2007 Issue 365

  1. Write $2005^{2006}$ as a sum of natural numbers, then calculate the sum of all the digits occurred in these summands. Can one obtain either $2006$ or $2007$ in this way? Why?
  2. Let $A B C$ be an isosceles right triangle, right angle at $A$. Choose a point $D$ in the half-plane on the side of $A B$ that does not contain $C$ such that $D A B$ is also an isosceles right triangle, with right angle at $D$. Let $E$ be a point (differs from $A$) on $A D .$ The perpendicular line with $B E$ through $E$ intersects with $A C$ at $F .$ Prove that $E F=E B$.
  3. Find the maximum and minimum values of the following expression $$\frac{x-y}{x^{4}+y^{4}+6}.$$
  4. Solve the equation $$\sqrt{2}\left(x^{2}+8\right)=5 \sqrt{x^{3}+8}.$$
  5. The two diagonals $A C$ and $B D$ of an inscribed quadrilateral $A B C D$ intersect at $O$. The circumcircles $\left(S_{1}\right)$ of $A B O$ and $\left(S_{2}\right)$ of $\mathrm{CDO}$ meet at $\mathrm{O}$ and $K$. Through $\mathrm{O},$ draw parallel lines with $A B$ and $C D$; they meet with $\left(S_{1}\right)$ and $\left(S_{2}\right)$ at $N$ and $M,$ respectively. Let $P$ and $Q$ be two points on $O N$ and $O M$ respectively such that $\dfrac{O P}{P N}=\dfrac{M Q}{Q}$. Prove that $O$, $K$, $P$ and $Q$ lie on the same circle.
  6. There are nine cards in a box, labelled from $1$ to $9$. How many cards should be taken from the box so that the probability of getting a card whose label is a multiple of 4 will be greater than $\dfrac{5}{6} ?$
  7. Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ arbitrary real numbers chosen in a given closed interval $[a ; b]$. Prove the inequality $$\sum_{i=1}^{n}\left(x_{i}-\frac{a+b}{2}\right)^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\frac{1}{n} \sum_{i=1}^{n} x_{i}\right)^{2}.$$ When does equality occur?
  8. Let $H$ be the orthocenter of an acute triangle $A B C$ whose edges are $B C=a, C A=b$, and $A B=c .$ Prove that $$\frac{H A^{2}+H B^{2}+H C^{2}}{a^{2}+b^{2}+c^{2}} \leq(\cot A \cdot \cot B)^{2}+(\cot B \cdot \cot C)^{2}+(\cot C \cdot \cot A)^{2}.$$
  9. Let $p$ be a prime number, greater than $3$ and $k=\left[\dfrac{2 p}{3}\right]$. (The notation $[x]$ denote the largest integer which is not exceeding $x$.) Prove that $\displaystyle \sum_{i=1}^{k} C_{p}^{i}$ is a multiple of $p^{2}$.
  10. Find the measures of the angles of triangle $A B C$, such that $$\cos \frac{5 A}{2}+\cos \frac{5 B}{2}+\cos \frac{5 C}{2}=\frac{3 \sqrt{3}}{2}.$$
  11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a boundedabove sequence such that $$x_{n+2} \geq \frac{1}{4} x_{n+1}+\frac{3}{4} x_{n},\,\forall n=1,2, \ldots$$ Prove that this sequence has limit.
  12. Let $\alpha, \alpha^{\prime} ; \beta, \beta^{\prime}$ and $\gamma, \gamma^{\prime}$ be the dihedral angles, opposite to the edges $B C$, $DA$, $CA$, $DB$ and $A B$, $D C$ respectively, of a tetrahedron $A B C D$. Prove the inequality $$\sin \alpha+\sin \alpha^{\prime}+\sin \beta+\sin \beta^{\prime}+\sin \gamma+\sin \gamma^{\prime} \leq 4 \sqrt{2}.$$ When does equality occur?




Mathematics & Youth: 2007 Issue 365
2007 Issue 365
Mathematics & Youth
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