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

## $show=home 1. Place the numbers from 1 to 20 on a circle such that the sum of any two numbers which are next to each other is a prime number. 2. Given a right triangle$ABC$with the right angle$A$and$\widehat{ABC}<45^{0}$. On the haft plane determined by$AB$, containing$C$, choose two points$D$and$E$such that$BD=BA$,$\widehat{DBA}=90^{0}$,$\widehat{EBC}=\widehat{CBA}$, and$ED$is perpendicular to$BD$. Prove that$BE=AC+DE$. 3. Given two real numbers$x$and$y$such that$x^{2}+2xy+2y^{2}=1$. Find the maximum and minimum values of the expression $P=x^{4}+y^{4}+(x+y)^{4}.$ 4. Given a triangle$ABC$with$AB<AC$. Let$(O)$be the incircle of$ABC$. The side$BC$is tangent to$(O)$at$D$. Choose$I$on$AD$such that$OI$is perpendicular to$AD$. The ray$IO$intersect the perpendicular bisector of$BC$at$K$. Prove that$BIKC$is an inscribed quadrilateral. 5. Solve the equation $\frac{\sqrt{x^{2}+28x+4}}{x+2}+8=\frac{x+4}{\sqrt{x-1}}+2x.$ 6. Given non-negative numbers$a,b$and$c$such that$ab+bc+ca=1$. Prove that $\sqrt{3}\leq\sqrt{1+a^{2}}+\sqrt{1+b^{2}}+\sqrt{1+c^{2}}-a-b-c\leq2.$ 7. Solve the system of equations $\begin{cases} x^{8} & =21y+13\\ \frac{(x+y)^{25}}{2^{18}} & =(x^{3}+y^{3})^{3}(x^{4}+y^{4})^{4}\end{cases}.$ 8. Given a triangle$ABC$and$P$is a point lying inside$ABC$. Let$E$(resp.$AC$and$AB$). Let$K$be the perpendicular projection of$K$on$BC$. On the side$AF$, choose an arbitrary point$M$. Assume that$N$is the intersection between$MK$and$PE$. Prove that$\widehat{KHM}=\widehat{KHN}$. 9. Given a bijection$f:\mathbb{N}^{*}\to\mathbb{N}^{*}$. Prove that there exist positive intergers$a,b,c$and$d$such that$a<b<c<d$and$f(x)+f(d)=f(b)+f(c)$. 10. Given a sequence$(u_{n})$whose terms are positive integers satisfying $0\leq u_{m+n}-u_{m}-u_{n}\leq2\quad\forall m,n\geq1.$ Prove that there exist two positive numbers$a_{1},a_{2}$such that $[a_{1}n]+[a_{n}n]-1\leq u_{n}\leq[a_{1}n]+[a_{2}n]+1$ for all$n\leq2017$. ($[x]$is the greatest integer which does not exceed$x$). 11. Find all continuous functions$f:(0,+\infty)\to(0,\infty)$such that $f(x+2016)=f\left(\frac{x+2017}{x+2018}\right)+\frac{x^{2}+4033x+4066271}{x+2018},\forall x>0.$ 12. Given a circle$(O)$and two fiexd points$B,C$on$(O)$. A point$A$is moving on$(O)$such that$ABC$is always an acute triangle and$AB<AC$. Choose$D$on the side$AC$such that$AB=AD$. The line$BD$intersects$(O)$at$E$which is different from$B$. The perpendicular projection of$E$(resp.$D$) on$AC$(resp.$AE$) is$H$(resp.$M$). The circumcircle of$AMH$intersects$(O)$at$K$. Let$N$be perpendicular projection of$K$on$AB$. a) Prove that$MN$goes through the midpoint$I$of$BD$. b) The circumcircles of$BCD$and$AMH$intersect each other at$P$and$Q$. Prove that the midpoint of$PQ$always lies on a fixed line when$A$is moving on$(O)$in the given way. ##$type=three$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: 2017 Issue 479
2017 Issue 479
Mathematics & Youth