2017 Issue 482

  1. For each natural number $n$, find the last digit of \[S_{n}=1^{n}+2^{n}+3^{n}+4^{n}.\]
  2. Suppose that $ABC$ is an isosceles right triangle with $B$ is the right angle. Let $O$ be the midpoint of $AC$. Choose $J$ on the segment $OC$ such that $3AJ=5JC$. Let $N$ be the midpoint of $OB$. Prove that $AN\perp NJ$.
  3. Solve the system of equations \[\begin{cases} 2(y^{2}-x^{2}) & =\dfrac{14}{x}-\dfrac{13}{y}\\ 4(x^{2}+y^{2}) & =\dfrac{14}{x}+\dfrac{13}{y} \end{cases}.\]
  4. Given a triangle $ABC$ with $\widehat{BAC}=120^{0}$. Suppose that on the side $BC$ there exists a point $D$ such that $\widehat{BAD}=90^{0}$ and $AB=DC=1$. Find the length of $BD$.
  5. Given three positive numbers $a,b,c$ such that $a+b+c=\sqrt{6051}$. Find the maximum calue of the expression \[P=\frac{2a}{\sqrt{a^{2}+2017}}+\frac{2b}{\sqrt{b^{2}+2017}}+\frac{2c}{\sqrt{c^{2}+2017}}.\]
  6. Solve the equation \[\dfrac{1}{\sqrt[3]{x}}+\dfrac{1}{\sqrt[3]{3x+1}}=\dfrac{1}{\sqrt[3]{2x-1}}+\dfrac{1}{\sqrt[3]{2x+2}}\] assuming that $x>\dfrac{1}{2}$.
  7. Suppose that $p,q,r$ are three distinct integer roots of the equation $x^{3}+ax^{2}+bx+c$ where $a,b,c$ are integers and $16a+c=0$. Prove that \[\frac{p+4}{p-4}\cdot\frac{q+4}{q-4}\cdot\frac{r+4}{r-4}\] is an integer.
  8. Let $(O)$ be the circumcircle of the equilateral triangle $ABC$ whose sides are equal to $a$. On the arc $BC$ which does not contain $A$ choose an arbitrary point $P$ ($P\ne B$, $P\ne C$). Suppose that $AP$ intersect $BC$ at $Q$. Prove that \[PQ\leq\frac{a}{2\sqrt{3}}.\]
  9. Let $a,b,c$ be postive numbers. Prove that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq6.\]
  10. For each natural number $n>2$, let $u_{n}$ be the number of $0$'s at the end in case representation of $n!$. Find the maximum value of the expression \[P=\frac{u_{n}}{n-1}.\]
  11. Given a convex decafon $A_{1}A_{2}\ldots A_{9}A_{10}$. We color its sides and its diagonals by 5 different colors as follows
    a) Each side of each diagonal is colored by at most 1 color.
    b) The sides and the diagonals which are colored have no common vertex and do not intersect (notice that the sides and the diagonals here are line segments, not he whole lines passing through them).
    In how many ways can we color by the above rule?. 
  12. Given a right triangle $ABC$ with the right angle $A$. A circle $(I,r)$ is tangent to the line segments $AB$, $BC$ and $CA$ at $P,Q$ and $R$ respectively. Let $K$ be the midpoint of $AC$. The line $IK$intersects $AB$ at $M$. The line segment $PQ$ intersect the altitude $AH$ (of $ABC$) at $N$. Prove that $N$ is the orthocenter of $MQR$.




Mathematics & Youth: 2017 Issue 482
2017 Issue 482
Mathematics & Youth
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