2014 Issue 446

  1. Find all prome numbers $p,q,r$ satisfying \[(p+1)(q+2)(r+3)=4pqr.\]
  2. Given a triangle $ABC$ with $\widehat{A}=75^{0}$, $\widehat{B}=45^{0}$. On the side $AB$, choose a point $D$ such that $\widehat{ACD}=45^{0}$. Prove that $DA=2DB$.
  3. Solve the following system of equations \[\begin{cases} \sqrt{x+y+2}+x+y & =2(x^{2}+y^{2})\\ \frac{1}{x}+\frac{1}{y} & =\frac{1}{x^{2}}+\frac{1}{y^{2}} \end{cases}.\]
  4. Given a triangle $ABC$. Let $(I)$ be the inscribed circle and $(J)$ the escribed circle corresponding to the angle $A$. Suppose that $(J)$ is tangent to the lines $BC$, $CA$ and $AB$ at $D,E$ and $F$ respectively. The line $JD$ meets the line $EF$ at $N$. The line which contains $I$ and is perpendicular to the line $BC$ intersects the line $AN$ at $P$. Let $M$ be the midpoint of $BC$. Prove that $MN=MP$.
  5. Find all the integer solutions of the following equation \[x^{3}=4y^{3}+x^{2}y+y+13.\]
  6. Let $$f(x)=\frac{4^{x+2}}{4^{x}+2}.$$ Find \[f(0)+f\left(\frac{1}{2014}\right)+f\left(\frac{2}{2014}\right)+\ldots+f\left(\frac{2013}{2014}\right)+f(1).\]
  7. Given a tetrahedron $ABCD$. Let $d_{1},d_{2},d_{3}$ be the distances between the pairs of opposite sides $AB$ and $CD$, $AC$ and $BD$, $AD$ and $BC$. Prove that \[V_{ABCD}\geq\frac{1}{3}d_{1}d_{2}d_{3}.\]
  8. Given an integer $n$ which is greater than $1$. Let $a_{1},a_{2},\ldots,a_{n}$ be arbitrary positive real numbers satisfying \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1.\] Prove that \[a_{1}^{a_{2}}+a_{2}^{a_{3}}+\ldots+a_{n-1}^{a_{n}}+a_{1}+a_{2}+\ldots+a_{n}>n^{3}+n.\]
  9. Let $T$ be a set of $n$ elements. What is the maximal number of subsets of $T$ which can be picked so that each subset has exactly 3 elements and any two subsets has nonempty intersection?.
  10. Let $p$ be a prime number. Find all the polynomials $f(x)$ with integer coefficients such that for every positive integer $n$, $f(n)$ is a divisor of $p^{n}-1$.
  11. Let $x,y$ be the positive real numbers satisfying $[x]\cdot[y]=30^{4}$, where $[a]$ is the greatest integer not wxceeding $a$. Find the minimum and maximum values of \[P=[x[x]]+[y[y]].\]
  12. Given a triangle $ABC$. Let $E,F$ be points on $CA$, $AB$ respectively such that $EF\parallel BC$. The perpendicular bisector of $BC$ intersects $AC$ at $M$ and the perpendicular bisector of $EF$ intersects $AB$ at $N$. The circle circumscribing the triangle $BCM$ meets $CF$ at $P$ which is different from $C$. The circle circumscribing the triangle $EFN$ meets $CF$ at $Q$ which is different from $F$. Prove that the perpecdicular bisector of $PQ$ contains the midpoint of $MN$.




Mathematics & Youth: 2014 Issue 446
2014 Issue 446
Mathematics & Youth
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