2012 Issue 424

  1. Find all $2$-digit numbers such that when multiplied by $2,3,4,$ $5,6,7,8,9,$ the sum of the digits of the resulting numbers are equal.
  2. Let $$S=\frac{2}{2013+1}+\frac{2^{2}}{2012^{2}+1}+\frac{2^{3}}{2013^{2^{2}}+1}+\ldots+\frac{2^{2014}}{2013^{2^{2013}}+1}.$$ Which number is greater? $S$ or $\dfrac{1}{1006}$?.
  3. Find all integer solutions of the equation $$(y-2) x^{2}+\left(y^{2}-6 y+8\right) x=y^{2}-5 y+62$$
  4. Let $x$, $y$ be two rational numbers such that $$x^{2}+y^{2}+\left(\frac{x y+1}{x+y}\right)^{2}=2 .$$ Prove that $\sqrt{1+x y}$. is also a rational number.
  5. Let $O$ denote the point of intersection of the two diagonals $A C$ and $B D$ of a convex quadrilateral $A B C D$. Let $E$, $F$, $H$ be the feet of the altitudes from $B$, $C$ and $O$ respectively onto $A D$. Prove that $$ A D \cdot B E \cdot C F \leq A C \cdot B D \cdot O H.$$ When does equality holds?
  6. $a, b, c$ are positive real numbers satisfying $a b c=1$. Prove that $$\frac{a^{3}+5}{a^{3}(b+c)}+\frac{b^{3}+5}{b^{3}(c+a)}+\frac{c^{3}+5}{c^{3}(a+b)} \geq 9$$
  7. Solve the equation $$\left(x^{3}+\frac{1}{x^{3}}+1\right)^{4}=3\left(x^{4}+\frac{1}{x^{4}}+1\right)^{3}$$
  8. Let $A B C$ be a triangle with acute angle $A$. Point $P$ inside the triangle $A B C$ such that $\widehat{B A P}=\widehat{A C P}$ and $\widehat{C A P}=\widehat{A B P}$. Let $M$ and $N$ be the incenters of triangles $A B P$ and $A C P$ respectively, $R$ is the circumradius of triangle $A M N$. Prove that $$\frac{1}{R}=\frac{1}{A B}+\frac{1}{A C}+\frac{1}{A P}.$$
  9. Solve the equation $$[x]^{3}+2 x^{2}=x^{3}+2[x]^{2}$$ where $[t]$ denotes the largest integer not exceeding $t$.
  10. In the interior of a unit square, there are $n\left(n \in \mathbb{N}^{*}\right)$ circles whose sum of areas is greater than $n-1$. Prove that the circles has at least a common point of intersection.
  11. Given that the following equation $$a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}=0 $$ has $n$ distinct roots. Prove that $$\frac{n-1}{n}>\frac{2 a_{0} a_{2}}{a_{1}^{2}}.$$
  12. Let $O$, $I$ and $I_{a}$ denote the circumcenter, incenter and excenter in the angle $A$ of a triangle $A B C$. $A I$ meets $B C$ at $D$. BI meets $C A$ at $E$. The line through $I$ and perpendicular to $O I_{a}$ intersects $A C$ at $M$. Prove that $D E$ passes through the midpoint of line segment $I M$.




Mathematics & Youth: 2012 Issue 424
2012 Issue 424
Mathematics & Youth
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