2011 Issue 407

  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?




Mathematics & Youth: 2011 Issue 407
2011 Issue 407
Mathematics & Youth
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