and $AB$) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that
\[\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}\]
$\fbox{2}.$ Determine all positive integers that are equal to $300$ times the sum of their digits.
$\fbox{3}.$ Let $n$ be an integer greater than or equal to $2$. There are $n$ people in one line, each of which is either a scoundrel (who always lie) or a knight (who always tells the truth). Every person, except the first, indicates the person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.
$\fbox{4}.$ Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation:
\[\left\{ \begin{array}{cc} x_1=4 \\ x_{n+1} = x_1x_2x_3 \cdots x_n+5 \ \ \text{for} \ n \ge 1\end{array} \right.\]
The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$
Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.
$\fbox{5}.$ $ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which
\[\widehat{APB}+\widehat{CPD}=180^\circ\]
$\fbox{6}.$ Determine all pairs $\{a, b\}$ of positive integers with the following property: in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers with the color $A$ difference of those two positive integers being $a$ or with color $B$ difference being $b$.
Bài viết đã được chỉnh sửa nội dung bởi Phạm Quang Toàn: 21-05-2012 - 19:12