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

## $show=home 1. Assume that$A$is a natural number which has two prime factors$p$and$q$only. Let$S$be the sum of all positive factors of$A$. Show that$S < 2 A$. 2. Denote$a_{n}$the integer which is closest to$\sqrt{n}(n \in \mathbb{N}^{*})$, for example $$\sqrt{1}=1=a_{1} ; \quad \sqrt{2} \approx 1,4 \Rightarrow a_{2}=1 ; \quad \sqrt{3} \approx 1,7 \Rightarrow a_{3}=2 ; \ldots.$$ Compute $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{2020}+\frac{1}{2021}.$$ 3. Find all$4$-tuples of positive integers$(x ; y, z, t)$satisfying $$\begin{cases}x+y+z+t=54 \\ \dfrac{x z}{(x+y)(z+t)} =\dfrac{23}{104} \\ x>z, x+y>z+t \end{cases}.$$ 4. Given an acute triangle$A B C$($CA < CB$) with the orthocenter$H$. Let$D$be the second intersection between$B C$and the circumcircle of$A B H$. Denote$P$the intersection between$D H$and$A C$. Let$M$be the second intersection between$A H$and the circumcircle of$A P D$; and$N$the second intersection between the circumcircle of$A B C$and the circumcircle of$A P D$. Show that three points$C, M$, and$N$are collinear. 5. Given positive numbers$a, b, c$satisfying$4 b^{2}+c^{2}=a c$. Find the minimum value of the expression $$P=\frac{2 a}{a+2 b}+\frac{b}{b+c}+\frac{c}{c+a} .$$ 6. Solve the equation $$\sin 3 x-\cos 3 x+\sin x+\cos x=\frac{1}{\sin 3 x+\cos x}-\frac{1}{\cos 3 x-\sin x} .$$ 7. Given positive numbers$a, b, c$satisfying$a^{2}+b^{2}+c^{2} \geq 2(a b+b c+c a)$. Find the minimum value of the expression $$P=a+b+c+\frac{8}{a b c} .$$ 8. Given two circles$(O ; R)$and$\left(O^{\prime} ; R^{\prime}\right)\left(R>R^{\prime}\right)$which is internally tangent to each other at$A$. Let$M$be a point on$\left(O^{\prime} ; R^{\prime}\right)(M \neq A) .$The tangent of$\left(O^{\prime} ; R^{\prime}\right)$at$M$intersects$(O ; R)$at$P$and$Q .$The circumcircle of$O^{\prime} P Q$intersects$\left(O^{\prime} ; R^{\prime}\right)$at$B$and$C$. Show that$A B M C$is a harmonic quadrilateral, i.e. a cyclic quadrilateral of which the products of opposite sides are equal. 9. Given a triangle$A B C$and$x, y, z$positive numbers. Show that $$\frac{x}{y+z}(1+\cos A)+\frac{y}{z+x}(1+\cos B)+\frac{z}{x+y}(1+\cos C) \geq \frac{\sqrt{3}}{2}(\sin A+\sin B+\sin C) .$$ 10. Two boxes contain$25$small white and black balls. From each box, pick randomly$1$ball. Find the probability that we get$2$balls with different colors assuming that the box with more balls has more black balls and the probability to get two black balls is$0,42$. 11. Find all functions$f: \mathbb{R} \rightarrow(0+\infty)$satisfying $$\frac{1}{2015} \leq\left(\frac{f(x)}{f(r)}\right)^{\frac{1}{(x-r)^{2}}} \leq 2015, \forall x \in \mathbb{R}, \forall r \in \mathbb{Q}, x \neq r$$ 12. Outside a triangle$A B C$, draw pairwise similar triangles$B C P$,$A C Q$,$A B R$, and$B A S$. Let$K$,$L$respectively be the midpoints of$B C$and$C A .$Prove that two triangles$R P K$and$S Q L$have the same area. ##$type=three$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: 2021 Issue 528
2021 Issue 528
Mathematics & Youth