# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 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$F(x)=0$##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2007 Issue 364
2007 Issue 364
Mathematics & Youth