The IMO experience

by Aiscrim, Jul 29, 2017, 11:50 AM

We arrived at the hotel at 2 AM on the 17th of July after roughly 24 hours of flights and connections. We had the check-in done, went to sleep and woke up at 8:30 AM because the breakfast was ending at 9 AM. I have no idea what happened next, I just remember that sometime during the day I visited my former MOP roommate, Dylan aka SGP 3. Oh, and they showed us the place in which the contest was going to take place next day. Went to sleep at 7:30 PM.

The next day we woke up at 6:30 AM. I was quite surprised to find out that I had no emotions. One might say that it was the Dunning-Kruger effect, but I actually was just hoping for this to end as soon as possible. I had reached satiety with Olympiad problems and was not very excited. As a result, in the last month before IMO I did not work very much. I was psychologically ready for bronze, even for honorable mention. Nevertheless, I entered the contest room trying to do my best. Number 1 seemed really easy. After playing with the small numbers for a bit I quickly realized that $a_0\equiv 2(\mathrm{mod}\ 3)$ does not work and $a_0\equiv 0(\mathrm{mod}\ 3)$ works. I was left with one more case. It was clear that there was only one general approach possible and that it had to work, otherwise the problem was way too hard for a number $1$ . However, I kept saying that if $a_0=4$ then $a_1=7$ and that if $a_0=16$ then $a_1=19$ ; at one point, seeing that I got that a lot of $a_0\equiv 1(\mathrm{mod}\ 3)$ work and that the question was asking for all solutions, I realized there was a mistake. I did not see anything wrong with my approach, so I checked my small cases. I realized that $16$ is actually a perfect square (woah) and that if $a_0=16$ then $a_1=4$ . However, I kept saying that if $a_0=4$ then $a_1=7$ . After two hours of despair that I could not find the mistake, I wrote my solution down, including the case $a_0\equiv 1(\mathrm{mod} \ 3)$ (which was almost identical to the case $a_0\equiv 0(\mathrm{mod}\ 3)$ ). I had about 3 hours left. As number three was looking horrendous, I decided to spend the remaining time on the FE. It was pretty easy to make significant progress, but pretty hard to find the ending. In the last 30 minutes, I tried to find my mistake in the remaining case of problem $1$ but did not manage to. After the first day ended I was very anxious because the progress I made thus far was not even assuring an honorable mention.

I spent the remaining of the day playing ping pong and singing karaoke. Went to sleep at 9 PM.

The next day we woke up at 7 AM. I was again relaxed knowing that a geo was going to appear. Almost all of the team wished for number 4 to be combo and number 5 geo, except for one guy who wanted them switched. Geometry as the first problem was not a pleasant sight, but I did not mind. After one hour I had almost no progress on the geo. It was a geo #1 and I was struggling with it. I would have laughed if I had not been so worried about not solving it. My first impression was that it was going to be easy after some similarities and PoP. I could not integrate that midpoint in the drawing. I tried to see if I could see any harmonic fascicle to use the midpoint, but the configuration was clearly not projective. As a last resort, I thought about inversion. I was aware that this was a problem 4 and inversion was very improbable as it was too complicated, but I had literally no idea. After analyzing the drawing, I decided that $R$ was the best inversion pole. After I finished the inverted drawing, the problem fell apart almost immediately. I obtained that $RKTB$ is cyclic, where $\{A,B\}=\ell\cap \Gamma$ and the rest was angle chasing. Yay, honorable mention. I spent 45 minutes on number 6 as it seemed very nice and I did not mind if I did not solve it, I just wanted to get a feel of it. I tried some Lagrange interpolation stuff and divisibility, but it was clear that the problem was harder than that. After that, I tackled number 5 almost entirely with inductive approaches. This is a good example of a time when I should have stopped the attacking and began scouting, to use Evan Chen’s preferred concepts. Anyway, the contest came to an end and I was very relieved. A little disappointed that after all of these years of wishing to go in the IMO I ended with such a poor performance, but mostly relieved.

Remember the guy who wanted P4 Geo and P5 Combo? He got his wish and went on to solve both of them and he also had substantial progress on P6. The other two guys who wanted P5 geo spent 3 hours on P4; one of them eventually solved it also using inversion and the other Cartesian bashed it.

The rest of the day was again spent playing ping pong, in the karaoke. I also played cards with the SGP team (or did this happen on another day? don’t really remember) and brushed up my Singaporean slang.

After the contest the atmosphere was pretty relaxed. In the next days we went to a talk about women in mathematics, visited Sugar Loaf, Maracana Stadium, the lagoon where the rowing contest took place in the Olympic Games, seen the Las Etnias (an amazing mural graffiti cca 15 meters tall), went to Artur Avila’s talk (with whom I had the pleasure to discuss later for about 15 minutes after meeting him randomly in the hotel lobby).

I got very lucky. I obtained $7$ points on number $1$ despite the $4\rightarrow 7$ mistake and this got me a silver medal. One might argue that the mistake was just the base case and it was clear that the solution worked. However, one might also argue that the solution was not perfect and it deserved a $6$ . The grading scheme on number $2$ was very loose and everybody got lots of points. Unfortunately, my luck was counterbalanced by my teammates’ misfortune: we had four people one point under the cutoff for a better medal.

I met babu2001, anantmudgal, Ankoganit, a Bulgarian and a Brazilian who are also coming at Princeton in fall, the USA team (Junyao puts an incredible amount of spin in table tennis), Alex Song, Evan Chen, befriended the other members of the SGP team, Kiril from Bulgaria (a friend from AMSP) and lots of other nice people to whom I apologize for not remembering their names.

College is arriving at a time when I don’t find Olympiad math as interesting as I used to, so I got this going for me, which is nice. There is plenty to be said about math, high school and life in general, maybe I will write another post sometime in the future.

I will most likely part ways with the Olympiad-type problems. A good thing to keep in mind is that you are left with the people you’ve met and the memories you share with them rather than with the problem you solved.

Thanks to every member of AoPS I have ever interacted with for a most interesting and formative journey!

High school stash, part 2

by Aiscrim, Jun 2, 2017, 5:11 PM

1. Prove that in any subset with $27$ elements of the set $\{1,2,...,2007\}$ there exist three distinct elements $a,b,c$ such that $\gcd{(a,b)}$ divides $c$ . (some Romanian contest?)
Proof.

2. Prove that for any two permutation $\sigma, \tau\in \mathcal{S}_n$ there is a function $f:\{1,2,...,n\} \rightarrow \{-1,1\}$ for which we simultaneously have
$\left | \displaystyle\sum\limits_{k=i}^{j} f(\sigma (k)) \right |\le 2,\ \left | \displaystyle\sum\limits_{k=i}^{j} f(\tau (k)) \right |\le 2$ for any $1\le i\le j\le n$ . (Danube Mathematical Competition 2009, a Romanian contest)
Proof.

3. Let $p\equiv 3 (\mathrm{mod}\ 4)$ be a prime number. Determine the number of distinct residues modulo $p$ given by $(x^2+y^2)^2$ when $(x,p)=(y,p)=1$ . (Bulgaria TST 2007)
Proof.

4. Given a cyclic quadrilateral $ABCD$ , let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$ . The midpoints of $AB$ and $CD$ are $G$ and $H$ , respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$ , $G$ and $H$ . (IMO Shortlist 2009)
Proof.

5. Prove that for any positive integer $m$ there exists a positive integer $N$ such that for all $2\le b\le 1389$ , the sum of the digits of $N$ in base $b$ is greater than $m$ . (Iran TST 2010)
Proof.

6. Let $a,b,n$ be positive integers with $b>1$ and $b^n-1$ dividing $a$ . Prove that there are at lest $n$ nonzero digits when writing $a$ in base $b$ . (IMO Shortlist 1993)
Proof.

7. Let $f$ be an irreducible polynomial with integer coefficients. Prove that if $f$ has two roots with product $1$ then the degree of $f$ is even. (Brazilian MO 2006)
Proof.

8. Let $a,b,c>1$ . Prove that $\dfrac{1}{a^3-1}+\dfrac{1}{b^3-1}+\dfrac{1}{c^3-1}\ge \dfrac{3}{abc-1}$ Proof.

9. Prove that the polynomial $x^n-x-1$ is irreducible in $\mathbb{Z}[X]$ for any positive integer $n$ . (Selmer's theorem)
Proof.

10. The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$ . Let $AD$ be an altitude of triangle $ABC$ , and let $M$ be the midpoint of the segment $AD$ . If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$ ), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$ . (IMO Shortlist 2002)
Proof.

This post has been edited 1 time. Last edited by Aiscrim, Jun 3, 2017, 5:26 AM

High school stash, part 1

by Aiscrim, May 30, 2017, 6:58 PM

As high school is over, I decided to post some of the nicer problems I encountered during these $4$ years. At first I was not really enthusiastic about the idea of writing so many problems and thoughts here. However, I really enjoy reading other blogs, so I though that there might be other users who would find this interesting. I won't access AoPS so much after IMO, so I decided to get started and begin posting. Also, here is a list of the blogs that I more or less follow(ed): yugrey,Zhero,pythag011,Ankoganit,randomusername,proglote,Wolstenholme,
v_Enhance,IDMasterz,mathocean97,TelvCohl,rkm0959.

1. Let $p$ be a prime number and $a$ an integer not divisible by $p$ . The polynomial $f(X)=X^p-X+a$ is irreducible in $\mathbb{Z}[x]$ .
Proof.

2. If $s(k)$ is the sum of the digits of the integer $k$ , then $s(2^n)$ goes to infinity as $n$ goes to infinity.
Proof.

3. Let $\mathcal{C}$ be a non-degenerate conic and let $ABC$ a triangle that is not self polar with respect to $\mathcal{C}$ . The triangle $ABC$ and its polar triangle are perspective. (Conway's theorem)
Proof.

4. For an odd prime $p$ , let $S_p$ be the set of positive integers $n$ with the property that $p$ divides $n!+1$ . Prove that there exists a constanct $c>0$ such that
$|S_p|<cp^{\frac{2}{3}}$ for all primes $p$ .
Proof.

5. Let $p$ be a prime number and $f$ a polynomial with integer coefficients with degree $d$ . If $f(0)=0,f(1)=1$ and $f(n)\equiv 0/1 (\mathrm{mod}\ p)$ for all integers $n$ , then $d\ge p-1$ . (IMO SHL 1997)
Proof.

6. Prove that there exists an infinite number of pairs of primes $(p,q)$ such that $p|2^{q-1}-1$ and $q|2^{p-1}-1$ . (Romania TST 2009)
Proof

7. Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A$ is included in $B$ and $\displaystyle\prod\limits_{x\in B} x=\displaystyle\sum\limits_{x\in B} x$ . (USA TST 2001)
Proof.

8. For a set $A$ of positive integers, let $\sigma_A(n)$ be the number of ways of writing $n$ as the sum of two distinct elements from $a$ . If two distinct finite sets $A$ and $B$ have the property that $\sigma_A(n)=\sigma_B(n)$ for all positive integers $n$ , prove that the number of elements in each of $A$ and $B$ is a power of two. (Erdos-Selfridge)
Proof.

9. The numbers $1,2,...,2n$ are partitioned in two sets with $n$ elements each. For each set, we consider the $n^2$ sums $a+b$ with $a,b$ from the set ( $a$ and $b$ can be equal). Each sum is reduced modulo $2n$ . Prove that the $n^2$ residues from a set are equal in some order to the other $n^2$ residues. (St. Petersburg, 1996)
Proof.

10. Let $A\subset \mathbb{Z}_{n^2}$ be a set of $n$ (distinct) residues mod $n^2$ . Prove that there exists $B\subset \mathbb{Z}_{n^2}$ with $|B|=n$ and $|A+B|\ge \dfrac{n^2}{2}$ (IMO Shl 1999)
Proof.

Days 3&4

by Aiscrim, May 5, 2016, 7:09 PM

1. (ARO 2016, grade 10, P2) Diagonals $AC,BD$ of cyclic quadrilateral $ABCD$ intersect at $P$ . Point $Q$ is on $BC$ (between $B$ and $C$ ) such that $PQ \perp AC$ . Prove that the line passes through the circumcenters of triangles $APD$ and $BQD$ is parallel to $AD$ .
Solution
Comment

2. (ARO 2016, grade 9, P2) $\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$ .An arbitrary line $\ell$ intersects with $AB,AC$ at $K,L$ ,respectively and intersect with $\omega$ at $P,Q$ .Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$ .Prove that $P,Q,S,T$ are concyclic.
Solution
Comment

3. (???) Let $XYZTUV$ be a convex hexagon. Let $\{A\}=XV\cap YZ,\ \{B\}=XV\cap UT,\ \{C\}=UT\cap YZ$ and $\{A^\prime \}=UV\cap ZT,\ \{B^\prime \}=XY\cap ZT,\ \{C^\prime \}=XY\cap UV$ . If $\dfrac{XY}{B^\prime C^\prime}=\dfrac{ZT}{A^\prime B^\prime}=\dfrac{UV}{A^\prime C^\prime}$ prove that
$\dfrac{YZ}{AC}=\dfrac{UT}{BC}=\dfrac{VX}{AB}$ Solution
Comment

Days 1&2, Russian geometry

by Aiscrim, May 3, 2016, 3:41 PM

1. (ARO 2016, grade 11, P8) Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$ .Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and $M_B,M_C$ .Define $\Omega_B$ and $\Omega_C$ analogusly. Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.
Solution
Comment

2. (Russia, 3rd round??, 2015) In an isosceles triangle $ABC$ internal and external angle bisector of angle $\angle B$ intersect with $AC$ at points $B_1,B_2$ ,respectively.From $B_1,B_2$ construct tangents to incircle of triangle $ABC$ ,other than $AC$ .They are tangent to incircle at points $K_1$ and $K_2$ respectively. Prove that the points $B, K_1$ and $K_2$ are on the same line.
Solution
Comment

3. (ARO 2016, grade 9,P7) In triangle $ABC$ , $AB<AC$ and $\omega$ is incirle.The $A$ -excircle is tangent to $BC$ at $A^\prime$ .Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$ .The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$ .Prove that the sum $XY+XZ$ not depends to point $X$ .
Solution
Comment

4. (ARO 2016, grade 10, P8) In acute triangle $ABC$ , $AC<BC$ , $M$ is midpoint of $AB$ and $\Omega$ is it's circumcircle.Let $C^\prime$ be antipode of $C$ in $\Omega$ . $AC^\prime$ and $BC^\prime$ intersect with $CM$ at $K,L$ ,respectively.The perpendicular drawn from $K$ to $AC^\prime$ and perpendicular drawn from $L$ to $BC^\prime$ intersect with $AB$ and each other and form a triangle $\Delta$ .Prove that circumcircles of $\Delta$ and $\Omega$ are tangent.
Solution
Comment

Submit

bump for four years!
by huashiliao2020, Jun 10, 2023, 5:18 PM
Best blog ever seen! Keep up the good work!
by tbn456834678, Mar 19, 2018, 12:37 PM
I scream/ We scream/ We all scream for... Aiscrim ! (Well sorry if I pronounced your username wrong )
by Kayak, Dec 15, 2017, 5:54 AM
updates please for us enthusiasts.
by vjdjmathaddict, Jul 10, 2017, 1:37 PM
Hello Aiscrim
by Ankoganit, May 16, 2017, 9:44 AM
Great Blog! More posts please!
by anantmudgal09, Dec 8, 2016, 7:51 PM
1st shout! This blog looks cool!
by Ankoganit, Nov 3, 2016, 2:25 PM

7 shouts

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