Skip to main content

TOP 10 problems of Week#2

This week was full Number Theory and algebra biased!! 😄

Do try all the problems first!! And if you guys get any nice solutions , do post in the comments section!

Here are the walkthroughs of this week's top 5 Number Theory problems!

5th position (1999 JBMO P2): For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.

Walkthrough: It doesn't require a walkthrough, I wrote this here, cause it's a cute problem for the person who has just started Contest math :P

a. What is $A_0$?

b. Find out $A_1$.

c. Show that $\boxed{7}$ is the required answer!

4th position (APMO, Evan Chen's orders modulo a prime handout): Let $a,b,c$ be distinct integers. Given that $a | bc + b + c, b | ca + c + a$

and $c | ab + a + b$, prove that at least one of $a, b, c$ is not prime.

Walkthrough: Fully thanks to MSE ! (Also one should try MSE, it has helped me a lot, ofc it's more tilted to College math but it's great! )

a. FTSOC let $a,b,c$ be primes. Then note that by simon's favorite factoring trick , we get $(b+1)(c+1)\equiv 1 \mod a$ , similarly for $b,c$.

b. Bound $\frac{(a+1)(b+1)(c+1)}{abc}$ and get a contradiction !

3rd position (IMO 2011 P1): Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq  i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

Walkthrough: This was clearly the hardest of all like I took  lot of hints :P.

a. Show that $n_A\ne 6,5$.

b. We will show that $n_A=4$ is possible. hmm...so what do we have ? 

c. Show that $a_1+a_4=a_2+a_3$.

d. So $a_1+a_2|2(a_2+a_3) \implies k_1(a_1+a_2)=2(a_2+a_3)$ , and similarly $k_2(a_1+a_3)=2(a_2+a_3)$ , where $k_1>k_2$.

e.But note that $2a_3+2a_1>2(a_2-a_1) \implies k_2=3$.

f. Show that $k(a_1+a_2)=6a_2-6a_1 \implies k_1=4,5$

g. Case bash!!!!!!!!! 

2nd position(IMO 2011 P5) : Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.

Walkthrough: Thanks to the Pr0est Mueller.25

a. Take $P(m,0)$ and show that $f(m)|f(0)$ for all $m$. This gives that $f(x)$ has finite solutions.

b. Take $P(0,n)$ and $P(0,-n) $ and show that $f(n)=f(-n)$.

c. Prove by induction that if $a|b \implies f(a)|f(b)$ [it's not required, we just want $f(1)|f(b)$, but it's cute, so one can try].

d. Since $f(x)$ has finite solution, let the solutions be $f(1)<f(a_1)< \dots <f(a_k)<f(0)$.

e. So.. the problem now reduces on showing $f(a_i)|f(a_j)$ when $i<j$. 

f. okay so we have $f(1)|f(a_i)$ , so let's try showing $f(a_1)|f(a_2)$.

g. Let's take $P(a_2,a_1)$ then show that $f(a_2-a_1)= f(a_1)$ or $f(1)$.

h. But we want to show that $f(1)|f(2)$ . But note that if we show that $f(a_2-a_1)= f(a_1)$, then we will be done! So let's try to show that!

i. Take $P(a_2-a_1, -a_1)$ and wola!

1st Position (IMO Shortlist 2011 N3):  Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$

Walkthrough: a. Try to guess what possible solutions can be by taking $n=1,3,6$ ( just guess, no need to prove :P) 

b. Note that if $f(x)$ works then $f(x)+c, -f(x)$ works too. So we can assume $f(0)=0$ and $f(1)=1$.

c. With $P(p,1)$ and $P(p,0)$, where $p$ is a prime, show that $f(p)=\pm p^k$ , where $k|n$ 

d. Show $f(p)\ne -p^k$ [ we here use the fact that when $a^b-1|a^c-1 \implies b|c$ and also $k\ne 0$ ]

e. But n has finitely many prime factors , and there are finitely many prime, so there will exist a prime factor of $n$ say $d$ , which will be used infinitely times. So let $q$ be a very( verrrry big) prime with $f(q)=q^d$

f. Take $P(x,q)$, and show that $f(x)=x^d$. And then conclude! 

g. Solutions are $\boxed{f(x)=\pm x^d+c}$ where $d|n$.

Next are the walkthroughs of this week's top 5 Algebra problems! This is only for beginners algebra people ( It's not my fault that people who are reading this blog are pr0s 😎). *Take it more like as a set of problems to motivate you to study algebra :P

5th position (2010 AMC 10 A P21) : The polynomial $x^3-ax^2+bx-2010$ has three positive integer roots. What is the smallest possible value of $a$?

Walkthrough: a. Use Vieta's formula and factorize $2010=2\cdot 3 \cdot 5\cdot 67$ 

b. okay to minimize , obviously 6,5,67 will be answer :P (idk how to explain this part , it actually follows trivially :P)

4th position (NIMO summer contest P9): The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r, s$, and $t$. Evaluate$ (r + s) ^4 (s + t) ^4 (t + r)^ 4$

Walkthrough: a. Use Viteta's , we get $r+s+t=0$ 

b. we just have to find $(rst)^4$.

3rd position (AIME 2008 II P7): Let $ r$, $ s$, and $ t$ be the three roots of the equation $8x^3+1001x+2008=0.$ Find $ (r+s)^3+(s+t)^3+(t+r)^3$.

Walkthrough: Ooo quite similar to the problem we did  previously.

a. Use viteta and get $r+s+t=0$. So $ (r+s)^3+(s+t)^3+(t+r)^3=-(t^3+r^3+s^3)$.

b. So for people who are in grade 9 or below India standard, or any beginner in algebra, there's a very well known formula, which says $a^3+b^3+c^3=3abc$ , when $a+b+c=0$, it's just $a^3 + b3^ + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$ . Use it and we are done!

2nd position (2017 AMC 12 A P17) : There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?

Walkthrough:  I don't think so any walkthrough is needed :P. Answer is $\boxed{12}$. Consider $z^{{\frac{2\pi\cdot i \cdot k}{24}}\cdot 6}$ , it will be real iff $k$ is even .

1st Position(JBMO 2012 Shortlist): Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that :

$\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}$

Walkthrough: a. Use AM-GM and get $a^3+bc\ge 2a$.

b. Try to get this $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le \frac{(ab+bc+ca)^2}{3}$ , conclude with AM-GM again !

So these were my top 10 ! I personally wanted to add JMO 2017 P4, but didn't ( due to some soul imbalance , you will understand what I mean once you try this problem) . Go ahead and try if u want to :P.

What are your top 10s ? do write in the comments section (at least write something ! I will be happy to hear your comments ). Follow this blog if you want to see more contest math problems! See you all soon 😊.

---

I also made a pdf compilation of these problems and hints. Here is the google drive link https://drive.google.com/file/d/1sFQBU1MDIYGXWKG_1Bb6TpUAfTZfT4en/view?usp=sharing

Sunaina 💜


Labels:

Comments

  1. AnandDecember 13, 2020 at 10:39 PM

    Nice Post!
    BTW which year's APMO was N4? Asking 'cause its almost same as IMO 1992/P1

    ReplyDelete
    Replies
    1. Sunaina Pati December 14, 2020 at 3:07 PM

      I actually don't know.. Evan sourced it as APMO in his orders modulo prime handout ..

      Delete

Post a Comment

Popular posts from this blog

My experiences at EGMO, IMOTC and PROMYS experience

Yes, I know. This post should have been posted like 2 months ago. Okay okay, sorry. But yeah, I was just waiting for everything to be over and I was lazy. ( sorry ) You know, the transitioning period from high school to college is very weird. I will join CMI( Chennai Mathematical  Institue) for bsc maths and cs degree. And I am very scared. Like very very scared. No, not about making new friends and all. I don't care about that part because I know a decent amount of CMI people already.  What I am scared of is whether I will be able to handle the coursework and get good grades T_T Anyways, here's my EGMO PDC, EGMO, IMOTC and PROMYS experience. Yes, a lot of stuff. My EGMO experience is a lot and I wrote a lot of details, IMOTC and PROMYS is just a few paras. Oh to those, who don't know me or are reading for the first time. I am Sunaina Pati. I was IND2 at EGMO 2023 which was held in Slovenia. I was also invited to the IMOTC or International Mathematical Olympiad Training Cam...
11 comments
Read more

Geometry ( Finally!!!)

 This is just such an unfair blog.  Like if one goes through this blog, one can notice how dominated  Algebra is!! Like 6 out of 9 blog post is Algebra dominated -_- Where as I am not a fan of Algebra, compared to other genres of Olympiad Math(as of now). And this was just injustice for Synthetic Geo. So this time , go geo!!!!!!!!!!!  These problems are randomly from A Beautiful Journey through Olympiad Geometry.  Also perhaps I will post geo after March, because I am studying combi.  Problem:  Let $ABC$ be an acute triangle where $\angle BAC = 60^{\circ}$. Prove that if the Euler’s line of $\triangle ABC$ intersects $AB$ and $AC$ at $D$ and $E$, respectively, then $\triangle ADE$ is equilateral. Solution:  Since $\angle A=60^{\circ}$ , we get $AH=2R\cos A=R=AO$. So $\angle EHA=\angle DOA.$ Also it's well known that $H$ and $O $ isogonal conjugates.$\angle OAD =\angle EAH.$ By $ASA$ congruence, we get $AE=AD.$ Hence $\triangle ADE$ is equilateral....
3 comments
Read more

Problems I did this week [Jan8-Jan14]

Yeyy!! I am being so consistent with my posts~~ Here are a few problems I did the past week and yeah INMO going to happen soon :) All the best to everyone who is writing!  I wont be trying any new problems and will simply revise stuffs :) Some problems here are hard. Try them yourself and yeah~~Solutions (with sources) are given at the end! Problems discussed in the blog post Problem1: Let $ABC$ be a triangle whose incircle $\omega$ touches sides $BC, CA, AB$ at $D,E,F$ respectively. Let $H$ be the orthocenter of $DEF$ and let altitude $DH$ intersect $\omega$ again at $P$ and $EF$ intersect $BC$ at $L$. Let the circumcircle of $BPC$ intersect $\omega$ again at $X$. Prove that points $L,D,H,X$ are concyclic. Problem 2: Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ= 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ ...
Post a Comment
Read more

Just spam combo problems cause why not

This post is mainly for Rohan Bhaiya. He gave me/EGMO contestants a lot and lots of problems. Here are solutions to a very few of them.  To Rohan Bhaiya: I just wrote the sketch/proofs here cause why not :P. I did a few more extra problems so yeah.  I sort of sorted the problems into different sub-areas, but it's just better to try all of them! I did try some more combo problems outside this but I tried them in my tablet and worked there itself. So latexing was tough. Algorithms  "Just find the algorithm" they said and they died.  References:  Algorithms Pset by Abhay Bestrapalli Algorithms by Cody Johnson Problem1: Suppose the positive integer $n$ is odd. First Al writes the numbers $1, 2,\dots, 2n$ on the blackboard. Then he picks any two numbers $a, b$ erases them, and writes, instead, $|a - b|$. Prove that an odd number will remain at the end.  Proof: Well, we go $\mod 2$. Note that $$|a-b|\equiv a+b\mod 2\implies \text{ the final number is }1+2+\dots ...
2 comments
Read more

IMO Shortlist 2021 C1

 I am planning to do at least one ISL every day so that I do not lose my Olympiad touch (and also they are fun to think about!). Today, I tried the 2021 IMO shortlist C1.  (2021 ISL C1) Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$. Suppose not. Then any $3$ elements $x,y,z\in S$ will be $(x,y)=(y,z)=(x,z)$ or $(x,y)\ne (y,z)\ne (x,z)$. There exists an infinite set $T$ such that $\forall x,y\in T,(x,y)=d,$ where $d$ is constant. Fix a random element $a$. Note that $(x,a)|a$. So $(x,a)\le a$.Since there are infinite elements and finite many possibilities for the gcd (atmost $a$). So $\exists$ set $T$ which is infinite such that $\forall b_1,b_2\in T$ $$(a,b_1)=(a,b_2)=d.$$ Note that if $(b_1,b_2)\ne d$ then we get a contradiction as we get a set satisfying the proble...
1 comment
Read more

Some Geometry Problems for everyone to try!

 These problems are INMO~ish level. So trying this would be a good practice for INMO!  Let $ABCD$ be a quadrilateral. Let $M,N,P,Q$ be the midpoints of sides $AB,BC,CD,DA$. Prove that $MNPQ$ is a parallelogram. Consider $\Delta ABD$ and $\Delta BDC$ .Note that $NP||BD||MQ$. Similarly, $NM||AC||PQ$. Hence the parallelogram. In $\Delta ABC$, $\angle A$ be right. Let $D$ be the foot of the altitude from $A$ onto $BC$. Prove that $AD^2=BD\cdot CD$. Note that $\Delta ADB\sim \Delta CDA$. So by similarity, we have $$\frac{AD}{BD}=\frac{CD}{AD}.$$ In $\Delta ABC$, $\angle A$ be right. Let $D$ be the foot of the altitude from $A$ onto $BC$. Prove that $AD^2=BD\cdot CD$. Let $D\in CA$, such that $AD = AB$.Note that $BD||AS$. So by the Thales’ Proportionality Theorem, we are done! Given $\Delta ABC$, construct equilateral triangles $\Delta BCD,\Delta CAE,\Delta ABF$ outside of $\Delta ABC$. Prove that $AD=BE=CF$. This is just congruence. N...
3 comments
Read more

Introduction

  Hey Everyone!! This is my first Blog post. So let me give a brief introduction about myself. I am Sunaina Pati. I love solving Olympiad math problems,  learning crazy astronomical facts , playing hanabi and anti-chess, listening to Kpop , love making diagrams in Geogebra and  teaching other people maths 😊 . I love geometry , number theory and Combinatorics . I am starting this blog to keep myself a bit motivated in doing studies 😎 . Right now, I am planning to write walkthroughs on some of the best problems I tried over the week which can refer for hints 'cause solutions contain some major spoilers and one learns a lot while solving the problem on his own rather than seeing solutions . Also, there will be some reviews about Kpop songs, study techniques, my day to day lifestyles,exam reviews and ofc some non-sense surprises 😂.  I am planning to  try  posting every week on Sundays or Saturdays ( most probably) ! Though there is no guarantee about when I ...
21 comments
Read more

Problems with meeting people!

Yeah, I did some problems and here are a few of them! I hope you guys try them! Putnam, 2018 B3 Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$. Proof We have $$n|2^n\implies n=2^a\implies 2^a-1|2^n-1\implies a|n\implies a=2^b$$ $$\implies 2^{2^b}-2|2^{2^a}-2\implies 2^b-1|2^a-1\implies b|a\implies b=2^c.$$ Then simply bounding. USAMO 1987 Determine all solutions in non-zero integers $a$ and $b$ of the equation $$(a^2+b)(a+b^2) = (a-b)^3.$$ Proof We get $$ 2b^2+(a^2-3a)b+(a+3a^2)=0\implies b = \frac{3a-a^2\pm\sqrt{a^4-6a^3-15a^2-8a}}{4}$$ $$\implies a^4-6a^3-15a^2-8a=a(a-8)(a+1)^2\text{ a perfect square}$$ $$\implies a(a-8)=k^2\implies a^2-8a-k^2=0\implies \implies a=\frac{8\pm\sqrt{64+4k^2}}{2}=4\pm\sqrt{16+k^2}. $$ $$ 16+k^2=m^2\implies (m-k)(m+k)=16.$$ Now just bash. USAMO 1988 Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1...
3 comments
Read more

How to prepare for RMO?

"Let's wait for this exam to get over".. *Proceeds to wait for 2 whole fricking years!  I always wanted to write a book recommendation list, because I have been asked so many times! But then I was always like "Let's wait for this exam to get over" and so on. Why? You see it's pretty embarrassing to write a "How to prepare for RMO/INMO" post and then proceed to "fail" i.e not qualifying.  Okay okay, you might be thinking, "Sunaina you qualified like in 10th grade itself, you will obviously qualify in 11th and 12th grade." No. It's not that easy. Plus you are talking to a very underconfident girl. I have always underestimated myself. And I think that's the worst thing one can do itself. Am I confident about myself now? Definitely not but I am learning not to self-depreciate myself little by little. Okay, I shall write more about it in the next post describing my experience in 3 different camps and 1 program.  So, I got...
7 comments
Read more

Some random problems

  I know, I know. Different font indeed. I have deleted a few of my MSE answers. I felt they weren't that good in quality. And a few questions are from my prev aops account which I have deactivated now. I also have posted 10 IOQM types of problems. These can be used while preparing for IOQM. Problem: Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$  are different positive real numbers.  Proof: Note that by AM-GM $$\frac{ab}{c^3}+\frac{bc}{a^3}\ge \frac{2b}{ac}$$ and we also have $$\frac {b}{ac}+\frac{c}{ab}\ge \frac{2}{a}$$. Hence, $$\sum_{cyc}\frac{ab}{c^3}\ge\sum_{cyc}\frac{b}{ac}\ge\sum_{cyc}\frac{1}{a}$$ where everything we got is by applying AM-GM on $2$ terms and then dividing by $2$. USA TST 2007: Triangle $ABC$ which is inscribed in circle $\omega$. The tangent lines to $\omega$ at $B$ and $C$ meet at $T$. Point $S$ lies on ray $BC$ such that $AS$ is perpendicular to $AT$. Points $B_1$ and $C_1...
17 comments
Read more