# Problems in Mathematics and Youth Magazine, 2007 , Issue 9

For Lower Secondary Schools

1. The first $100$ positive integer numbers are written consecutively in a certain order. Call the resulting number $A$. Is $A$ a multiple of $2007$?

2. Let $ABC$ be a nonisosceles triangle, where $AB$ is the shortest side. Choose a point $D$ in the opposite ray of $BA$ such that $BD=BC$. Prove that $\angle ACD<90^\circ$.

3. Let $a,b,c$ be positive reals such that $a+b+c=1$. Prove that $\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3.$

4.  Solve the equation $(x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16$ in $\mathbb{R}$.

5. Let $AH$ denote the altitude of a right triangle $ABC$, right angle at $A$ and suppose that $AH^2=4AM\cdot AN$, where $M,N$ are the feet of the altitude from $H$ to $AB$ and $AC$, respectively. Find the measures of the angles of triangle $ABC$.

For Upper Secondary Schools
1. Find all $(x,y)\in\mathbb{Z}^2$ such that $x^{2007}=y^{2007}-y^{1338}-y^{669}+2$.

2. Let $(x_n)$ be a sequence given by $x_1=5\; , x_{n+1}=x_n^2-2\forall n\geq 1.$
Calculate $\lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.$

3. Let $a,b,c$ and denote the three sides of a triangle $ABC$. Its altitudes are $h_a,h_b,h_c$ and the radius of its three escribed circles are $r_a,r_b,r_c$. Prove that $\dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.$

1. In a quadrilateral $ABCD$, where $AD=BC$ meets at $O$, and the angle bisector of the angles $DAB,CBA$ meets at $I$. Prove that the midpoints of $AB,CD,OI$ are colinear.
2. Prove that $\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \\ \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right)\forall a,b,c\in [1,+\infty).$
3. Find the number of binary strings of length $n(n>3)$ in which the substring $01$ occurs exactly twice.
4. Let $f:\mathbb{N}\to\mathbb{R}$ be a function such that $f(1)=\dfrac{2007}{6}$ and $\dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}.$ Find the limit $\lim_{n\to\infty} (2008+n)f(n)$.