Olympiad problems

2019 Ramnicean Hope, 2

Calculate $ \inf_{x> 0} \sqrt{(1+x)^2+4/x} . $ [i]Constantin Rusu[/i] and [i]Mihai Neagu[/i]

2024 All-Russian Olympiad Regional Round, 9.2

Tags: quadratic , function graphing , conic , parabola , trapezoid , analytic geometry , geometry

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

2005 Moldova National Olympiad, 11.2

Tags: algebra , function , function graphing , inequalities

Let $a$ and $b$ be two real numbers. Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.

2007 Bulgarian Autumn Math Competition, Problem 9.1

Tags: algebra , function , geometry , function graphing

We're given the functions $f(x)=|x-1|-|x-2|$ and $g(x)=|x-3|$. a) Draw the graph of the function $f(x)$. b) Determine the area of the section enclosed by the functions $f(x)$ and $g(x)$.

2009 Kosovo National Mathematical Olympiad, 1

Tags: algebra , function graphing

Find the graph of the function $y=x-|x+x^2|$

1964 Spain Mathematical Olympiad, 6

Tags: function graphing

Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$.