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2018 Issue 493

  1. Find $2018$ numbers so that each of them is the square of the sum of all remaining numbers.
  2. Find the following sum $$S=(1+2.3+3.5+\ldots+101.201) +\left(1^{2}+2^{2}+3^{2}+\ldots+100^{2}\right).$$
  3. Find all pairs of positive integers $(m, n)$ such that $$n^{3}-5 n+10=2^{m}.$$
  4. Given a triangle $A B C$ with $B C=a$, $A C=b$, $A B=c$ and $3 \hat{B}+2 \hat{C}=180^{\circ}$. Prove that $$b+c \leq \dfrac{5}{4} a.$$
  5. Solve the system of equations $$\begin{cases}x^{2}-y^{2}+\sqrt{x}-y+2&=0 \\ x+8 y+4 \sqrt{x}-8 \sqrt{y}-4 \sqrt{x y} &=0\end{cases}$$
  6. Given three positive numbers $a, b, c$ satisfying $a+b+c=3 .$ Show that $$\frac{1}{(a+b)^{2}+c^{2}}+\frac{1}{(b+c)^{2}+a^{2}}+\frac{1}{(c+a)^{2}+b^{2}} \leq \frac{3}{5}$$
  7. Solve the system of equations $$\begin{cases}x-1&=\sqrt[4]{9+12 y-6 y^{2}} \\ y-1&=\sqrt[4]{9+12 x-6 x^{2}}\end{cases}.$$
  8. Given a right prism with equilateral bases $A B C . A^{\prime} B^{\prime} C$. Let $\alpha$ be the angle between the line $B C$ and the plane $\left(A^{\prime} B C\right)$. Prove that $\sin \alpha \leq 2 \sqrt{3}-3$
  9. Given positive numbers $a$, $b$. Show that $$\frac{1}{2}\left[1-\frac{\min (a, b)}{\max (a, b)}\right]^{2} \leq \frac{b-a}{a}-\ln b+\ln a \leq \frac{1}{2}\left[\frac{\max (a, b)}{\min (a, b)}-1\right]^{2}.$$
  10. Find the maximal positive number $k$ so that the following inequality $$a^{2}+b^{2}+c^{2}+k(a+b+c) \geq 3+k(a b+b c+c a)$$ holds true for all positive numbers $a, b, c$.
  11. Given the sequence $\left(x_{n}\right)$ $$x_{2}=x_{3}=1,\quad (n+1)(n-2) x_{n+1}=\left(n^{3}-n^{2}-n\right) x_{n}-(n-1)^{3} x_{n-1},\, \forall n \geq 3.$$ Find all indices $n$ so that $x_{n}$ is an integer.
  12. Given a cyclic quadrilateral $A B C D .$ Let $K$ be the intersection between $A C$ and $B D .$ Let $M$, $N$, $P$ and $Q$ respectively be the perpendicular projection of $K$ on $A B$, $B C$, $C D$ and $D A$. And then, let $X$, $Y$, $Z$ and $T$ respectively be the perpendicular projection of $K$ on $M N$, $N P$, $P Q$, $Q M$. Prove that $A X C Z$ and $B Y D T$ have equal areas.

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Mathematics & Youth: 2018 Issue 493
2018 Issue 493
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-493.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-493.html
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