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2020 Issue 513

  1. Find prime numbers $p$, $q$, $r$ satisfying $$p+q^{2}+r^{3}=200.$$
  2. Consider the number $$P=\frac{2^{2}+1}{2^{2}+3.2+4}+\frac{3^{2}+1}{3^{2}+3.3+4}+\ldots+\frac{98^{2}+1}{98^{2}+3.98+4}$$ (including 97 terms). Show that $$\frac{6}{10^{6}}<P<\frac{1}{83325}.$$
  3. Find all positive integers $a$, $b$, $c$, $d$ satisfying $$\begin{cases}a+b+c+d-3 &=a b \\ a+b+c+d-3 &=c d\end{cases}.$$
  4. Given a quadrilateral $A B C D$ with $A B=A D$, $C B=C D$ and $\widehat{A B C}=90^{\circ}$. Let $R$ and $r$ be the circumradius and inradius of $A B C D$ respectively. Prove that $R \geq r \sqrt{2}$.
  5. Given positive numbers $x$, $y$, $z$ satisfying $$\sqrt{x^{2}+y^{2}}+\sqrt{y^{2}+z^{2}}+\sqrt{z^{2}+x^{2}}=2020.$$ Find the minimum value of the expression $$T=\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y}.$$
  6. Suppose that $a$, $b$, $c$ are positive numbers and $a b c=1$. Show that $$\sqrt{\frac{a b}{b c^{2}+1}}+\sqrt{\frac{b c}{c a^{2}+1}}+\sqrt{\frac{c a}{a b^{2}+1}} \leq \frac{a+b+c}{\sqrt{2}}.$$
  7. Find the real solutions of the system of equations $$\begin{cases} x &\notin(-\pi ; \pi) \\ \sin y-\sin x &=\dfrac{2 x y(\pi+x)}{\pi^{2}+x^{2}} \\ y^{3}+\pi^{3} & =x^{3}-3 \pi x y \end{cases}.$$
  8. Given a circle $(O)$ and a point $P$ inside the circle and is different from $O$. A moving line $\Delta$ passing through $P$ but not $O$ intersects $(O)$ at $E$ and $F .$ The tangents at $E$, $F$ to the circle $(O)$ meet at $T$. Let $S$ be the intersection between the line segment $T P$ and $(O)$. Let $\omega$ be the circle which passes through $S$, $T$ and is tangent to $(O)$. Show that the circle $\omega$ always passing through a fixed point when $\Delta$ varies.
  9. Given real numbers $x$, $y$, $z$ satisfying $x+y+z=0$. Find the minimum value of the expression $$S=\frac{1}{4 e^{2 x}-2 e^{x}+1}+\frac{1}{4 e^{2 y}-2 e^{y}+1}+\frac{1}{4 e^{2 z}-2 e^{z}+1}.$$
  10. Three sequences $\left(a_{n}\right)$, $\left(b_{n}\right)$, $\left(c_{n}\right)$ are determined as follows $a_{0}=2$, $b_{0}=9$, $c_{0}=2020$ and $$\begin{cases}a_{n} & =-\dfrac{1}{4} a_{n-1}+\dfrac{1}{2} b_{n-1}+\dfrac{1}{2} c_{n-1} \\ b_{n} &=\dfrac{1}{2} a_{n-1}-\dfrac{1}{4} b_{n-1}+\dfrac{1}{2} c_{n-1} \\ c_{n} &=\dfrac{1}{2} a_{n-1}+\dfrac{1}{2} b_{n-1}-\dfrac{1}{4} c_{n-1}\end{cases}$$ for all $n=1,2, \ldots$. Find the limits $\displaystyle \lim_{n\to\infty} a_{n}$, $\displaystyle \lim_{n\to\infty}b_{n}$, $\displaystyle \lim_{n\to\infty}c_{n}$. Can you generalize this problem?
  11. Let $h$ be a positive integer so that $p:=2^{h}+1$ is a prime number. Find the smallest positive integer $k$ so that $2^{k}-1$ is divisible by $p$.
  12. Suppose that $(D)$ and $(O)$ are two circles which tangent to each other at $X$. In the case of internally tangent then $(D)$ is inside $(O)$. Let $A$ be a point on $(D)$ which is different from $X$ so that the tangent to $(D)$ at $A$ intersects $(O)$. Let $B$ be any point of that intersection. Show that the radical line of $(O)$ and the circle $(B, B A)$ is a tangent line to $(D)$.

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Mathematics & Youth: 2020 Issue 513
2020 Issue 513
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-513.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-513.html
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