2014 Issue 443

  1. $21$ distinct integers are chosen so that the sum of any subset of $11$ numbers among them is always greater than the sum of the remaining $10$. If one of them is $101$, and the largest number is $2014$, find the other $19$ numbers.
  2. In a triangle $ABC$ where $\widehat{BAC}=40^{0}$ and $\widehat{ABC}=60^{0}$, point $D$ and $E$ are chosen on the sides $AC$ and $AB$ respectively such that $\widehat{CBD}=40^{0}$ and $\widehat{BCE}=70^{0}$. $BD$ and $CE$ intersect at point $F$. Prove that $AF$ is perpendicular to $BC$.
  3. Solve the following system of equations \[\begin{cases} 2\sqrt{2x}-\sqrt{y} & =1\\ \sqrt[3]{8x^{3}+y^{3}} & =\sqrt[3]{2}(\sqrt{x}+\sqrt{y}-1) \end{cases}.\]
  4. In a triangle $ABC$, points $E,D$ on the sides $AB$ and $AC$ respectively such that $\widehat{ABD}=\widehat{ACE}$. The circumcircle of triangle $ADB$ meets $CE$ at $M$ and $N$. The circumcircle of triangle $AEC$ meets $BD$ at $I$ and $K$. Prove that the points $M,I,N,K$ lie on a circle.
  5. Prove that for all positive rel numbers $a,b,c$ the following inequality holds \[\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\geq\frac{\sqrt{2}}{4}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}).\]
  6. Determine all real solutions of the equation \[(x^{5}+x-1)^{5}+x^{5}=2.\]
  7. Let $M$ be a point inside a given triangle $ABC$ and let $x,y,z$ denote the distance from $M$ onto $BC,CA,AB$ respectively. Prove that $\widehat{BAM}=\widehat{CBM}=\widehat{ACM}$ if and only if \[\frac{bx}{c}=\frac{cy}{a}=\frac{az}{b}\] where $BC=a$, $CA=b$, $AB=c$.
  8. Let $x,y,z$ be theree arbitrary numbers from the interval $[0,1]$. Determine the maximum value of $P$, where \[P=\frac{x}{y+z+1}+\frac{y}{z+x+1}+\frac{z}{z+y+1}+(1-x)(1-y)(1-z).\]




Mathematics & Youth: 2014 Issue 443
2014 Issue 443
Mathematics & Youth
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