# Mathematics and Youth Magazine Problems - November 2003, Issue 317

1. Calculate the following sums
a) $A=1.2+2.3+\ldots+n(n+1)+\ldots+98.99$.
b) $B=1.99+2.98+\ldots+n(100-n)+\ldots+98.2+99.1$
2. Find all pairs of rational numbers $x, y$ such that both numbers $x+y$ and $\dfrac{1}{x}+\dfrac{1}{y}$ are integers.
3. Find the remainder of the division of $13376^{2003 !}$ by $2000$, where $n !$ is the product of the $n$ integers from 1 to $n$.
4. Prove that $$\frac{1}{a^{2}+2 b^{2}+3}+\frac{1}{b^{2}+2 c^{2}+3}+\frac{1}{c^{2}+2 a^{2}+3} \leq \frac{1}{2}$$ where $a, b, c$ are real numbers satisfying the condition $a b c=1$. When does equality occur?
5. Let be given the real numbers $a, b, c$, $x, y,z$ satisfying the conditions $a x^{2003}=b y^{2003}=c z^{2003}$ and $x y+y z+z x=x y z \neq 0$. Prove that $$\sqrt[203]{a x^{2002}+b y^{2002}+c z^{2002}}=\sqrt[2003]{a}+\sqrt[2003]{b}+\sqrt[2003]{c}$$
6. Let be given a triangle $A B C$. Construct outside of $A B C$ the parallelograms $A B E F$ and $A C P Q$ so that $A F=A C$, $A Q=A B$. Let $D$ be the point of intersection of $B P$ and $C E$. The lines $Q D$ and $F D$ cut $B C$ respectively at $M$ and $N$. Calculate the ratio $M N: B C$.
7. A quadrilateral $A B C D$ with $A B>A C$ circumscribes about a circle with center $O$. Let $E$ and $F$ be the points of intersection of $B D$ with the circle. The line $OH$ passing through $O$ cuts orthogonally $A C$ at $H$. Prove that $\widehat{B H E}=\widehat{D H F}$.
8. Let be given positive integers $m, n, k$ with $n>m$. Prove that the number of positive integral solutions of the system of equations $$\begin{cases}x_{1}+x_{2}+\ldots+x_{n} &=y_{1}+y_{2}+\ldots+y_{m}+1 \\ x_{1}+x_{2}+\ldots+x_{n} &\leq n k\end{cases}$$ is equal to $\displaystyle\sum_{i=0}^{n(k-1)} C_{n-1+i}^{n-1} \cdot C_{n-2+i}^{m-1}$.
9. For every positive integer $k$, consider the sequence of numbers $\left(x_{n}^{k}\right)$ $(n=1,2, \ldots)$ defined by $$x_{1}^{k}=1,\quad x_{n}^{k}=\sum_{i=1}^{n} \frac{i^{k}}{i !},\,\forall n=2,3, \ldots$$ a) Prove that the sequence $\left(x_{n}^{k}\right)$ $(n=1,2, \ldots)$ has a finite limit for every positive integer $k$.
b) Put $\displaystyle E_{k}=\lim _{n \rightarrow \infty} x_{n}^{k}$. Prove that $y_{k}=E_{k} / E_{1}$ is a positive integer for every positive integer $k$.
10. Let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers distinct from $0$ and $u_{1}, u_{2}, \ldots, u_{n}$ be positive real numbers such that $u_{1}<u_{2}<\ldots<u_{n}$, and let $f(x)=\sum_{i=1}^{n} a_{i} \cos \left(u_{i} x\right)$ be a periodic function defined on $R$. Prove that $u_{i} / u_{1}$ is a rational number for every $i=2,3, \ldots, n$.
11. Let be given a convex polygon $A_{1} A_{2} \ldots A_{n}$ $(n \geq 3)$ and $M$ be a point inside the polygon. Let $\alpha_{i}=M A_{i} A_{i+1}$ $\left(i=1,2, \ldots, n\right)$; $A_{n+1}$ is considered as $\left(A_{1}\right)$. Prove that $$\min_{1 \leq i \leq n}\left\{\alpha_{i}\right\} \leq \frac{(n-2) \pi}{2 n}$$
12. The sides $O A$, $O B$, $O C$ of a tetrahedron $O A B C$ are orthogonal each to the others. The incircle of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Let $\alpha$, $\beta$, $\gamma$ be respectively the measures of the dihedral angles with sides $B C$, $C A$, $A B$ of the tetrahedron. Prove that
a) $V_{O D E F}=\dfrac{1}{3} O A \cdot O B \cdot O C \cdot \sin \dfrac{A}{2} \cdot \sin \dfrac{B}{2} \cdot \sin \dfrac{C}{2}$ where $A$, $B$, $C$ denote the angles of triangle $A B C$.
b) $\tan \alpha \cdot \tan \beta \cdot \tan \gamma \geq 2 \sqrt{2}$. When does equality occur?
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