Slični zadaci

IMO Shortlist 1997 problem 13

In town $A,$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $B,$ there are $n$ girls $g_1, g_2, \ldots, g_n$ and $2n - 1$ boys $b_1, b_2, \ldots, b_{2n-1}.$ The girl $g_i,$ $i = 1, 2, \ldots, n,$ knows the boys $b_1, b_2, \ldots, b_{2i-1},$ and no others. For all $r = 1, 2, \ldots, n,$ denote by $A(r),B(r)$ the number of different ways in which $r$ girls from town $A,$ respectively town $B,$ can dance with $r$ boys from their own town, forming $r$ pairs, each girl with a boy she knows. Prove that $A(r) = B(r)$ for each $r = 1, 2, \ldots, n.$

Slični zadaci

Lista Tekst Dva stupca

Zadaci

IMO Shortlist 1997 problem 17

Find all pairs $(a,b)$ of positive integers that satisfy the equation: $a^{b^2} = b^a$ .

IMO Shortlist 1997 problem 18

The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E, F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R,$ respectively. The line $EF$ meets $BC$ at $P.$ Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC.$

IMO Shortlist 1997 problem 19

Let $a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that
$\sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}).$
Proposed by Romania

IMO Shortlist 1997 problem 20

Let $ABC$ be a triangle. $D$ is a point on the side $(BC)$ . The line $AD$ meets the circumcircle again at $X$ . $P$ is the foot of the perpendicular from $X$ to $AB$ , and $Q$ is the foot of the perpendicular from $X$ to $AC$ . Show that the line $PQ$ is a tangent to the circle on diameter $XD$ if and only if $AB = AC$ .

IMO Shortlist 1997 problem 21

Let $x_1$ , $x_2$ , $\ldots$ , $x_n$ be real numbers satisfying the conditions:
$|x_1 + x_2 + \dots + x_n| = 1$ and $|x_i| \leq \frac{n+1}{2}$ , for $i = 1, 2, \dots, n$
Show that there exists a permutation $y_1$ , $y_2$ , $\ldots$ , $y_n$ of $x_1$ , $x_2$ , $\ldots$ , $x_n$ such that
$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$

IMO Shortlist 1997 problem 22

Does there exist functions $f,g: \mathbb{R}\to\mathbb{R}$ such that $f(g(x)) = x^2$ and $g(f(x)) = x^k$ for all real numbers $x$

a) if $k = 3$ ?

b) if $k = 4$ ?