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2012 Issue 426

  1. Prove that for any natural number $n>4$ there exists a pair of natural numbers $x, y$ with $\dfrac{n}{2} \leq x<n$ and $\dfrac{n}{2} \leq y<n,$ such that $x^{2}-y$ is divisible by $n$ 
  2. Let $A B C$ be an isosceles right triangle with right angle at $A$. On the ray $A C$ choose two points $E$ and $F$ such that $\widehat{A B E}=15^{\circ}$ and $C E=C F$. What is the measure of the angle $C B F$?.
  3. Find all positive integer solutions of the equation $$65\left(x^{3} y^{3}+x^{2}+y^{2}\right)=81\left(x y^{3}+1\right).$$
  4. Solve the system of equations $$\begin{cases} 9 x^{2}+9 x y+5 x-4 y+9 \sqrt{y} &=7 \\ \sqrt{x-y+2}+1 &=9(x-y)^{2}+\sqrt{7 x-7 y} \end{cases}.$$
  5. Let $A B C$ be an isosceles triangle where the angle $B A C$ is obtuse. Suppose $D$ is a point on edge $A B$ such that $B C=C D \sqrt{2}$. The line through $D$ and perpendicular to $A B$ meets $C A$ at $E$. Prove that $C D$ passes through the midpoint of $B E$.
  6. Let $$S=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\ldots+2013 \sqrt[2]{\frac{2013}{2012}}.$$ Find the integer part of $S$.
  7. Given that $a, b, c$ are edge lengths of a triangle. Prove the following inequality $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \leq \frac{3 \sqrt{3} a b c}{(a+b+c) \sqrt{a+b+c}}.$$
  8. Let $A B C D$ be a trirectangular tetrahedron where the edges $A B$, $A C$, $A D$ are pairwise perpendicular. $M$ is an arbitrary point in the space. Given that $A B=4$, $A C=8$, $A D=12$. Find the minimum value of the expression $$P=\sqrt{7} M A+\sqrt{11} M B+\sqrt{23} M C+\sqrt{43} M D.$$
  9. Let $\left(a_{n}\right)$ be a sequence of positive integers where $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=4 a_{n+1}+a_{n},\,\forall n \geq 1 .$$ Prove that
    a) $a_{n} a_{n+2}+(-1)^{n} .5$ is a perfect square for all $n \geq 1$.
    b) The equation $x^{2}-4 x y-y^{2}=5$ has infinitely many positive integer solutions.
  10. Let $a$ be a real number from $(0,1)$ and $b$ is a complex number, $|b|<1$. Prove that $$|b|+\left|\frac{a-b}{1-a b}\right| \geq a.$$
  11. Let $0<\alpha<\dfrac{\pi}{2}$. Prove that $$(\cot \alpha)^{\cos 2 \alpha} \geq \frac{1}{\sin 2 \alpha}$$
  12. A tetrahedron $A B C D$ is inscribed in a sphere centered at $O$. Point $G$ does not belong to planes $(B C D)$, $(C D A)$, $(D A B)$, $(A B C)$ and the sphere $(O)$. $X$, $Y$, $Z$, $T$ are the centers of the circumscribed spheres of the tetrahedron $G B C D$, $G C D A$, $G D A B$, $G A B C$ respectively. Prove that $G$ is the centroid of tetrahedron $A B C D$ if and only if $O$ is the centroid of tetrahedron $X Y Z T$.

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Mathematics & Youth: 2012 Issue 426
2012 Issue 426
Mathematics & Youth
https://www.molympiad.org/2020/09/2012-issue-426.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2012-issue-426.html
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