Posted on Categories:数学代写, 数学竞赛代写

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PROBLEM 1
Show that if $\alpha_{1} / b_{1}=a_{2} / b_{2}=a_{3} / b_{3}$ and $p_{1} \cdot p_{2} \cdot p$ are not all zeto, then
$$\left(\frac{a_{1}}{b_{1}}\right)^{n}=\frac{p_{1} a_{1}^{n}+p a_{2}^{n}+p a a_{3}^{n}}{p b_{1}^{n}+p p b_{2}^{n}+p s b_{3}^{n}}$$
for every positive integer $n$.
PROBLEM 2
Deternine which of the tro number $\sqrt{c+1}-\sqrt{c} \cdot \sqrt{c}-\sqrt{c-1}$ is greater for any $c \geq 1$.
PROBLEM 3
Let e be the length of the lypotense of a right angle triagle whone of her two wides lave lengths a and b. Prove that $a+b \leq \sqrt{2} e$. When does the $\propto$. Wality hold?
PROBLEM 4
Let $A B C$ be an «uilateral triangle, and $P$ be an arbitrary point within the triangle, Perpendiculan $P D, P E, P F$ aw drawn to the thre siden of the triangle. Show that, no natter where $P$ is chosen,
$$\frac{P D+P E+P F}{A B+B C+C A}=\frac{1}{2 \sqrt{3}}$$
PROBLEM 5
Let $A B C$ be a triangle with sides of lengths $a, b$ and $e$. Let the bisector of the angle $C$ ent $A B$ in $D$. Prove that the length of $C D$ is
$$\frac{2 a b \cos \frac{C}{2}}{a+b}$$
PROBLEM 6
Find the sum of $1 \cdot 1 !+2 \cdot 2 !+3 \cdot 3 !+\cdots+(n-1)(n-1) !+n \cdot n !$, where $n !=$ $n(n-1)(n-2) \cdots 2 \cdot 1$
PROBLEM 7
Show that there are no integers $a, b$, e for which $a^{2}+b^{2}-8 c=6$.