2015 Issue 453

  1. Find all natural numbers $x,y,z$ such that $x!+y!+z!=\overline{xyz}$ and $\overline{xyz}$ is a three-digit number. Recall that $n!=1.2.3\ldots n$ and $0!=1$.
  2. Given a triangle $ABC$ with $\widehat{BAC}=50^{0}$, $\widehat{ABC}=60^{0}$. On the sides $AB$ and $BC$, choose $D$ and $E$ respectively such that $\widehat{ACD}=\widehat{CAE}=30^{0}$. Find the angle $\widehat{CDE}$.
  3. Given three positive numbers $a,b,c$. Find the maximum value of the expression \[T=\frac{a+b+c}{(4a^{2}+2b^{2}+1)(4c^{2}+3)}.\]
  4. Suppose that $ABC$ is an acute triangle. Its altitudes $AD$, $BE$ and $CF$ are concurrent at the point $H$. Let $K$ be a point on the side $DC$ and choose $S$ on $HK$ such that $AS\perp HK$. Let $I$ be the intersection between $EF$ and $AH$. Prove that $SH$ is the angle bisector of the angle $\widehat{DSI}$.
  5. Let $a,b,c$ be positive numbers satisfying \[a^{4}+b^{4}+c^{4}\leq2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).\] Prove that at lease one of the following equations has no solution \[ax^{2}+2bx+2c=0,\] \[bx^{2}+2cx+2a=0,\] \[cx^{2}+2ax+2b=0.\]
  6. Solve the system of equations \[\begin{cases} x^{3}+y^{3} & =1\\ x^{5}+y^{5} & =1 \end{cases}.\]
  7. Given a quadrilateral pyramid $S.ABCD$ whose base $ABCD$ is a parallelogram. A point $M$ varies on the side $AB$ ($M$ is different from $A$ and $B$) Let $(\alpha)$ be a plane which goes through $M$ and is parallel to $SA$ and $BD$. Determine the cross section of $S.ABCD$ determined by $(\alpha)$ and find the position for $M$ so that the cross section has maximal area.
  8. Solve the equation \[3\sin^{3}x+2\cos^{3}x=2\sin x+\cos x.\]
  9. Find all pairs of positive integers $(a,b)$ so that $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
  10. Given a polynomial \[P(x)=a_{n}x^{n}+\ldots+a_{1}x+a_{0}\quad(a_{n}\ne0,\,n\geq2)\] with integer coefficients. Prove that there are infinitely many integers $k$ such that $P(x)+k$ cannot be factorized as a product of two positive degree polynomials with integral coefficients.
  11. Suppose that the funtion $f:\mathbb{N\to\mathbb{N}}$ is onto and the funtion $g:\mathbb{N}\to\mathbb{N}$ is one-to-one. Assume furthermore that $f(x)\geq g(n)$ for all $n\in\mathbb{N}$. Show that \[f(n)=g(n),\,\forall n\in\mathbb{N}.\]
  12. Given a triangle $ABC$ inscribed in a circle $(O)$. $M$ and $N$ are two fixed points on $(O)$ so that $MN\parallel BC$. Let $P$ vary on the line $AM$. The line through $P$ which is parallel to $BC$ intersects $CA$, $AB$ at $E,F$ respectively. The circumscribed circle of the triangle $NEF$ intersects $(O)$ at the second point $Q$ (different from $N$). Prove that the line $PQ$ always goes through a fixed point when $P$ moves.




Mathematics & Youth: 2015 Issue 453
2015 Issue 453
Mathematics & Youth
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