AB]$](http://latex.artofproblemsolving.com/6/7/0/67097a379cb4ca1b13259c0d563ebe25abf4bd18.png)
respectively. Observe that the quadrilaterals 
and 
are cyclically 
,
. Since 
, 
obtain that the quadrilateral 
is a parallelogram. In conclusion,
. Proposed problem 5. Let
be a convex quadrilateral for which 
and 
. Prove that 
.Proof 1 (synthetic). Observe that
is -right and isosceles triangle. Denote 
for which 
.Observe that
, i.e. the quadrilateral 
is cyclically. Since 
obtain that
. In conclusion, 
.Proof 2 (trigonometric). Denote
.Apply the trigonometric form of Ceva's theorem in the convex quadrilateral 
.Proposed problem 6 (IMO 1985). A circle
has center on the side ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
of the cyclical .The other three sides are tangent to the circle. Prove that
.Proof 1 (trigonometric). Denote the tangent points
of 
with 
resperctively and 
.Suppose w.l.o.g. that
. Prove easily that 
. Therefore, 
, which is truly because
(prove easily). From the remarks 
can find a synthetical proof of this problem.Proof 2 (synthetic). ?!
Proposed problem 7.
and 
. Proof.
![$\sum_{i=1}^{n}\left(a_i+\frac{1}{a_i}\right)^2 \ge \frac{1}{n} \cdot \left[ \sum_{i=1}^{n}\left(a_i+\frac{1}{a_i}\right) \right]^2=$](http://latex.artofproblemsolving.com/e/8/9/e89a376551130df8caac190aecbe23629df17557.png)
.Proposed problem 8. Let
be an -isosceles such that the orthocentre lies on its incircle 
. Ascertain 
.Proof. Denote the midpoint
of the side ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Show easily that 
.Denote
. Thus, 
.Proposed problem 9. Let
be an arbitrary point in . Through construct 
, 
and 
, where 
,
and 
. Prove that ![$[AIPF] + [BGPD] + [CEPJ] \le \frac{2}{3}\cdot [ABC] $](http://latex.artofproblemsolving.com/d/c/2/dc2faf50b70e613c1bf0db1cc57eb5441b2c5b56.png)
, where through ![$[T]$](http://latex.artofproblemsolving.com/f/4/2/f42aef7deb65ea67b2d33762ac74fbe5b3b83bfb.png)
denote area of 
.Proof. Denote
, 
and 
.Observe that 
and ![$\frac {[PDI]}{[ABC]}+\frac {[PGJ]}{[ABC]}+\frac {[PFE]}{[ABC]}=$](http://latex.artofproblemsolving.com/5/e/8/5e88e294df2e67366927e832f384147d0c5b8e65.png)
![$[PDI]+[PGJ]+[PFE]\ge $](http://latex.artofproblemsolving.com/a/f/6/af6cd6f949f463dc8b695ebba1b45892ec9e0f71.png)
![$\frac 13\cdot [ABC]$](http://latex.artofproblemsolving.com/7/e/0/7e0988eec58f2d19c57ba9b386a1f662929fe9a7.png)
. In conclusion,![$[AIPF] + [BGPD] + [CEPJ]=$](http://latex.artofproblemsolving.com/e/e/9/ee981669ce82df941912d44d6b0f14fb50012ec5.png)
![$[ABC]-\left([PDI]+[PGJ]+[PFE]\right)\ \le\ [ABC]-\frac 13\cdot [ABC]=\frac{2}{3}\cdot [ABC] $](http://latex.artofproblemsolving.com/6/7/8/678d87d7de0baccb939216051448d0cde92c1f97.png)
.Proposed problem 10. Prove that
.Proof. I"ll use the well-known relation
a.s.o. Therefore, 
.Proposed problem 11. Let
and 
be two positive real numbers. Prove the inequality 
.Proof.
.PP11. Let
be a trapezoid with 
. Denote the midpoints , of , ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
and 
, 
. Prove that 
.Proof[/u].[/b] Denote
and 
. Observe that 
and 
. Since 
obtain that
, i.e. 
. Analogously denote 
and 
.Observe that
and 
. Since 
obtain that 
, i.e. 
. In conclusion, and 
and/or 
.PP12. Let
be a triangle. Prove that 
.Proof 1.
.Proof 2.
so that 
and 
.Proof 3.
so that 
and 
and 
. Prove easily that in the isosceles trapezoid have
, i.e. . Otherwise, note 
and from 
obtain that 
. Since 
obtain 
, i.e. 
.PP13. Let an equilateral
and for a point 
construct the parallelogram 
, where 
. Denote theprojections
of the points 
on the line 
respectively and 
. Prove that 
and find 
.Proof. Let
and 
, i.e. 
. Apply theorem of Sines to ![$[NP]$](http://latex.artofproblemsolving.com/4/b/f/4bf4a2a9ab593d9ee3bd6f3316f38ad187f4cbb9.png)
in 
and 
exists 
so that 
Since ![$2\cdot [NAP]=MN\cdot MP\cdot\sin\widehat{NMP}=lh$](http://latex.artofproblemsolving.com/d/3/6/d368e29b077b8caf85e5335cf97b08939087eb54.png)
get 
and 
.This post has been edited 106 times. Last edited by Virgil Nicula, Nov 26, 2015, 7:04 PM
August 2018
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