Olympiad problems

2004 Korea Junior Math Olympiad, 1

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.

2014 Contests, 2

Tags: function , algebra proposed , algebra , functional equation , KMO , Korea

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

2014 Korea National Olympiad, 2

Tags: function , algebra proposed , algebra , functional equation , KMO , Korea

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

2020 Korea Junior Math Olympiad, 6

Tags: KJMO , KMO

for a positive integer $n$, there are positive integers $a_1, a_2, ... a_n$ that satisfy these two. (1) $a_1=1, a_n=2020$ (2) for all integer $i$, $i$satisfies $2\leq i\leq n, a_i-a_{i-1}=-2$ or $3$. find the greatest $n$

2020 Korean MO winter camp, #8

Tags: graph theory , KMO , combinatorics

I've come across a challenging graph theory problem. Roughly translated, it goes something like this: There are n lines drawn on a plane; no two lines are parallel to each other, and no three lines meet at a single point. Those lines would partition the plane down into many 'area's. Suppose we select one point from each area. Also, should two areas share a common side, we connect the two points belonging to the respective areas with a line. A graph consisted of points and lines will have been made. Find all possible 'n' that will make a hamiltonian circuit exist for the given graph