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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 $$3\left(\frac{G A}{G A_{1}}+\frac{G B}{G B_{1}}+\frac{G C}{G C_{1}}+\frac{G D}{G D_{1}}\right) = \frac{G A}{G A_{2}}+\frac{G B}{G B_{2}}+\frac{G C}{G C_{2}}+\frac{G D}{G D_{2}}$$

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Mathematics & Youth: 2007 Issue 358
2007 Issue 358
Mathematics & Youth
https://www.molympiad.org/2020/09/2007-issue-358.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2007-issue-358.html
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