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

## $show=home 1. Find all single digit numbers$a,b,c$such that the numbers$\overline{abc}$,$\overline{acb}$,$\overline{bca}$,$\overline{cab}$,$\overline{cba}$are prime numbers. 2. Given a right triangle$XYZ$with the right angle$X$($XY<XZ$). Draw$XU$perpendicular to$YZ$. Let$P$be the midoint of$YU$. Choose$K$on the half plane determined by$YZ$which does not contain$X$such that$KZ$is perpendicular to$XZ$and$XY=2KZ$. The line$d$which passes through$Y$and is parallel to$XU$intersects$KP$at$V$. Let$T$be the intersection of$XP$and$d$. Prove that$\widehat{VTK}=\widehat{XKZ}+\widehat{VXY}$. 3. There are 3 people wanting to buy sheeps from Mr. An. The first one wants to buy$\dfrac{1}{a}$of the herd, the second one wants to buy$\dfrac{1}{b}$of the herd, and the third one wants to buy$\dfrac{1}{c}$of the herd and it happens that •$a,b,c\in\mathbb{N}^{*}$and$a<b<c$, • the numbers of sheeps each person wants to buy are positive integers, • after shelling, Mr. An still has exactly one sheep left. What are the possible numbers of sheeps Mr. An has?. 4. Given a circle$(O)$with a diameter$AB$. On$(O)$choose a point$C$such that$CA<CB$. On the open line segment$OB$choose$E$.$CE$intersect$(O)$at$D$. The line which goes through$A$and is parallel to$BD$intersects$BC$at$I$. The lines$OI$and$CE$meet at$F$. Prove that$FA$is a tangent to the circle$(O)$. 5. Given real numbers$a,b,c$satisfying$a+b+c=3$and$abc\geq-4$. Prove that $3(abc+4)\geq5(ab+bc+ca).$ 6. Solve the following system of equations ($a$is a parameter) $\begin{cases}2x(y^{2}+a^{2}) & =y(y^{2}+9a^{2})\\ 2y(z^{2}+a^{2}) & =z(z^{2}+9a^{2})\\ 2z(x^{2}+a^{2}) & =x(x^{2}+9a^{2})\end{cases}.$ 7. Suppose that$x=\dfrac{2013}{2015}$is a solution of the polynomial $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}\in\mathbb{Z}[x].$ Can the sum of the coefficients of$f(x)$be 2017?. 8. Given three circles$(O_{1}R_{1})$,$(O_{2}R_{2})$,$(O_{3}R_{3})$which are pariwise externally tangent to each other at$A,B,C$. Let$r$be the radius of the incircle of$ABC$. Prove that $r\leq\frac{R_{1}+R_{2}+R_{3}}{6\sqrt{3}}.$ 9. Given positive numbers$x,y,z$satisfying $x^{2}+y^{2}-2z^{2}+2xy+yz+zx\leq0.$ Find the minimum value of the expression $P=\frac{x^{4}+y^{4}}{z^{4}}+\frac{y^{4}+z^{4}}{x^{4}}+\frac{z^{4}+x^{4}}{y^{4}}.$ 10. Find the maximum value of the expression $T=\frac{a+b}{c+d}\left(\frac{a+c}{a+d}+\frac{b+d}{b+c}\right)$ where$a,b,c,d$belong to$[\frac{1}{2},\frac{2}{3}]$. 11. Let$R(t)$be a polynomial of degree 2017. Prove that there exist infinitely many polynomials$P(x)$such that $P((R^{2017}(t)+R(t)+1)^{2}-2)=P^{2}(R^{2017}(t)+R(t)+1)-2.$ Find a relation between those polynomials$P(x)$. 12. Given a triangle$ABC$. The incircle$(I)$of$ABC$is rangent to$AB$,$BC$and$CA$at$K,L$and$M$. The line$t$which passes through$B$and is different from$AB$and$BC$intersects$MK$at$ML$respectively at$R$and$S$. Prove that$\widehat{RIS}$is an acute angle. ##$type=three$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 483
2017 Issue 483
Mathematics & Youth