2014 Issue 441

  1. How many triples of positive integers $(a,b,c)$ are there such that $$\text{lcm}(a,b)=1000,\quad \text{lcm}(b,c)=2000,\quad \text{lcm}(a,c)=2000?.$$
  2. Let $ABC$ be an isosceles triangle $A$ with $\widehat{BAC}=100^{0}$, point $D$ on segment $BC$ such that $\widehat{CAD}=20^{0}$, point $E$ on the ray $AD$ such that triangle $ACE$ is isosceles at vertex $C$. Determine the measure of all angles of triangle $BDE$. 
  3. The sum of $m$ distinct even positive integers and $n$ distinct odd positive integers equal $2014$. Find the greatest possible value of $3m+4n$.
  4. Triangle $ABC$ is inscribed in circle center at $O$. Parallel lines are drawn through vertices $A,B,C$ such that they are not parallel to any of the sides of triangle $ABC$. These parallel lines intersect $(O)$ at $A_{1},B_{1},C_{1}$ respectively. Prove that the orthocenters of triangles $A_{1}BC$, $B_{1}CA$, $C_{1}AB$ are collinear. 
  5. Solve the system of equations \[ \begin{cases} (1+x)(1+x^{2})(1+x^{4}) & =1+y^{7}\\ (1+y)(1+y^{2})(1+y^{4}) & =1+x^{7} \end{cases}.\]
  6. Find all polynomials with real coefficients $P(x)$ such that the following conditions are satisfied \[\begin{cases} P(x)-10 & =\sqrt{P(x^{2}+3)}-13\quad(x\geq0)\\ P(2014) & =2024 \end{cases}.\]
  7. Let $h_{a},h_{b},h_{c}$ and $l_{a},l_{b},l_{c}$ denote the altitudes and inner angle-bisectors of a triangle $ABC$. Prove that \[\frac{1}{h_{a}h_{b}}+\frac{1}{h_{b}h_{c}}+\frac{1}{h_{c}h_{a}}\geq\frac{1}{l_{a}^{2}}+\frac{1}{l_{b}^{2}}+\frac{1}{l_{c}^{2}}.\]
  8. Given that $0<x<\frac{\pi}{2}$. Prove that at least one of the two numbers $\left(\frac{1}{\sin x}\right)^{\frac{1}{\cos^{2}x}}$, $\left(\frac{1}{\cos x}\right)^{\frac{1}{\sin^{2}x}}$ is greater than $\sqrt{3}$.




Mathematics & Youth: 2014 Issue 441
2014 Issue 441
Mathematics & Youth
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