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2010 Isue 402

  1. Compare $\dfrac{5}{24}$ ưith the sum $$\frac{1}{1.2 .4}+\frac{1}{2.5 .7}+\frac{1}{3.8 .10}+\ldots+\frac{1}{1000.2999 .3001}$$
  2. Let $A B C$ be an isosceles right triangle, with right angle at vertex $A .$ Let $D$ be a point inside the triangle such that $\Delta A B D$ is an isosceles triangle and $\widehat{A D B}=150^{\circ} .$ Let $A C E$ be an equilateral triangle so that points $D$ and $E$ are on the different side of the half-plane $A C .$ Prove that $B$, $D$, $E$ are collinear.
  3. Find all positive integer numbers $m$ such that the following equation $$x^{2}-m x y+y^{2}+1=0$$ has positive integer roots.
  4. Let $A B C$ be an acute triangle and let $A H$, $B K$, $C L$ be its three altitudes. Prove that $$A K \cdot B L \cdot C H=A L \cdot B H \cdot C K=H K \cdot K L \cdot L H.$$
  5. Solve the equation. $$\frac{8 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}}-\frac{2 \sqrt{2} x(x+3)}{1+x^{2}}=5-\sqrt{2}$$
  6. Let $a$, $b$, $c$ be three positive real numbers such that $a b c=1$. Prove that $$\dfrac{1}{\sqrt{a^{5}-a^{2}+3 a b+6}}+\dfrac{1}{\sqrt{b^{5}-b^{2}+3 b c+6}} +\frac{1}{\sqrt{c^{5}-c^{2}+3 c a+6}} \leq 1.$$
  7. Solve the equation $$2 \sin \left(x+\frac{\pi}{3}\right) +2^{2} \sin \left(x+\frac{2 \pi}{3}\right)+\ldots +2^{2010} \sin \left(x+\frac{2010 \pi}{3}\right)=0.$$
  8. Let $O A B C$ be a tetrahedron where $O A$, $O B$, $O C$ are pairwise orthogonal. Let $M$ be a point on the plane containing the base $A B C$. Let $G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of triangles $O A B$, $O B C$ and $O C A$. Put $O A=a$, $O B=b$, $O C=c$. Prove the inequality $$M G_{1}^{2}+M G_{2}^{2}+M G_{3}^{2} \geq \frac{a^{2} b^{2} c^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}.$$ When does equality occur?
  9. Let $A B C$ be an acute triangle with altitude $A D$. $M$ is a point on $A D$. The lines $B M$, $C M$ meets $A C$, $A B$ at $E$, $F$ respectively. $D E$, $D F$ intersects with the circles whose diameters $A B$, $A C$ at $K$, $L$ respectively. Prove that the line connecting the midpoints of $E F$, $K L$ goes through $A$.
  10. Prove that there exist infinitely many triples of positive integers $(a, b, c)$ such that $a b+1$, $b c+1$, $c a+1$ are all square numbers.
  11. Find all positive integers $a$, $b$, $c$ so that the equation $$x+3 \sqrt[4]{x}+\sqrt{4-x}+3 \sqrt[4]{4-x}=a+b+c$$ has solution and the expression $P=a b+2 a c+3 b c$ is greatest possible.
  12. Find all positive real numbers $a$ such that there exists a positive real number $k$ and a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$\frac{f(x)+f(y)}{2} \geq f\left(\frac{x+y}{2}\right)+k|x-y|^{a}$$ for all real numbers $x$, $y$.

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Mathematics & Youth: 2010 Isue 402
2010 Isue 402
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-isue-402.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-isue-402.html
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