Mathematics and Youth Magazine Problems - Jun 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 $$\begin{align*} & f(x^{5})+f(y^{5})\\ = & (x+y)[f^{4}(x)-f^{3}(x)f(y)+f^{2}(x)f^{2}(y)-f(x)f^{3}(y)+f^{4}(y)] \end{align*}$$ 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 $G_{1}G_{2}$.

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