2007 Issue 363

  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 in $\mathbb{R}$ $$(x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16$$
  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 $ABC$.
  6. Find all $(x,y)\in\mathbb{Z}^2$ such that $$x^{2007}=y^{2007}-y^{1338}-y^{669}+2.$$
  7. Let $(x_n)$ be a sequence given by $$x_1=5,\quad x_{n+1}=x_n^2-2,\,\forall n\geq 1.$$ Calculate $\displaystyle\lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.$
  8. 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 $$\dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.$$
  9. 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.
  10. Prove that for all $a,b,c\in [1,+\infty)$ we have $$\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right).$$
  11. Find the number of binary strings of length $n$ $(n>3)$ in which the substring $01$ occurs exactly twice.
  12. Let $f:\mathbb{N}\to\mathbb{R}$ be a function such that $$f(1)=\dfrac{2007}{6},\quad \dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}.$$ Find the limit $\displaystyle\lim_{n\to\infty} (2008+n)f(n)$.




Mathematics & Youth: 2007 Issue 363
2007 Issue 363
Mathematics & Youth
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