IMO Math

It is the last moment of 2021. The very last. Do your last farewells. I thank God and my parents for all the opportunities they have given me during this year.  And I hope me and everyone get even better ones in the coming year.  After this, no more 2021.  I would like to share a meme at this moment.  On this occasion, I am solving 2021 IMO #6; as the last of 2021.  Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements. Let $A=\{a_1, a_2, a_3, \dots, a_n\}$. Note that for some number $0 \leq N \leq m^{m+1} - m$ with $m | N$, we can choose integers $x_{1}, x_{2}, \ldots x_{m}$ so that $$0 \leq x_{i} < m$$ and \[N = x_{1}m + x_{2}m^{2} + \ldots + x_{m}m^{m}.\] We know this by dividing both sides by $m$ and then writing $N$ in base $m$. Next, notice that we can write $N$ as the sum

I won't be posting any more problems today until 11:59 PM EST.  At 11:59 PM, I will post a very unique problem that is suitable for this day.  Take a wild guess for what it could be  😉

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$.

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that $$AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$$

 Prove that for all reals $a,b,c,d\in(0,1)$ we have$$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$

Let $ABCD$ be a cyclic quadrilateral, and let diagonals $AC$ and $BD$ intersect at $X$.Let $C_1,D_1$ and $M$ be the midpoints of segments $CX,DX$ and $CD$, respectively. Lines $AD_1$ and $BC_1$ intersect at $Y$, and line $MY$ intersects diagonals $AC$ and $BD$ at different points $E$ and $F$, respectively. Prove that line $XY$ is tangent to the circle through $E,F$ and $X$.

Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?

There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear. A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.

Hello,  Thanks for 1000 views of this blog over just a few days!  Be sure to read my solutions to the problems posted by clicking "Read more" on the post. I'm sure you will find them very interesting.  Please keep visiting my blog!  

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$.

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$\[f\left( f(x)^2y \right) = x^3 f(xy).\]

December 29th, 1979 is when my mom was born, and yesterday was her birthday. I am solving the 1989 IMO SL #29 in honor of her 🙂 (oops i forgot to do this yesterday). I would solve 1979 IMO SL #29, but it doesn't exist because there were only 26 problems for that shortlist.  Happy Birthday Mama!

Let $A_1$, $A_2$, $\ldots\,$, $A_n$ be finite sets. Prove that \[ \Bigl| \bigcup_{1 \le i \le n} A_i \Bigr| \ge \frac{1}{2} \sum_{1 \le i \le n} \left| A_i \right| - \frac{1}{6} \sum_{1 \le i < j \le n} \left| A_i \cap A_j \right| \, . \]Recall that if $S$ is a finite set, then its cardinality $|S|$ is the number of elements of $S$.

In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$ prove that $\angle KNB= \angle BNL$ .