# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Find prime numbers$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$(D)$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2020 Issue 513
2020 Issue 513
Mathematics & Youth