2012 Issue 422

  1. Let $$A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2011}-\frac{1}{2012},\quad B=\frac{1}{1007}+\frac{1}{1008}+\ldots+\frac{1}{2012}.$$ Compute the value of $\left(\dfrac{A}{B}\right)^{2012}$.
  2. Let $f(x)$ be a polynomial with integer coefficients such that $f(3) \cdot f(4)=5 .$ Prove that $f(x)-6$ does not have any integer solution.
  3. Find all triple of integers $a, b, c$ such that $$2^{a}+8 b^{2}-3^{c}=283.$$
  4. Given a triangle $A B C$, $B C=a$, $C A=b$, $A B=c$, $\widehat{A B C}=45^{\circ}$ and $\widehat{A C B}=120^{\circ}$. Point $I$ is taken on the opposite ray of $C B$ such that $\widehat{A I B}=75^{\circ} .$ Find the length of $A I$ in term of $a$, $b$ and $c$
  5. Point $K$ lies on side $B C$ of a triangle $A B C$. Prove that $$A K^{2}=A B \cdot A C - K B \cdot K C$$ if and only if $A B=A C$ or $\widehat{B A K}=\widehat{C A K}$.
  6. A non-isosceles triangle $A B C$ has $B C=a$, $C A=b$, $A B=c$. Let $\left(A A_{1}, A A_{2}\right)$, $\left(B B_{1}, B B_{2}\right)$, $\left(C C_{1}, C C_{2}\right)$ be the median and the altitude from vertices $A$, $B$ and respectively. Prove that $$\frac{a^{2}}{b^{2}-c^{2}} \overline{A_{1} A_{2}}+\frac{b^{2}}{c^{2}-a^{2}} \overrightarrow{B_{1} B_{2}}+\frac{c^{2}}{a^{2}-b^{2}} \overrightarrow{C_{1} C_{2}}=\overrightarrow{0}$$
  7. Let $a, b, c \in(0 ; 1)$ and $$a b+b c+c a+a+b+c=1+a b c.$$ Prove that $$\frac{1+a}{1+a^{2}}+\frac{1+b}{1+b^{2}}+\frac{1+c}{1+c^{2}} \leq \frac{3}{4}(3+\sqrt{3})$$
  8. Let $A B C$ be an acute triangle with all angles greater than $45^{\circ}$. Prove that $$\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \leq 3(\sqrt{3}-1).$$ When does equality occur?
  9. Two sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are defined inductively as follows $$a_{0}=3, b_{0}=-3,\quad a_{n}=3 a_{n-1}+2 b_{n-1},\,b_{n}=4_{n-1}+3 b_{n-1},\,\forall n \geq 1.$$ Find all natural numbers $n$ such that $\displaystyle\prod_{k=0}^{n}\left(b_{k}^{2}+9\right)$ is a perfect square.
  10. Let $n$ be a positive integer. How many strings of length $n: a_{1} a_{2} \ldots a_{n}$ where $a_{i}$ is chosen from $\{0,1,2, \ldots, 9\}(i=1,2, \ldots, n)$ are there such that the number of occurrences of 0 is even?
  11. Let $\left(u_{n}\right)$ be a sequence defined by $u_{0}=a \in[0 ; 2), u_{n}=\dfrac{u_{n-1}^{2}-1}{n}$ for all $n=1,2,$ $3, \ldots$ Find $\displaystyle\lim _{n \rightarrow+\infty}\left(u_{n} \sqrt{n}\right)$.
  12. Let $A B C$ be a triangle, inscribed in the circle $(O)$ with altitudes $A D$, $B E$ and $C F$. $A A^{\prime}$ is a diameter of $(O)$. $A^{\prime} B$, $A^{\prime} C$ intersect $A C$, $A B$ at $M$, $N$ respectively. Points $P$, $Q$ are in $E F$, such that $P B$, $Q C$ are perpendicular to $B C$. The line passing through $A$ and orthogonal to $Q N$, $P M$ cuts $(O)$ at $X$, $Y$ respectively. The tangents to circle $(O)$ at $X$ and $Y$ meet at $J$. Prove that $J A^{\prime}$ is perpendicular to $B C$.




Mathematics & Youth: 2012 Issue 422
2012 Issue 422
Mathematics & Youth
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