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

## $show=home 1. Denote$T(a)$the number of digits of the natural number$a$. If$T\left(5^{n}\right)-T\left(2^{n}\right)$is even, is$n$necessarily odd or even? 2. A triangle$A B C$has$\widehat{B A C}<90^{\circ}$and$H A=B C$where$H$is its orthocenter. Find the measure of angle$B A C$. 3. Find all pair of integers$x, y$such that $$7^{x}+24^{x}=y^{2}$$ 4. Let$x, y, z$be arbitrary positive real numbers, prove the inequality $$\frac{x^{2}-z^{2}}{y+z}+\frac{z^{2}-y^{2}}{x+y}+\frac{y^{2}-x^{2}}{z+x} \geq 0.$$ When does equality occur? 5. From point$M$outside circle$(O),$draw tangent$M A$and secant$M B C$($B$is between$M$and$C$). Let$H$be the projection of$A$onto$M O, K$is the intersection of segment$M O$with$(O)$. Prove that a) The quadrilateral$O H B C$is cyclic. b)$B K$is the internal angle-bisector of angle$H B M$6. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}} &=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3 \end{cases}$$ 7. Let$\left(x_{n}\right)$be a sequence defined by $$3 x_{n+1}=x_{n}^{3}-2,\,n=1,2, \ldots.$$ For what values of$x_{1}$does the sequence$\left(x_{n}\right)$converge? Determine this limit when it converges. 8.$S . A B C$is a tetrahedron with isosceles perpendicular to plane$(A B C)$.$D$is the midpoint of$B C .$Let$\alpha$is the angle between edge$S B$and plane$(A B C)$;$\beta$is the angle between edge$S B$and plane$(S A D)$. Prove that $$\cos ^{2} \alpha+\cos ^{2} \beta>1$$ 9. Let$x, y, z$be positive real numbers such that$x^{2}+y^{2}+z^{2}+2 x y z=1 .$Prove the inequality $$8(x+y+z)^{3} \leq 10\left(x^{3}+y^{3}+z^{3}\right) + 11(x+y+z)(1+4 x y z)-12 x y z.$$ 10. Let$p$be an odd prime,$n$is a positive integer so that$(n, p)=1$. Find the number of tuples$\left(a_{1}, a_{2}, \ldots, a_{p-1}\right)$such that the sum$\displaystyle\sum_{k=1}^{p-1} k a_{k}$is divisible by$p,$and$a_{1}, a_{2}, \ldots, a_{p-1}$are natural numbers which do not exceed$n-1$. 11. Find all the functions$f$which is defined on$\mathbb{R},$take value on$\mathbb{R}$and satisfying the equation$f(x+y+f(y))=f(f(x))+2 y,$for all real numbers$x$,$y$. 12. Let$p, r, r_{a}, r_{b}, r_{c}$be semiperimeter, inradius, and exradius opposite angles$A$,$B$,$C$of triangle$A B C$having side lengths$B C=a$,$C A=b$,$A B=c$. Prove the inequality $$\sqrt{a b}+\sqrt{b c}+\sqrt{c a} \geq p+\sqrt{r r_{a}}+\sqrt{r r_{b}}+\sqrt{r r_{c}}$$ When does equality hold? ##$type=three$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: 2011 Issue 407
2011 Issue 407
Mathematics & Youth