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2007 Issue 364

  1. Find the last two digit numbers of $3^{9999}-2^{9999}$.
  2. Compare the sum (consisting of $n+1$ terms) $$S_{n}=\frac{2}{101+1}+\frac{2^{2}}{101^{2}+1}+\ldots+\frac{2^{n+1}}{101^{2 \prime}+1}$$ with $0,02$.
  3. Let $a$, $b$ and $c$ be positive numbers such that $a b+b c+c a=1$. Prove the inequality $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a} \geq 3+\sqrt{\frac{1}{a^{2}}+1}+\sqrt{\frac{1}{b^{2}}+1}+\sqrt{\frac{1}{c^{2}}+1}.$$ When does equality occur?
  4. Solve the equation $$\sqrt{7 x^{2}-22 x+28}+\sqrt{7 x^{2}+8 x+13}+\sqrt{31 x^{2}+14 x+4}=3 \sqrt{3}(x+2).$$
  5. Let $A B C D$ be a rectangular with $A B<B C$. Let $M$ be a point, different from $A$ and $B$, on the half-circle with $A B$ as its diameter and on the same side with $CD$. $MA$ and $MB$ meet $CD$ at $P$ and $Q$ respectively. $M C$ and $M D$ meet $A B$ at $E$ and $F$ respectively. Find the position of the point $M$ on the half-circle such that the sum $P Q+E F$ is smallest possible. Calculate this smallest value.
  6. Find all pairs of positive integers $a$, $b$ such that $q^{2}-r=2007,$ where $q$ and $r$ are respectively the quotient and the remainder obtained when dividing $a^{2}+b^{2}$ by $a+b$
  7. Consider the equation $$a x^{3}-x^{2}+b x-1=0$$ where $a, b$ are real numbers, $a \neq 0$ and $a \neq b$ such that all of its roots are positive real numbers. Find the smallest value of $$P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}.$$
  8. Choose five points $A$, $B$, $C$, $D$ and $E$ on a sphere with radius $R$ such that $$\widehat{B A C}=\widehat{C A D} =\widehat{D A E}=\widehat{E A B}=\frac{2}{3} \widehat{B A D}=\frac{2}{3} \widehat{C A E}.$$ Prove the inequality $$A B+A C+A D+A E \leq 4 \sqrt{2} R.$$
  9. Suppose that $M$, $N$ and $P$ are three points lying respectively on the edges $A B$, $B C$. $C A$ of a triangle $A B C$ such that $$A M+B N+C P=M R+N C+P A.$$ Prove the inequality $S_{M N P} \leq \dfrac{1}{4} S_{A B C}$.
  10. Find the limit $$\lim_{n \rightarrow+\infty}\sqrt{2-\sqrt{2}} \cdot \sqrt{2-\sqrt{2+\sqrt{2}}} \cdots \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}}_{n \text { signsor railiad }}$$
  11. Denote by $[x]$ the largest integer not exceeding $x$ and write $\{x\}=x-[x] .$ Find the limit $\displaystyle \lim _{n \rightarrow+\infty}(7+4 \sqrt{3})^{n}$.
  12. Let $n$ be a positive integer and $2 n+2$ real numbers $a, b, a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ such that $a_{i} \neq 0(i=1,2, \ldots, n)$ and the function $$F(x)=\sum_{i=1}^{n} \sqrt{a_{i} x+b_{i}}-(a x+b)$$ satisfies the following property: There exists distinct real numbers $\alpha, \beta$ such that $F(\alpha)=F(\beta)=0$ Prove that $\alpha$ and $\beta$ are the only real solutions of the equation $F(x)=0$

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Mathematics & Youth: 2007 Issue 364
2007 Issue 364
Mathematics & Youth
https://www.molympiad.org/2020/09/2007-issue-364.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2007-issue-364.html
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