Slični zadaci

skakavac 2013 prvo kolo ss3 3

Za sustav novčanica $1 = n_1 < n_2, ... < n_k$ kažemo da je dobar ako "greedy" postupak vraćanja novca uvijek rezultira s najmanjim brojem isplaćenih novčanica. Dokažite da je sustav $1 < b < a$ dobar ako i samo ako postoji $k \in \mathbb{N}$ takav da je $\lceil\frac{a}{k}\rceil = b$ .
Napomena: U greedy postupku u svakom koraku vraćamo najviše moguće najvećih mogućih novčanica. Nadalje, $\lceil x\rceil$ označava najmanji cijeli broj veći od (ili jednak) $x$ .

Slični zadaci

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Zadaci

IMO Shortlist 2003 problem A1

Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$ ) be real numbers such that

$a_{ij}>0$ for $i = j$ ;
$a_{ij}<0$ for $i\neq j$ .

Prove the existence of positive real numbers $c_{1}$ , $c_{2}$ , $c_{3}$ such that the numbers

$a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}$ ,
$a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}$ ,
$a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$

are either all negative, or all zero, or all positive.

IMO Shortlist 1999 problem A2

The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ( $n \geq 2$ ). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?

IMO Shortlist 1998 problem A2

Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that

$\frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}.$

IMO Shortlist 1996 problem A1

Suppose that $a, b, c > 0$ such that $abc = 1$ . Prove that $\frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1.$

IMO Shortlist 1993 problem A2

Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

Državno natjecanje 1999 SŠ2 4

na jednom turniru sudjelovalo je $n$ kosarkaskih ekipa. svaka ekipa odigrala je sa svakom drugom tocno jednu utakmicu. nerijesenih ishoda nije bilo. ako na kraju turnira $i$ -ita ekipa ima $x_i$ pobjeda i $y_i$ poraza $(i = 1, 2, \dots, n)$ dokazite da je
$x_1^2 + x_2^2 + \dots + x_n^2 = y_1^2 + y_2^2 + \dots + y_n^2$