Skip to main content

Top 10 problems of the week!

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. 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

Sunaina 💜


Popular posts from this blog

EGMO solutions, motivations and reviews ft. Atul, Pranjal and Abhay

The  European Girls' Mathematical Olympiad a.k.a EGMO 2022 just ended. Congrats to Jessica Wan from USA, Taisiia Korotchenko, and Galiia Sharafetdinova for the perfect scores! Moreover, the Indian girls brought home 4 bronze medals! By far, this is the best result the EGMO India Team has ever achieved! To celebrate the brilliant result, here's a compilation of EGMO 2022 solutions and motivations written by my and everyone's favorite IMOTCer Atul ! And along with that, we also have reviews of each problem written by everyone's favorite senior, Pranjal !  These solutions were actually found by Atul, Pranjal,  and Abhay  during the 3-hour live solve. In the live solve, they solved all the 6 problems in 3 hours 😍!!! Okie Dokie, I think we should get started with the problems! Enjoy! Problem 1:  Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and

Kőnig-Egerváry theorem

Graph theory has been my most favourite thing to learn in maths and and in this blog i hope to spread the knowledge about Kőnig's theorem. It is advised that the readers are aware about basic graph theory terminologies including bipartite graphs. Before going on to the theorem i would like to go on about matchings and vertex cover which we are going to use in the theorem  Matchings A matching $M$ of a graph $G$ is a subset of the edges of a graph in which no two edge share a common vertex (non adjacent edges). For example :- The Matching $M$ over here is edge $\{ 3 - 5 , 1-2 , \}$ or $\{ 1 - 2 , 4 - 3 \}$ etc .  Maximum Matching is a matching that contains the largest possible number of edges for instance in the above example the maximum matching is 2 edges as there cannot be a subset of non adjacent edges having greater than 2 edges (Readers are advised to try so they can convince themselves) A Perfect Matching  is a matching that matches all vertices of the graph or in other sen

Algorithms, or Mathematics?!

Hi everyone! In my last blog, I spoke about a couple of very interesting algorithmic techniques and how they can be used to solve a variety of very interesting problems. This blog however, is going to be completely different.   When we’re young we begin by learning the steps to add – we’re given the rules and we must learn to repeat them – no questions asked. Why does carry forward work the way it does? Well, no questions asked. To some extent, it is important to know how exactly to do a certain set of things while first learning maths. We may not be able to get anywhere in a subject if we’re unable to learn a few basic rules and know how to use them. However, after a certain point it is important to bring in the spirit of mathematical thinking within each student too – something missing in almost every form of school math education. Mathematical miseducation is so common, we wouldn’t even see it. We practically expect a math class to look like repetition and memorisation of disjointed