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 $\angle KNB= \angle BNL$ .

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove $$\frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$$

 For a positive integer $n>2$, consider the $n-1$ fractions$$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.

 Find all triplets of positive integers $ (a,m,n)$ such that $ a^m + 1 \mid (a + 1)^n$.

Determine all integers $a, b, c$ satisfying the identities $$a + b + c = 15$$  $$(a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540.$$

I have been working on this Python program that randomizes and gives a math Olympiad problem per run. This is what I use to determine which problem I will do.  You can see the code below and run it either on your computer or at repl.it . 

Find all pairs $ (x; y) $ of positive integers such that $$xy | x^2 + 2y -1.$$

$S$ is a circle with $AB$ a diameter and $t$ is the tangent line to $S$ at $B$. Consider the two points $C$ and $D$ on $t$ such that $B$ is between $C$ and $D$. Suppose $E$ and $F$ are the intersections of $S$ with $AC$ and $AD$ and $G$ and $H$ are the intersections of $S$ with $CF$ and $DE$. Show that $AH=AG$.

Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

Let $a > b > c > d$ be positive integers and suppose that\[ ac + bd = (b+d+a-c)(b+d-a+c).  \]Prove that $ab + cd$ is not prime.

 Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that\[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\] lol I know I just said I'll be posting geo less frequently but looking at the 2015 Canadian MO problems this one was way too tempting 

Hello,  Just letting you know that I will be posting less geometry problems, as I am currently reading EGMO (Euclidean Geometry in Mathematical Olympiads) and have not finished the whole book yet. So, there are many topics which I haven't covered yet and a few geometry problems require the use of those new topics, which I am therefore unable to solve.  I will still be posting geometry problems, but a tad less frequently. 

Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$.

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...).\]If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

 Let $ABC$ be a triangle with orthocentre $H$, and let $D$, $E$, and $F$ denote the respective midpoints of line segments $AB$, $AC$, and $AH$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively. (a) Show that lines $PE$ and $QD$ intersect on the circumcircle of triangle $ABC$. (b) Prove that lines $PD$ and $QE$ intersect on line segment $AH$.

Let $m,n$ be positive integer numbers. Prove that there exist infinitely many couples of positive integers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

Let $a_i,b_i,i=1,\cdots,n$ are nonnegative numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

$\textbf{(a)}$ Find all real solutions of the equation$$\Big(x^2-2x\Big)^{x^2+x-6}=1$$Explain why your solutions are the only solutions. $\textbf{(b)}$ The following expression is a rational number. Find its value.$$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$

 Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.

 The incircle $\Omega$ of the acute triangle $ABC$ is tangent to $\overline{BC}$ at a point $K$. Let $\overline{AD}$ be an altitude of triangle $ABC$, and let $M$ be the midpoint of the segment $\overline{AD}$. 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 $N$.

 Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying$$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$

Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by \[\dfrac{a_1+a_2+\ldots+a_n}{n}.\] Among all $n$ and partitions of $S$, determine the minimum possible score.

 Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by $$a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1}, \dots \text{for }n=2,3,4,\ldots,$$ $$b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1}, \dots \text{for }n=2,3,4,\ldots.$$ Prove that, besides the number $1$, no two numbers in the sequences are identical.

Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality \[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\] By the AM-GM inequality for 3-variables, we have $$a^3+2=a^3+1+1 \ge 3\sqrt[3]{a^3 \cdot 1 \cdot 1 }=3a.$$ Thus, it is enough to show that $$\frac{a}{3a+6}+\frac{b}{3b+6}+\frac{c}{3c+6}+\frac{d}{3d+6} \le \frac{4}{9}.$$ Note that we can write the last inequality as $$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}+\frac{1}{d+2} \le \frac{4}{3}. \space \space \space \space \space \space \space \space (*)$$ By HM-AM inequality, we can say that $$\frac{1}{4}(\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}+\frac{1}{d+2}) \ge \frac{4}{(a+2)+(b+2)+(c+2)+(d+2)}$$ $$=\frac{4}{4+2+2+2+2}=\frac{1}{3}.$$ This implies the desired inequality, namely $(*)$. $\square$

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:  1) The total weights of both piles are the same.  2) Each pile contains two pebbles of each colour.

Let $a,b$ and $c$ be the length of sides of a triangle. Then prove that $S<\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle. 

Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]

Let $ABCD$ be a parallelogram. Let $O$ be a point in its interior such that $\angle AOB+\angle DOC=180^\circ$. Show that $\angle ODC = \angle OBC$. 

Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $\angle PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following array is an anti-Pascal triangle with four rows which contains every integer from 1 to 10.  Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to 1+2+...+2018?

Determine the largest number which is the product of positive integers with sum 1976. The answer is \(\boxed{2 \cdot 3^{658}}\).  Note that there cannot be any integers \(n>4\) in the maximal product, because we can just replace \(n\) by \(3,n-3\) to achieve a greater product. Also, we would not want any \(1\)s in the maximal product as they would not help in increasing the product and would just take up space in the sum. Notice that we can replace any \(4\)s by two \(2\)s leaving the product and the sum unchanged. Lastly, we don't want more than two \(2\)s, because we can just replace three \(2\)s by two \(3\)s to obtain a larger product. Therefore, the maximum product must consist of \(3\)s and zero, one or two \(2\)s. Since \(1976 = 3\cdot 658 + 2\), we have one \(2\) and \(658\) \(3\)'s in our maximum product, giving the answer of \(2\cdot 3^{658}\), as desired. \(\square\)