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

## $show=home 1. Let$a_{1},a_{2},\ldots,a_{1964}$be integers such that $$a_{1964}^{2}+a_{1963}^{2}=a_{1962}^{2}-a_{1961}^{2}+a_{1960}^{2}-a_{1959}^{2}+\ldots+a_{2}^{2}-a_{1}^{2}.$$ Prove that$a_{1}\cdot a_{2}\cdots a_{1964}+2015$can be written as a difference of two perfect squares. 2. Let$ABC$be a triangle with its side$BC$is fixed and its vertex$A$varies such that the triangle is not isosceles at$A$. Draw the internal angle bisector$AD$of the triangle. On the ray$CA$choose$E$such that$CE=AB$. Let$I$be the midpoint of$AE$. Prove that the line which passes through a fixed point. 3. Find all natural numbers$x,y$such that $$x^{5}=y^{5}+10y^{3}+20y+1.$$ 4. Given a triangle$PQR$inscribed in a circle$\left(O\right)$. Let$S$be the midpoint of the arc$QR$which does not contain$P$. Draw a circle with center$O'$passing through$P$and$S$. This circle intersects$PQ$and$PR$respectively at$M$and$N$. Let$H$and$K$respectively be the midpoint of$MN$and$QR$. Prove that$HK$is perpendicular to$PS$. 5. Solve the system of equations $$\begin{cases}x^{3}-x^{2}+x\left(y^{2}+1\right) &=y^{2}-y+1 \\ 2y^{3}+12y^{2}+18y-2+z&=0 \\ 3z^{3}-9z+x-7 &=0\end{cases}$$ where$x,y,z\in\mathbb{R}$. 6. Given positive real numbers$x,y$and$z$such that $$\left(x+y\right)\left(y+z\right)\left(z+x\right)=1.$$ Show that $$\frac{\sqrt{x^{2}+xy+y^{2}}}{\sqrt{xy}+1}+\frac{\sqrt{y^{2}+yz+z^{2}}}{\sqrt{yz}+1}+\frac{\sqrt{z^{2}+zx+x^{2}}}{\sqrt{zx}+1}\geq\sqrt{3}.$$ 7. Solve the equation $$\left(\log\left(x^{2}\left(2-x\right)\right)\right)^{3}+\left(\log x\right)^{2}\cdot\log\left(x\left(2-x\right)^{2}\right)=0.$$ 8. Let$d_{a},d_{b},f_{c}$respectively be the distances from the orthocenter$H$of an acute triangle$ABC$to the sides$BC,CA,AB$. Let$r$and$R$respectively be the inradius and circumradius of$ABC$. Prove that $$d_{a}+d_{b}+d_{c}\leq\frac{3}{4}\cdot\frac{R^{2}}{r}.$$ 9. Solve the system of equations $$\begin{cases}x\left(x-2\right)+\left(y-2\right)\left(2z+1\right)&=0 \\ x\left(y+1\right)+\left(y-2\right)\left(5z+1\right)&=0 \\ \sqrt{\left(y+1\right)^{2}+\left(5z+1\right)^{2}}&=2\sqrt{\left(x-2\right)^{2}+\left(2z+1\right)^{2}}\end{cases}$$ 10. Find the maximal number$m$such that the following inequality holds for all non-negative real numbers$a,b,c,d$$$\left(ab+cd\right)^{2}+\left(ac+bd\right)^{2}+\left(ad+bc\right)^{2}\geq ma\left(b+c\right)\left(b+d\right)\left(c+d\right).$$ 11. Given natural numbers$m,n>2$. Prove that the number$\dfrac{m^{2^{n}-1}-1}{m-1}$always has a divisor of the form$m^{a}+1$where$a$is a natural number. 12. Given an acute triangle$ABC$with the circumcenter$O$. Let$D,E$and$F$respectively be the midpoints of$BC,CA$and$AB$. Choose a point$M$on the line through$BC$. Let$N$be the intersection between$AM$and$EF$and$P$the second intersection between$ON$and the circumcircle of the triangle$ODM$. Let$Q$be the reflection point of$M$through the midpoint of$DP$. Prove that$Q$belongs to the circumcircle of the triangle$DEF$. ##$type=three$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: 2016 Issue 463
2016 Issue 463
Mathematics & Youth