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

## $show=home 1. Does there exist a positive integer$k$such that$2^{k}+3^{k}$is a perfect square? 2. Let$A B C$be a triangle and let$M$be the midpoint of$B C$such that$A B=5cm$,$ A M=6cm$and$A C=13cm$. The line through$B$and perpendicular to$B C$meets$A M$at$D,$the line through$C$and perpendicular to$B C$meets$A B$at$E$. Prove that$C D$is perpendicular to$M E$3. Find all pair of real numbers$a$and$b$so that$a+b=\dfrac{\sqrt{8}}{2}$and$A=a^{4}-6 a^{2} b^{2}+b^{4}$is a positive integer. 4. Let$O$be the midpoint of a line segment$A B=2 a$. In the half-plane with edge$A B$, draw two rays$A x$,$B y$, both perpendicular to$A B$. Choose$M$and$N$on$A x$and$B y$respectively such that$M N=A M+B N$. Let$H$be the foot of the altitude from$O$onto$M N$. Find the positions of$M$and$N$such that the area of the triangle$H A B$is greatest possible. 5. Without using trigonometry formula, prove the following equalities a)$\cos 36^{\circ} \cdot \cos 72^{\circ}=\dfrac{1}{4}$. b)$\tan 36^{\circ} \cdot \tan 72^{\circ}=\sqrt{5}$. 6. Solve the equation $$3 x^{4}-4 x^{3}=1-\sqrt{\left(1+x^{2}\right)^{3}}$$ 7. Does there exist a polynomial$P(x)$of degree$2010$such that$P\left(x^{2}-2010\right)$is divisible to$P(x)$?. 8. Let$Oxyz$be a right trihedral with right angle at$O$and$A$,$B$,$C$move on the sides$O x$,$O y$and$O z$respectively so that the area of the triangle$A B C$is a constant$S$. Let$S_{1}$,$S_{2}$,$S_{3}$be the areas of the triangles$O B C$,$OCA$,$OAB$respectively. Find the greatest value of the expression $$P=\frac{S_{1}}{S+2 S_{1}}+\frac{S_{2}}{S+2 S_{2}}+\frac{S_{3}}{S+2 S_{3}}$$ 9. Let$a, b, c$be positive real numbers. Prove that $$\min \left\{\frac{a b}{c^{2}}+\frac{b c}{a^{2}}+\frac{c a}{b^{2}} ; \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\right\} \geq \max \left\{\frac{a}{b}+\frac{b}{c}+\frac{c}{a} ; \frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right\}$$ 10. For a given positive integer$n$, how many$n$-digit natural numbers can be formed from five possible digits$1,2,3,4$and$5$so that an odd numbers of$1$and even numbers of$2$are used? 11. sequence$\left(x_{n}\right)$,$n=0,1, \ldots$is given by $$x_{0}=\alpha,\quad x_{n}=\sqrt{1+\frac{1}{x_{n}+1}},\, n=0,1, \ldots$$ where$\alpha$is greater than$1$. Determine$\displaystyle\lim_{n\to\infty} x_{n}$. 12. Let$A B C$be a triangle with the altitudes$A A^{\prime}$,$B B^{\prime}$,$C C^{\prime}$meet at$H$. Prove that $$\frac{H A}{H A^{\prime}}+\frac{H B}{H B^{\prime}}+\frac{H C}{H C^{\prime}}+6 \sqrt{3} \geq 6+\frac{a}{H A^{\prime}}+\frac{b}{H B^{\prime}}+\frac{c}{H C^{\prime}}$$ ##$type=three$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: 2010 Issue 391
2010 Issue 391
Mathematics & Youth