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

## $show=home 1. Show that $$A=10^{10}+10^{10^1}+10^{10^2}+\ldots+10^{10^{10}}-5$$ is divisible by$7$. 2. Given integers$a$,$b$and$c$. Find the natural pumber$d$so that $$|a-b|+|b-c|+|c-a|=2018^{d}+2019.$$ 3. Given positive oumbers$a, b, c$such that$a^2+b^2+c^{2}=\dfrac{3}{4}$. Find the maximum value of the expression $$P=\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)\left(1-\frac{1}{c}\right).$$ 4. Given a triangle$A B C$and$M$is inside the triangle so that$\widehat{A B M}=\widehat{A C M}$. Let$D$be the symmetrical point of$M$about the line$A C$. Draw the parallelogram$B M C E$. The ray$B M$intersects$A C$at$l$. The ray$D l$intersects$B E$at$F$. Show that the points$A$.$D$,$C$,$E$,$F$are on the same circle. 5. Solve the equation $$x^{6}-7 x^{2}+\sqrt{6}=0.$$ 6. Solve the system of equations $$\begin{cases}\log_{\frac{1}{4}} x &=\left(\dfrac{1}{4}\right)^{2}+1 \\ 128 x^{3} \sqrt{4 x^{2}+y^{2}} &=y^{3} \sqrt{64 x^{2}+y^{2}}\end{cases}.$$ 7. Suppose that $$(1+x+x^2+\ldots+x^{2018})^{2019} = a_0 +a_1 x +a_2 x^2 + \ldots +a_{4074342}x^{4074342}.$$ Prove that $$C_{2019}^0 a_{2019}-C_{2019}^1 a_{2018}+C_{2019}^{2} a_{2017}-C_{2019}^{3} a_{2016}+\ldots+C_{2019}^{2018} a_{1}-C_{2019}^{2019} a_{0}=-2019.$$ 8. Given an aculc triangle$A B C$inscribed in a circle$(O)$. Let$H$be the orthocenter of the triangle,$P$the midpoint of the minor are$\widehat{B C}$. Assume that$E$is the intersection between$B H$and$A C$.$F$is the intersection between$\mathrm{CH}$and$A B$. Let$Q$,$R$respectively be the second interscctions between$P E$,$P F$and$(O)$. Let$K$be the interscetion between$B Q$and$C R$. Show that$K H \parallel A P$. 9. Given non-ncgative numbers$a, b, c$with at most one of them is equal to$0$. Prove that $$\frac{a^{3}}{b^{2}-b c+c^{2}}+\frac{b^{3}}{c^{2}-c a+a^{2}}+\frac{c^{3}}{a^{2}-a b+b^{2}} \geq a+b+c.$$ 10. Let$\displaystyle S_n = \sum_{k=2}^n k\cos\dfrac{\pi}{k}$. Find$\displaystyle \lim_{n \rightarrow+\infty} \dfrac{S_{n}}{n^{2}}$. 11. There are$100$marbles distributed into$k$groups. A distribution is called "special" if any two groups have different numbers of marbles and if we divide any group into two smaller ones then among$k+1$new groups there are groups with equal numbers of marbles. Find the maximum and minimum values for$k$so that there exist corresponding special distributions. 12. Outside a cyclic quadrilateral$A B C D$, draw the squares$A B M N$,$B C P Q$,$C D R S$,$D A U V$. Let$B^{\prime}$be the intersection between$P Q$and$M N$,$D^{\prime}$the intersection between$U V$and$RS$. Show that the midpoint of$B^{\circ} D^{\circ}$belongs to$B D$. ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2019 Issue 502
2019 Issue 502
Mathematics & Youth