2015 Issue 454

  1. Given $n$ numbers $a_{1},a_{2},\ldots,a_{n}$ and $n$ distinct primes $p_{1},p_{2},\ldots,p_{n}$ ($n\geq2$) satisfying \[ p_{1}|a_{1}-a_{2}|=p_{2}|a_{2}-a_{3}|=\ldots=p_{n}|a_{n}-a_{1}|.\] Prove that $a_{1}=a_{2}=\ldots=a_{n}$.
  2. Choose 100 different natural numbers so that each of them is less than or equal to 2015 and has remainder 10 when divided by 17. Prove that, among these ones, we can always choose three numbers of which the sum is less than or equal to 999.
  3. Prove that, for every positive integer $n$, the value of the expression \[\sqrt{\frac{(1^{4}+4)(2^{4}+4)\ldots(n^{4}+4)}{2}}\]is an irrational number.
  4. Given a triangle $ABC$ and $D$ is any point on the side $BC$ ($D$is different from $B$ and $C$). The perpendicular bisector of $BD$and $CD$ intersect $AB$ and $AC$ respectively at $M$ and $N$. Let $H$ be the orthogonal projection of $D$ on the line $MN$ and $E,F$ respectively the midpoints of $BD$ and $CD$. Prove that $\widehat{EHF}=\widehat{BAC}$.
  5. Given the equation $ax^{2}+bx+c=0$ where the coefficients $a,b,c$are integers and $a>0$. Suppose that the equation has two distinctpositive roots which are less than $1$. Find the smallese possible value for the coefficient $a$.
  6. Solve the following system of equations \[\begin{cases} \sqrt{x-\sqrt{y}} & =\sqrt{z}-1\\ \sqrt{y-\sqrt{z}} & =\sqrt{x}-1\\ \sqrt{z-\sqrt{x}} & =\sqrt{y}-1 \end{cases}.\]
  7. Let $P$ be a point on the plane containing a triangle $ABC$. Suppose that $A_{1}=BC\cap AP$, $B_{1}=AC\cap BP$, $C_{1}=AB\cap CP$, $A_{2}=BC\cap B_{1}C_{1}$, $B_{2}=AC\cap A_{1}C_{1}$, $C_{2}=AB\cap A_{1}B_{1}$; $A_{3}=B_{1}C_{1}\cap AP$, $B_{3}=BP\cap A_{1}C_{1}$, $C_{3}=A_{1}B_{1}\cap CP$. Prove that $A_{2}$, $B_{2}$ and $C_{2}$ respectively are on the lines going through $B_{3}C_{3}$, $A_{3}C_{3}$ and $A_{3}B_{3}$.
  8. Given three positive numbers $a,b,c$. Prove that \[\left(\frac{a}{a+b}\right)^{2}+\left(\frac{b}{b+c}\right)^{2}+\left(\frac{c}{c+a}\right)^{2}+3\] \[\geq\frac{5}{2}\left(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\right).\]
  9. Let $d$ be a line in the coordinate plane with the equation $y=\frac{3}{2}x+\frac{1}{3}$. Let $a_{1}$ and $a_{2}$ be two distinct lines which are parallel to $d$. Suppose furthermore that the distances from $a_{1}$ and $a_{2}$ to $d$ both are equal to $\frac{1}{12}$. Is there any integer point, i.e. point with both coordinates are integers, between or on two lines $a_{1}$ and $a_{2}$?.
  10. Prove that, for each positive integer $n$, the equation $2014^{x}+nx=2013$ has a unique solution, say $x_{n}$. Find $\lim_{n\to\infty}x_{n}$.
  11. Find all continuous funtions $f:\mathbb{R}\to\mathbb{R}$ which satisfy \[ (x+y)f(x+y)=xf(x)+yf(y)+2xy,\quad\forall x,y\in\mathbb{R}.\]
  12. Given a triangle $ABC$. A point $M$ varies on the side $BC$. Let $(I_{1})$ and $(I_{2})$ be the inscribed circles of the triangles $ABM$ and $ACM$ respectively. A common tangent line $XY$ of $(I_{1})$ and $(I_{2})$, which is different from $BC$, intersects $AM$ at $N$ ($X\in(I_{1})$ and $Y\in(I_{2})$). Let $Z$ and $T$ respectively be the tangent points between $AM$ and $(I_{1})$, $(I_{2})$. $XT$ cuts $YZ$ at $K$. Prove that $NK$ always goes through a fixed point.




Mathematics & Youth: 2015 Issue 454
2015 Issue 454
Mathematics & Youth
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