Olympiad problems

AND:

OR:

NO:

Found problems: 78

First
Previous
1
2
3
4
Next
Last

KoMaL A Problems 2023/2024, A. 883

Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

1967 Vietnam National Olympiad, 1

Tags: analysis , algebra , graph

Draw the graph of the function $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$.

1976 Spain Mathematical Olympiad, 8

Tags: function , algebra , analysis

Given the function $$y =|x^2 - 4x + 3|.$$ Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.

First
Previous
1
2
3
4
Next
Last