Olympiad problems

2024 Korea Junior Math Olympiad (First Round), 12.

For reals $x,y$, find the maximum of A. $ A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12} $

2003 Junior Balkan Team Selection Tests - Romania, 4

Tags: square , geometry , area , min , maximum value , distance

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

2017 South East Mathematical Olympiad, 4

Tags: algebra , maximum value

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

1975 Vietnam National Olympiad, 5

Tags: algebra , function , minimum value , maximum value , analysis

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

2012 IFYM, Sozopol, 4

Tags: algebra , maximum value

The numbers $x_i,i=1,2…6\in \mathbb{R}^+$ are such that $x_1+x_2+...+x_6=1$ and $x_1 x_3 x_5+x_2 x_4 x_6\geq \frac{1}{540}$. Let $S=x_1 x_2 x_3+x_2 x_3 x_4+...+x_6 x_1 x_2$. If $max\, S=\frac{p}{q}$ , where $gcd(p,q)=1$, find $p+q$.

2016 India Regional Mathematical Olympiad, 4

Tags: inequality , algebra , three variable inequality , maximum value , inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$