Problems done in August

 Welcome back! So today I will be sharing a few problems which I did last week and some ISLs. Easy ones I guess.

Happy September 2021! 

Problem[APMO 2018 P1]: Let $ABC$ be a triangle with orthocenter $H$ and let $M$ and $N$ denote the midpoints of ${AB}$ and ${AC}$. Assume $H$ lies inside quadrilateral $BMNC$, and the

circumcircles of $\triangle BMH$ and $\triangle CNH$ are tangent. The line through $H$ parallel to ${BC}$ intersects $(BMH)$ and $(CNH)$ again at $K$, $L$ respectively.

Let $F = {MK} \cap {NL}$, and let $J$ denote the incenter of $\triangle MHN$.

Prove that $FJ = FA$.

Proof: 

• By angle chase, we get $\angle FKL=\angle FLK.$
•    Hence $KL||MN\implies FM=FN.$
•   And we get $\angle MFN=2A\implies F$ is circumcentre if $(AMN)\implies FA=FM=FN.$
•   And we get $\angle MHN=180-2A$
•    Hence $MFHN$ is cyclic.
•    By fact 5, $ME=FJ=FN\implies FJ=FA.$

Problem[Shortlist 2007 G3]:

Let $ABCD$ be a trapezoid whose diagonals meet at $P$. Point $Q$ lies between parallel lines $BC$ and $AD$, and line $CD$ separates points $P$ and $Q$.

Given that $\angle AQD = \angle CQB$, prove that $\angle BQP = \angle DAQ$.

Proof:    (With Erijon)

•  consider the homothety centered at $P$ which  takes $B$ to $D$ and $C$ to $A$

•     Note that $BL||QD,$ $L$  is the image of $Q$ We also have $AL||CQ$
•    And we get $$\angle DLA=\angle CLB=\angle CDB=\angle AQB$$
•   Hence we get $QCBL,QADL$ cyclic.
•   So we have $BQ||DL$ then we get

$$\angle BQL=\angle QLD =\angle QAD$$

and we are done.

Problem[EGMO 2013/1]:

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that if $AD=BE$ then the triangle $ABC$ is right-angled.

Proof:

•      So introduce the midpoint of $AE,AB$ as $M,J.$
•      Note that $JC=\frac{1}{2}AD$
•     And we have $JM=\frac{1}{2}AB$
•     $AM=AC\implies JA\perp MC\implies BAC=90.$
  
  

Problem[APMO 2013 P5 ]:

 Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.

Proof: Note that $$-1=(A,E;C,D)=(A,RE\cap \omega; C,D)$$ But we know $$(A,C;B,D)=-1\implies RE\cap \omega=B$$

Problem[2008 G1]:

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$.

Prove that the six points $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are concyclic.

Proof: 

•  Let $(M_c)\cap (M_B)=X,H.$
•      Note that $XH$ is the radical axis, so we have $XH\perp M_CM_B.$ But we have $AH\perp BC||M_CM_B\implies A-X-H$
•      So $A$ lies on the radical axis $\implies Ac_1\cdot AC_2=AB_2\cdot AB_1\implies (C_1B_2B_1C_2)$ is cyclic, similarly we get $C_1C_2B_1B_2$ cyclic, $B_1A_2A_1B_2$ is cyclic.
•    Now note that the pairwise radical axis doesn't concur.
  
  

Problem 6: Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares

Solution: I am not gonna lie this question felt really hard to me :( 

The idea took time, but it's simple.

So let $4n+1=a^2$ then note $b^2=9n+1= 2\cdot (4n+1)+n-1= 2\cdot a^2+ \frac{a^2-1}{4} -1= \frac{9a^2-5}{4}\implies 9a^2-5\text{~~is a square}$

So both $a^2,9a^2-5$ is a square. Let $9a^2-5=k^2, 9a^2=l^2\implies l^2-k^2= (l+k)(l-k)=5\implies l=2,k=\implies a^2=1=4n+1\implies n=0.$

Problem 7: Find all $n \in \mathbb{N}$ for which $\exists$ $x,y,k \in \mathbb{N}$ such that $(x,y)=1$ , $k>1$ and $3^{n}=x^k+y^k$

Solution: The second form of Zigmondy Theorem states that : there’s a prime dividing $a^k + b^k$ and not $a^l + b^l$ for $l < k$ ( The exception is $(a,b,k)=(2,1,3)$).

So now if $(a,b,k)\ne(2,1,3)$ then it is not possible since we will have two distinct primes dividing $x^k+y^k=3^n.$ Not possible.

If we have $(a,b,k)=(2,1,3)$ we get $3^n=2^k+1.$ Using Mihăilescu theorem ( which states that the only consecutive perfect powers are

$8$ and $9$) 

Hence $\boxed{n=2, k=3, x=2, y=1}$ is the only solution.

Problem 8: Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

Proof: We know that $$a-b|P(a)-P(b)\implies km|P(n+km)-P(n)\implies m|P(n+km)-P(n)\implies m|P(n+km).$$

Problem 9: The sequence of integers $a_n$ is given by $a_0 = 0, a_n= p(a_{n-1})$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.

Proof : We assume $m\le k.$

Note that $a_{m-1}-a_{k-1}|P(a_{m-1})-P(a_{k-1})$

Similarly we get $a_{m-2}-a_{k-2}|P(a_{m-2})-P(a_{k-2})$

And we continue we get $a_0-a_{k-m}| P(a_{m-1})-P(a_{k-1}).$

So we get $gcd(a_m,a_k)=gcd(a_{k-m},a_k).$ And this continues.

By the euclidean algorithm, we are done.

Problem 10:Given a polynomial $P(x)$ with  natural coefficients;

Let us denote with $a_n$ the sum of the digits of $P(n)$ value.

Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.

Proof : Let $(b_n,b_{n-1},\dots, b_0)$ be the coefficients of the polynomial $P(x)=b_n\cdot x^n+\dots+x_0.$

Then surely there exist a natural number $k$ such that $P(10^k)$ is of the form $\overline{b_n0\dots 0b_{n-1}0\dots 0 b_0}.$

Then consider $10^k,10^{k+1},\dots .$

Yep, this is it for this week's blog! Autumn is about to come! I didn't even realise it but 2021 is about to end in 4 months!

Also, I have exported a few contacts from Feedburner to Mailchimp and I am in the process of removing FeedBurner completely from this blog. And people will be getting mails once a week only!

P.S. Do comment on how you liked the problems, which one was easy and what was your favourite!

P.P.S. Since IOQM is close, would you all like it if I post 10 IOQM problems each week? That would be great I think? It would probably be a series of collaborative posts.

See you soon 

Sunaina 💜