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

## $show=home 1. Which number is bigger,$2^{3100}$or$3^{2100}$?. 2. Let$ABC$be an isosceles triangle with$AB=AC$.$BM$is the median from$B$.$N$is a point on$BC$such that$\widehat{CAN}=\widehat{ABM}$. Prove that$CM\geq CN$. 3. Let$a,b,c$be positive numbers such that $|a+b+c|\leq1,\,|a-b+c|\leq1,\,|4a+2b+c|\leq8,\,|4a-2b+c|\leq8.$ Prove the inequality $|a|+3|b|+|c|\leq7.$ 4. Solve the equation $(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.$ 5. Let$ABC$be a right triangle, with right angle at$A$,$AH$is the altitude from$A$and$I,J$ae the incenters of triangles$HAB$and$HAC$, respectively.$IJ$cuts$AB$at$M$and meets$AC$at$N$. Let$X$and$Y$be the intersections of$HI$with$AB$and$HJ$with$AC$;$BY$,$CX$cuts$MN$at$P$and$Q$respectively. Prove that $\frac{AI}{AJ}=\frac{HP}{HQ}.$ 6. Let$x,y,z$be real numbers such that$x^{2}+y^{2}+z^{2}=3$. Find the minimum and maximum value of the expression $P=(x+2)(y+2)(z+2).$ 7. In a triangle$ABC$, let$m_{a},m_{b},m_{c}$be its median lengths, and$l_{a},l_{b},l_{c}$be the lengths of its inner bisectors,$p$is half of its perimeter. Prove the inequality $m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.$ 8. Let$S.ABC$be a pyramid where surface$SAB$is a isosceles triangle at$S$and$\widehat{BSA}=120^{0}$, the plane$(SAB)$is perpendicular to$(ABC)$. Prove that$\dfrac{S_{ABC}}{S_{SAC}}\leq\sqrt{3}$, when does the equality occur?. (Denote by$S_{DEF}$the area of triangle$DEF$) 9. A natural number$n$is a good number if it is possible to partition any square into$n$smaller squares such that at least two of them are not equal. a) Prove that if$n$is a good number, then$n\geq4$. b) Prove that both$4$and$5$are not good. c) Find all good numbers. 10. A sequence$a_{0},a_{1},\ldots,a_{n}$($n\geq2$) is defined by $a_{0}=0,\quad a_{k}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+k},\,k=1,2,\ldots,n.$ Prove the inequality $\sum_{k=0}^{n-1}\frac{e^{a_{k}}}{n+k+1}+(\ln2-a_{n})e^{a_{n}}<1$ where${\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$. 11. Find all functions$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$satisfying $f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.$ 12. Given a triangle$ABC$inscribed in a circle$(O,R)$, with center$G$and area$S$. Prove that $a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.$ ##$type=three$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: 2012 Issue 417
2012 Issue 417
Mathematics & Youth