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

## $show=home 1. Find all positive integers$n$such that there exist prime numbers$p, q$satisfying $$1 !+2 !+3 !+\ldots+n !=p^{2}+q^{2}+5895$$ Notice that$n !=1.2 .3 \ldots n$. 2. Given an acute angle$A B C$with the altitude$A H,$the median$B M,$the angle bisector$C K$. Show that if$HMK$is an equilateral triangle then so is$A B C$. 3. Suppose that$n$is a positive integer so that$3^{n}+7^{n}$is divisible by$11$. Find the remainder in the divison of$2^{n}+17^{n}+2018^{n^{2}}$by$11$4. Given a circle$(O)$and suppose that$B C$is a fixed chord of$(O) .$Let$A$be a point moving on the major arc$B C ;$and$M,N$respectively the midpoints of$A B$and$A C$. Show that each of the altitudes$M M^{\prime}$and$N N^{\prime}$of$\triangle A M N$contains some fixed point. 5. Solve the equation $$13 x^{2}-5 x-13=(16 x-11) \sqrt{2 x^{2}-3}$$ 6. Solve the system of equations $$\begin{cases}\sqrt{4 x^{2}+5}+\sqrt{4 y^{2}+5}&=6|x y| \\ 2^{\dfrac{1}{x^{8}}+\dfrac{2}{y^{4}}}+2^{\dfrac{1}{y^{8}}+\dfrac{2}{x^{4}}}&=16 \end{cases}.$$ 7. Show that the following inequality holds for all positive numbers$a, b, c$$$a^{2}+b^{2}+c^{2} \geq \frac{1}{2}(a b+b c+c a) + \sqrt{\frac{2(a+b+c)\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\right)}{(a+b)(b+c)(c+a)}}$$ 8. Given triangle$A B C$with the incenter$I$, the centroid$G$. There lines$A G$,$B G$and$CG$respectively intersect the circumcircle of$A B C$at$A_{2}$,$B_{2}$,$C_{2}$. Show that $$G A_{2}+G B_{2}+G C_{2} \geq I A+I B+I C.$$ 9. Given continuous functions$f,g:[a, b] \rightarrow[a, b]$such that $$f(g(x))=g(f(x)) \quad \forall x \in[a, b]$$ where$a, b$are real numbers. Show that the equation$f(x)=g(x)$has at least one solution. 10. Find all pairs of integers$(m, n)$so that both$m^{5} n+2019$and$m n^{5}+2019$are cubes of integers. 11. Let$\quad S=\{1,2,3, \ldots, 2019\}$and suppose that$S_{1}, S_{2}, \ldots, S_{410}$are subsets of$S$such that for every$i \in S$, there exist exactly$40$of them containing i. Prove that there exist different indices$j, k, l \in\{1,2, \ldots, 410\}$such that$\left|S_{j} \cap S_{k} \cap S_{l}\right| \leq 1$12. Given a cyclic quadrilateral$A B C D$with$E$is the intersection between$A C$and$B D .$The circumcircles of$E A D$and$E B C$intersect at another point$F$(besides$E$). The perpendicular bisectors of$B D$,$A C$respectively intersect$F B$,$F A$at$K$,$L$. Show that$K L$goes through the midpoint of$A B$. ##$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: 2018 Issue 497
2018 Issue 497
Mathematics & Youth