2018 JBMO SL #A6

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

2013 BAMO-8 #4

 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.

2000 IMO SL #N4

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

2009 JBMO Shortlist #A1

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

[RELEASED] Random math olympiad problems

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 . 

2018 European Mathematical Cup Junior #2

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

2003 CentroAmerican #2

$S$ 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 $AH=AG$.

2003 CentroAmerican #1

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.

2001 IMO SL NT #5

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.

2015 Canada MO #2

 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 

About posting geometry

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. 

2021 USA TST #1

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 $abc+abd+acd+bcd$.

2013 APMO #2

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 $r$.

2018 Benelux #3

 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 $AH$.

2011 China National Olympiad #6

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