2015 Issue 456

  1. Find all finite sets of primes such that for each set, the product of its elements is 10 times the sum of its element.
  2. Let $ABC$ be an isosceles triangle with the vertex angle $\widehat{BAC}=80^{0}$. Choose $D$ and $E$ on the sides $BC$ and $CA$ respectively such that $\widehat{BAD}=\widehat{ABE}=30^{0}$. Find the angle $\widehat{BED}$.
  3. Solve the equation \[\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{2x-1}}=\sqrt{5}\left(\frac{1}{\sqrt{6x-1}}+\frac{1}{\sqrt{9x-4}}\right).\]
  4. Let $ABCD$ be a square and let $a$ be the length each side. On the sides $AB$ and $BC$, choose $M$ and $N$ respectively such that $\widehat{MDN}=45^{0}$. Find the positions of $M$ and $N$ so that the length $MN$ is minimal.
  5. Find all positive integers $x$ and $y$ such that \[x^{4}+y^{2}+13y+1\leq(y-2)x^{2}+8xy.\]
  6. Solve the following system of equations \[ \begin{cases} x+y+z & =3\\ x^{2}y+y^{2}z+z^{2}x & =4\\ x^{2}+y^{2}+z^{2} & =5 \end{cases}.\]
  7. Given a quadrilateral pyramid $S.ABCD$ with the following properties: the base $ABCD$ is a rectangle and $SA$ is perpendicular to the plane $(ABCD)$. Suppose that $G$ is the centroid of the triangle $SBC$ and let $d$ be the distance from $G$ to the plane $(SBD)$. Let $SB=a$, $BD=b$ and $SD=c$. Prove that \[a^{2}+b^{2}+c^{2}\geq162d^{2}.\]
  8. Prove that the following equation \[(x+1)^{\frac{1}{x+1}}=x^{\frac{1}{x}}\] has a unique solution.
  9. Given positive integers $a_{1},a_{2},\ldots a_{15}$ satisfying
    a) $a_{1}<a_{2}<\ldots<a_{15}$,
    b) for each $k$ ($k=1,\ldots,15$), if we denote $b_{k}$ the largest divisor of $a_{k}$ such that $b_{k}<a_{k}$, then $b_{1}>b_{2}>\ldots>b_{15}$.
    Prove that $a_{15}>2015$.
  10. Given the following polynomial \[f(x)=x^{3}+3x^{2}+6x+1975.\] In the interval $[1,3^{2015}]$, how many are there integers $a$ such that $f(a)$ is divisible by $3^{2015}$?.
  11. Find all injections $f:\mathbb{R\to\mathbb{R}}$ satisfying $$ f(x^{5})+f(y^{5}) = (x+y)[f^{4}(x)-f^{3}(x)f(y)+f^{2}(x)f^{2}(y)-f(x)f^{3}(y)+f^{4}(y)]$$ for all $x,y\in\mathbb{R}$.
  12. Given a triangle $ABC$ and let $G$ be its centroid. Choose a point $M$, which is different from $G$, inside the triangle. Suppose that $AM$, $BM$, and $CM$ intersect $BC$, $CA$ and $AB$ at $A_{0},B_{0},C_{0}$ respectively. Choose $A_{1},A_{2}$ on $B_{0}C_{0}$ such that $A_{0}A_{1}\parallel CA$ and $A_{0}A_{2}\parallel AB$. We choose four points $B_{1},B_{2},C_{1},C_{2}$ similarly. Let $G_{1},G_{2}$ be the centroids of the triangles $A_{1}B_{1}C_{1}$, $A_{2}B_{2}C_{2}$ respectively. Prove that
    a) $A_{1}B_{2}\parallel B_{1}C_{2}\parallel C_{1}A_{2}$,
    b) $MG$ goes through the midpoint of $G_{1}G_{2}$.




Mathematics & Youth: 2015 Issue 456
2015 Issue 456
Mathematics & Youth
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