2007 Issue 355

  1. Let $a, b$ be two natural numbers satisfying $$2006 a^{2}+a=2007 b^{2}+b.$$ Prove that $a-b$ is a perfect square.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}, \widehat{A B C}$ $=60^{\circ} .$ Take the point $M$ on the side $B C$ such that $A B+B M=A C+C M$. Caculate the measure of $\widehat{C A M}$
  3. Find all positive integers $x, y$ greater than 1 so that $2 x y-1$ divisible by $(x-1)(y-1)$.
  4. Prove that $$\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a \leq 3 a b c$. When does equality occur?
  5. Let be given two circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ with centers $O_{1}$, $O_{2}$ with distinct radii, externally touching each other at a point $T$. Let $O_{1} A$ be a tangent to $\left(O_{2}\right)$ at a point $A,$ let $O_{2} B$ be a tangent to $\left(O_{1}\right)$ at a point $B$ so that the points $A, B$ are on the same side with respect to the line $O_{1} O_{2} .$ Let $H$ be the point on $O_{1} A, K$ be the point on $O_{2} B$ so that the lines $B H, A K$ are perpendicular to $O_{1} O_{2}$. The line $T H$ cuts $\left(O_{1}\right)$ again at $E,$ the line $T K$ cuts $\left(O_{2}\right)$ againt at $F$. The line $E F$ cuts $A B$ at $S$. Prove that the lines $O_{1} A$, $O_{2} B$ and $T S$ are concurrent.
  6. Let $S$ be a set consisting of 43 distinct positive integers not exceeding $100 .$ For each subset $X$ of $S$ let $t_{X}$ be the product of its elements. Prove that there exist two disjoint substs $A$ and $B$ of $S$ such that $t_{A} t_{B}^{2}$ is the cube of a natural numbers.
  7. Find the greast value of the expression $$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$$ where $a, b, c d$ are distinct real numbers satisfying the conditions $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $a c=b d$
  8. a) Let $f(x)$ be a polynomial of degree $n$ with leading coefficient $a$. Suppose that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n}$ all not equal to zero. Prove that $$\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$$ b) Does there exist a polynomial $f(x)$ of degree $n,$ with leading coefficient $a=1,$ such that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n},$ all not equal to zero, satisfying the condition $$\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$$
  9. Let $A D$, $B E$, $C F$ be the altitudes and $H$ be the orthocenter of an acute triangle $A B C .$ Let $M$, $N$ be respectively the points of intersection of $D E$ and $C F$ and of $D E$ and $B E$. Prove that the line passing through $A$ perpendicular to the line $M N$ passes through the circumcenter of triangle $B H C$.




Mathematics & Youth: 2007 Issue 355
2007 Issue 355
Mathematics & Youth
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