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

## $show=home 1. Prove that for any natural number$n>4$there exists a pair of natural numbers$x, y$with$\dfrac{n}{2} \leq x<n$and$\dfrac{n}{2} \leq y<n,$such that$x^{2}-y$is divisible by$n$2. Let$A B C$be an isosceles right triangle with right angle at$A$. On the ray$A C$choose two points$E$and$F$such that$\widehat{A B E}=15^{\circ}$and$C E=C F$. What is the measure of the angle$C B F$?. 3. Find all positive integer solutions of the equation $$65\left(x^{3} y^{3}+x^{2}+y^{2}\right)=81\left(x y^{3}+1\right).$$ 4. Solve the system of equations $$\begin{cases} 9 x^{2}+9 x y+5 x-4 y+9 \sqrt{y} &=7 \\ \sqrt{x-y+2}+1 &=9(x-y)^{2}+\sqrt{7 x-7 y} \end{cases}.$$ 5. Let$A B C$be an isosceles triangle where the angle$B A C$is obtuse. Suppose$D$is a point on edge$A B$such that$B C=C D \sqrt{2}$. The line through$D$and perpendicular to$A B$meets$C A$at$E$. Prove that$C D$passes through the midpoint of$B E$. 6. Let $$S=\sqrt{2}+\sqrt{\frac{3}{2}}+\sqrt{\frac{4}{3}}+\sqrt{\frac{5}{4}}+\ldots+2013 \sqrt{\frac{2013}{2012}}.$$ Find the integer part of$S$. 7. Given that$a, b, c$are edge lengths of a triangle. Prove the following inequality $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \leq \frac{3 \sqrt{3} a b c}{(a+b+c) \sqrt{a+b+c}}.$$ 8. Let$A B C D$be a trirectangular tetrahedron where the edges$A B$,$A C$,$A D$are pairwise perpendicular.$M$is an arbitrary point in the space. Given that$A B=4$,$A C=8$,$A D=12$. Find the minimum value of the expression $$P=\sqrt{7} M A+\sqrt{11} M B+\sqrt{23} M C+\sqrt{43} M D.$$ 9. Let$\left(a_{n}\right)$be a sequence of positive integers where $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=4 a_{n+1}+a_{n},\,\forall n \geq 1 .$$ Prove that a)$a_{n} a_{n+2}+(-1)^{n} .5$is a perfect square for all$n \geq 1$. b) The equation$x^{2}-4 x y-y^{2}=5$has infinitely many positive integer solutions. 10. Let$a$be a real number from$(0,1)$and$b$is a complex number,$|b|<1$. Prove that $$|b|+\left|\frac{a-b}{1-a b}\right| \geq a.$$ 11. Let$0<\alpha<\dfrac{\pi}{2}$. Prove that $$(\cot \alpha)^{\cos 2 \alpha} \geq \frac{1}{\sin 2 \alpha}$$ 12. A tetrahedron$A B C D$is inscribed in a sphere centered at$O$. Point$G$does not belong to planes$(B C D)$,$(C D A)$,$(D A B)$,$(A B C)$and the sphere$(O)$.$X$,$Y$,$Z$,$T$are the centers of the circumscribed spheres of the tetrahedron$G B C D$,$G C D A$,$G D A B$,$G A B C$respectively. Prove that$G$is the centroid of tetrahedron$A B C D$if and only if$O$is the centroid of tetrahedron$X Y Z T$. ##$type=three$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: 2012 Issue 426
2012 Issue 426
Mathematics & Youth