2007 Issue 358

  1. Let $a, b, c$ be positive numbers such that $a^{3}+b^{3}=c^{3}$. Compare $a^{2007}+b^{2007}$ and $c^{2007}$
  2. Let $A B C$ be an isosceles triangle at $A$. Let $E$ be an arbitrary point on the side $B C$. The line through $E$ perpendicular to $A B$ meets with the line through $C$ perpendicular to $A C$ at a point denoted by $D$. Let $K$ be the midpoint of $B E$ Find the measure angle $\widehat{A K D}$.
  3. Find the least positive integer $n$ $(n>1)$ such that $\dfrac{1^{2}+2^{2}+\ldots+n^{2}}{n}$ is a perfect square number.
  4. Prove the following inequality $$\left(1+\frac{2 a}{b}\right)^{2}+\left(1+\frac{2 b}{c}\right)^{2}+\left(1+\frac{2 c}{a}\right)^{2} \geq \frac{9(a+b+c)^{2}}{a b+b c+c a}$$ where $a, b, c$ are arbitrary positive real numbers. When does equality occur?
  5. Solve the equation $$x^{3}-\sqrt[3]{6+\sqrt[3]{x+6}}=6.$$
  6. Let $A B C$ be a right triangle at $A$. Draw the circle $(B)$ with center at $B$ and radius $B A$ and draw a diameter $A D .$ Choose two points $E$ and $F$ on the line $B C$ such that $B$ is their midpoint. $D E$ and $D F$ meets with $(B)$ at $M$ and $N$ respectively. Prove that $M, N$ and $C$ are colinear.
  7. Let $A B C$ be an equilateral triangle with circumcircle $(O) .$ Draw a circle $\left(O_{1}\right)$ which is tangent to both side $B C$ and arc $\widehat{B C}$. Construct the circles $\left(O_{2}\right)$, $\left(O_{3}\right)$ similarly for the remaining sides $C A$ and $A B .$ Prove that $A O_{1}=B O_{2}=C O_{3}$ iff (ie. if and only if) the circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ and $\left(O_{3}\right)$ all have the same radius. (Notation: $(X)$ is a circle with center at $X$.)
  8. Let $\left(a_{n}\right)$ $(n=0,1,2, \ldots)$ be a sequence given by $$a_{0}=29,\, a_{1}=105,\,a_{2}=381,\, a_{n+3}=3 a_{n+2}+2 a_{n+1}+a_{n},\,\forall n=0,1,2, \ldots .$$ Prove that for each positive integer $m,$ there exists a number $n$ such that $a_{n}$, $a_{n+1}-1$, $a_{n+2}-2$ are all divisible by $m$
  9. Let $f$ be a continuous function on the interval [0,1] and satisfies the following properties $$f(0)=0,\,f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y \in [0,1].$$ Find $f\left(\dfrac{8}{23}\right)$.
  10. Let $a, b, c$ be non-negative real numbers with sum equal to $1 .$ Prove that $$\sqrt{a+(b-c)^{2}}+\sqrt{b+(c-a)^{2}}+\sqrt{c+(a-b)^{2}} \geq \sqrt{3}$$
  11. Let $H$ and $I$ be, respectively, the orthorcenter and the incenter of an acute triangle $A B C$. The lines $A H$, $B H$ and $C H$ meet the circumcircles of triangles $HBC$, $HCA$ and $HAB$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. $A I$, $B I$ and $C I$ meet the circumcircles of triangles $I B C$, $I C A$ and $I A B$ at $A_{2}$, $B_{2}$, $C_{2}$ respectively. Prove that $$H A_{1} \cdot H B_{1} \cdot H C_{1}+64 R^{3} \geq 9 \cdot I A_{2} \cdot I B_{2} \cdot I C_{2}.$$
  12. Let $G$ be the centroid of a tetrahedron $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be arbitrary points on $G A$, $G B$, $G C$ and $G D$ respectively. GA meets the plane $\left(B_{1} C_{1} D_{1}\right)$ at $A_{2} .$ Construct the points $B_{2}$, $C_{2}$, $D_{2}$ similarly. Prove that $$3\left(\frac{G A}{G A_{1}}+\frac{G B}{G B_{1}}+\frac{G C}{G C_{1}}+\frac{G D}{G D_{1}}\right) = \frac{G A}{G A_{2}}+\frac{G B}{G B_{2}}+\frac{G C}{G C_{2}}+\frac{G D}{G D_{2}}$$




Mathematics & Youth: 2007 Issue 358
2007 Issue 358
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy