2010 Issue 394

  1. Find all pair of natural numbers $x, y$ such that $$\left(2^{x}+1\right)\left(2^{x}+2\right)\left(2^{x}+3\right)\left(2^{x}+4\right)-5^{y}=11879$$
  2. Let $n$ be a positive integer and let $U(n)=\left\{d_{1} ; d_{2} ; \ldots ; d_{m}\right\}$ be the set of all positive divisors of $n$. Prove that $$d_{1}^{2}+d_{2}^{2}+\ldots+d_{m}^{2} \leq n^{2} \sqrt{n}$$
  3. Prove that $$\frac{1}{a^{4}(a+b)}+\frac{1}{b^{4}(b+c)}+\frac{1}{c^{4}(c+a)} \geq \frac{3}{2}$$ where $a$, $b$, $c$ are three positive numbers satisfying $a b c=1$.
  4. Solve the equation $$3 \sqrt{x^{3}+8}=2 x^{2}-6 x+4$$
  5. Let $A B C D$ be a square, $M$ is a point lying on $C D$ ($M \neq C$, $M \neq D$).  Through the point $C$ draw a line perpendicular to $A M$ at $H$ $B H$ meets $A C$ at $K$. Prove that a) $M K$ is always parallel to a fixed line when $M$ moves on the side $C D$.
    b) The circumcenter of the quadrilateral $ADMK$ lies on a fixed line.
  6. Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that $$\frac{1}{\sqrt{a^{3}+2 b^{3}+6}}+\frac{1}{\sqrt{b^{3}+2 c^{3}+6}}+\frac{1}{\sqrt{c^{3}+2 a^{3}+6}} \leq 1$$
  7. Consider all triangles $A B C$ where $A<B<C \leq \frac{\pi}{2}$. Find the least value of the expression $$M=\cot ^{2} A+\cot ^{2} B+\cot ^{2} C +2(\cot A-\cot B)(\cot B-\cot C)(\cot C-\cot A)$$
  8. Let $A B C$ be a triangle with $B C=a$, $A C=b$, $A B=c$. A line $d$ passing through its incenter meets $A B$, $A C$, $B C$ respectively at $M$, $N$, $P$. Prove that $$\frac{a}{\overline{B P} \cdot \overline{P C}}+\frac{b}{C N \cdot N A}+\frac{c}{\overline{A M} \cdot \overline{M B}}=\frac{(a+b+c)^{2}}{a b c}$$
  9. Let $x, y, z$ be non-zero real numbers such that $x+2 y+3 z=5$ and $2 x y+6 y z+3 x z=8$. Prove that $$1 \leq x \leq \frac{7}{3} ; \frac{1}{2} \leq y \leq \frac{7}{6} ; \frac{1}{3} \leq z \leq \frac{7}{9}$$
  10. Solve the system of equations $$\begin{cases} \sqrt[3]{x}+\sqrt[3]{y} &=\sqrt[3]{3(x+y)} \\ 4 x^{3}+6 x^{2}+4 x+1 &=15 y^{4}\end{cases}$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(f(x)+y)=f(x+y)+x f(y)-x y-x+1$$
  12. Suppose that the tetrahedron $A B C D$ satisfies the following conditions: All faces are acute triangles and $B C$ is perpendicular to $AD$. Let $h_{a}$, $h_{d}$ be respectively the lengths of the altitudes from $A$, $D$ onto the opposite faces, and let $2 \alpha\left(0^{\circ}<\alpha<45^{\circ}\right)$ be the measure of the dihedral angle at edge $B C$, $d$ is the distance between $B C$ and $A D$. Prove the inequality $$\frac{1}{h_{a}}+\frac{1}{h_{d}} \leq \frac{1}{d \cdot \sin \alpha}$$




Mathematics & Youth: 2010 Issue 394
2010 Issue 394
Mathematics & Youth
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