IMO Math

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.  a) Prove that $0\le b_n<2$.  b) Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

 Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n.$$

Hello,  I am making changes to this site today, so please visit after 6pm EST. The content might look messed up and weird before that.  Thank you!

Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that  a) Using medians of that triangle it is possible to construct a rectangular triangle.  b) The following inequality:\[5(a^2+b^2-c^2) \geq 8ab,\]is valid, where $a,b$ and $c$ are side length of the given triangle.

Let $a$ be a postive integer and let ${a_n}$ be defined by $a_0=0$ and \[a_{n+1}=(a_n+1)a+(a+1)a_n+2\sqrt{a(a+1)a_n(a_n+1)}\]Show that for each positive integer $n,a_n$ is a positive integer.

Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles $ DAB, ABC, BCD, CDA,$ respectively. Prove that $R_A + R_C > R_B + R_D$ if and only if $ \angle A + \angle C > \angle B + \angle D.$

Prove that from $x + y = 1 \  (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\] 

This problem was requested to be solved by a user. If you would like to request a problem too, please use the form on the left menu of the blog.  Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

This problem was requested to be solved by a user. If you would like to request a problem too, please use the form on the left menu of the blog.  Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$. 

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$  (a) Find a pair $(a,b)$ which is 51-good, but not very good.  (b) Show that all 2010-good pairs are very good.

$$ne^n = \sum_{k=0}^{\infty}\frac{n^k(n - k)^2}{k!}.$$   I dreamt of this identity yesterday night but I don't even know how to prove it lol

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) Suppose that among all projections of points in $\mathcal{S}$ onto some line $m$, the maximum possible distance between two consecutive projections is $\delta$. We need to prove $\delta \ge \Omega(n^{-1/3})$.  At this point, define $A,B$ as the two farthest points in $\mathcal{S}$. So, all points lie in the intersections of the circles centered at $A,B$ with radius $AB \geq 1$.  Choose chord $XY$ in circle $B$ where $XY \perp AB$ and $d(A, XY)=\frac{1}{2}$. Also, let $\mat

Due to the request of a few users, I got a custom domain at  imomath.xyz . The  math4l.blogspot.com  link will still work, but you can just go to  imomath.xyz  now. 

If $a,b,c$ are the sides of a triangle, prove that \[\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 \]

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Let $ \alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$.

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with\[\sum^{2011}_{j=1} j  x^n_j = a^{n+1} + 1\]

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]

I.  Am.  Completely.  Shocked.  I couldn't even think, in my dreams, for this blog to blow up so much. I'm glad to know that my solutions are helping people! Keep visiting this blog, more solutions are to come! As of now, I have no plans of closing this blog, and I plan to continue posting daily.  Btw, I won't be posting much of these "X views" anymore, since they're getting so frequent lol. I'll only post at some major number of views, like 100K, 150K, 200K, etc.

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.  a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;  b) Calculate the distance of $p$ from either base;  c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.  Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+  a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

Consider a prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as the top and bottom faces is given. Each side of the two pentagons and each of the 25 segments $A_iB_j$ is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Prove that all 10 sides of the top and bottom faces have the same color.

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,10^6\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100  \]are pairwise disjoint.

I am speechless. Thank you so much for 10K views! I will try to keep posting interesting content every day. Keep visiting! By the way, I have recently added a new feature in this blog. If you have a specific contest math problem you would like me to solve, please use the "Problem/Advice request" form on the left side of the homepage. It doesn't have to be an olympiad problem, but as long as it is a contest problem you can submit it through the form. You can use the same form if you want to ask for any advice. I will make a blog post regarding your problem/advice request.

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that\[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \]

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrilateral such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$  (a) Prove that $ABCD$ and $A''B''C''D''$ are similar.  (b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M = \left[\frac {a + b}{2} \right]$. Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\ n - b, & \text{if } n >M. \end{cases} \]Let $ f^1(n) = f(n),$ $ f_{i + 1}(n) = f(f^i(n)),$ $ i = 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) = 0$

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that \[ a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}. \]

Thanks everyone for 3K views on this blog! I didn't expect it to blow up this much. Keep viewing it, more interesting content is to come!

 Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying$$f(x^2f(y)^2)=f(x)^2f(y)$$for all $x,y\in\mathbb{Q}_{>0}$

At this point,  I have decided that I am only going to do IMO Shortlist problems.  Once I finish EGMO, I will keep doing this if I don't feel like the problems have started to repeat.  If I do feel that way, then I will then go towards research and most likely close this blog.  These plans might change in the future.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$has no positive roots.

Since winter break is now over for me, although I will still be posting daily, I will post 3-6 problems instead of more.  However, the content will still be high quality! You won't ever run out of posts to view, and you won't ever be bored. 

Let $a,b,c,d$ be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

From now on, I will post one IMO shortlist, then two of these below.        1 : "EGMO" ,     2 : "Romanian Masters of Mathematics" ,     3 : "Tournament of Towns" ,     4 : "USA ELMO Shortlist" ,     5 : "USA Math Prize for Girls Olympiad" ,     7 : "USAJMO" ,     8 : "Canada National Olympiad" ,     9 : "China TST" ,     10 : "India IMO Training Camp" ,     11 : "India National Olympiad" For example, it could go like: 2021 IMO #1, 2019 EGM0 #2, China TST #3.

Thanks for 1600 views, everybody! Keep viewing this blog!

Everyone has trouble with math olympiad problems, but what makes the difference is what you do  after .  Below are steps you should do (in order) if you do not know how to solve a problem.  1. Keep thinking about the problem until you completely run out of ideas, and you have tried every single one and it hasn't worked. Don't let any idea go; e 2. Okay, now it's time to give up. Read the solution carefully and understand every step of it. Don't think about the motivation for it yet.  3. When you have finished reading the solution and have verified that it is correct, in your mind summarize what they did, yet without thinking about the motivation. 4. Now, think about what the 'key' in the solution was, or what the 'heart of the problem' was. Basically, think about the main insight in the solution after which it got easy. Make sure to remember this 'insight', because it will apply to many, many other problems than just this one. 5. After a couple d

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d = \max \{d_{i}\mid 1 \leq i \leq n \}$.  (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \]  (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). 

Let $ a, b, c$ be positive real numbers so that $ abc = 1$. Prove that \[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1. \]

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that\[ f(m+n)f(m-n) = f(m^2)  \]for $m,n \in \mathbb{N}$.

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$

I will be posting higher quality math olympiad problems from now on. Using the random math olympiad problem generator Python program (which I had posted before), most problems are not high quality.  So, in the Python code, I have now removed the olympiads that do not have many good problems.  My updated list from which the computer randomizes looks like this:    1 : "IMO Shortlist" ,   2 : "IMO Shortlist" ,   3 : "APMO" ,   4 : "Austrian-Polish" ,     5 : "Balkan MO Shortlist" ,     6 : "Balkan MO Shortlist" ,     7 : "Baltic Way" ,     8 : "Benelux" ,     9 : "EGMO" ,     10 : "Iranian Geometry Olympiad" ,     11 : "Romanian Masters of Mathematics" ,     12 : "Silk Road" ,     13 : "Tournament of Towns" ,     14 : "USA BAMO" ,     15 : "USA ELMO Shortlist" ,     16 : "USA ELMO Shortlist" ,     17 : "USA USEMO" ,     1

Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

 In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

 It is now officially the very first moment of 2022 (in EST).  Welcome, 2022! I hope you bring great happiness and health to everyone.  Time to eat cake.  via GIPHY