2008 Issue 374

  1. Find all triple of natural numbers $a, b, c$ less than $20$ such that $$a(a+1)+b(b+1)=c(c+1)$$ where $a$ is a prime number and $b$ is a multiple of $3 .$
  2. Let $f(n)=\left(n^{2}+n+1\right)^{2}+1,$ where $n$ is a positive integer and let $$P_{n}=\frac{f(1) \cdot f(3) \cdot f(5) \ldots f(2 n-1)}{f(2) \cdot f(4) \cdot f(6) \ldots f(2 n)}.$$ Prove the inequality $$P_{1}+P_{2}+\ldots+P_{n}<\frac{1}{2}.$$
  3. Find the maximum value of the expression $$T=\frac{(y+z)^{2}}{y^{2}+z^{2}}-\frac{(x+z)^{2}}{x^{2}+z^{2}},$$ where $x$, $y$, $z$ are real numbers such that $x>y$, $z>0$ and $z^{2} \geq x y$.
  4. Solve for $x$ $$\sqrt[3]{14-x^{3}}+x=2\left(1+\sqrt{x^{2}-2 x-1}\right).$$
  5. An acute triangle $A B C$ is inscribed in a fixed circle with center at $O$. Let $A I$, $B D$ and $C E$ denote the altitudes through $A$, $B$ and $C$ respectively. Prove that the perimeter of the triangle $I D E$ does not change when $A, B,$ and $C$ move on the circle $(O)$ such that the area of the triangle $A B C$ is always equal to $a^{2}$.
  6. Solve the system of equations $$\begin{cases} \sqrt{x^{2}+91} &=\sqrt{y-2}+y^{2} \\ \sqrt{y^{2}+91} &=\sqrt{x-2}+x^{2}\end{cases}.$$
  7. Let $a, b, c$ be real numbers such that $4(a+b+c)-9=0 .$ Find the maximum value of the sum $$S=\left(a+\sqrt{a^{2}+1}\right)^{b}\left(b+\sqrt{b^{2}+1}\right)^{c}\left(c+\sqrt{c^{2}+1}\right)^{a}.$$
  8. Let $I$ and $O$ denote respectively the incenter and the circumcenter of a triangle $A B C .$ Given that $\widehat{A I O}=90^{\circ},$ prove that the area of the triangle $A B C$ is less than $\dfrac{3 \sqrt{3}}{4} A I^{2}$.
  9. Let $a, b, n$ be positive integers, $b>1$ and $a$ is a multiple of $b^{n}-1 .$ Rewritten $a$ to the base $b,$ prove that the resulting contains at least $n$ non-zero digits.
  10. Let $x, y, z$ be real numbers such that $0<z \leq y \leq x \leq 8$ and $$3 x+4 y \geq \max \left\{x y ; \frac{1}{2} x y z-8 z\right\}.$$ Find the maximum value of $$A=x^{5}+y^{5}+z^{5}.$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(f(x)+y^{2}\right)=f^{2}(x)-f(x) f(y)+x y+x.$$
  12. Let $K$ denote the intersection of the two diagonals of a quadrilateral $A B C D$ where $\widehat{A B C}=\widehat{A D C}=90^{\circ} ;$ and $A C=A B+A D .$ Prove that the radii of the inscribed of the triangles $A B K$ and $A D K$ are equal.




Mathematics & Youth: 2008 Issue 374
2008 Issue 374
Mathematics & Youth
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