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

## $show=home 1.$21$distinct integers are chosen so that the sum of any subset of$11$numbers among them is always greater than the sum of the remaining$10$. If one of them is$101$, and the largest number is$2014$, find the other$19$numbers. 2. In a triangle$ABC$where$\widehat{BAC}=40^{0}$and$\widehat{ABC}=60^{0}$, point$D$and$E$are chosen on the sides$AC$and$AB$respectively such that$\widehat{CBD}=40^{0}$and$\widehat{BCE}=70^{0}$.$BD$and$CE$intersect at point$F$. Prove that$AF$is perpendicular to$BC$. 3. Solve the following system of equations $\begin{cases} 2\sqrt{2x}-\sqrt{y} & =1\\ \sqrt{8x^{3}+y^{3}} & =\sqrt{2}(\sqrt{x}+\sqrt{y}-1) \end{cases}.$ 4. In a triangle$ABC$, points$E,D$on the sides$AB$and$AC$respectively such that$\widehat{ABD}=\widehat{ACE}$. The circumcircle of triangle$ADB$meets$CE$at$M$and$N$. The circumcircle of triangle$AEC$meets$BD$at$I$and$K$. Prove that the points$M,I,N,K$lie on a circle. 5. Prove that for all positive rel numbers$a,b,c$the following inequality holds $\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\geq\frac{\sqrt{2}}{4}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}).$ 6. Determine all real solutions of the equation $(x^{5}+x-1)^{5}+x^{5}=2.$ 7. Let$M$be a point inside a given triangle$ABC$and let$x,y,z$denote the distance from$M$onto$BC,CA,AB$respectively. Prove that$\widehat{BAM}=\widehat{CBM}=\widehat{ACM}$if and only if $\frac{bx}{c}=\frac{cy}{a}=\frac{az}{b}$ where$BC=a$,$CA=b$,$AB=c$. 8. Let$x,y,z$be theree arbitrary numbers from the interval$[0,1]$. Determine the maximum value of$P$, where $P=\frac{x}{y+z+1}+\frac{y}{z+x+1}+\frac{z}{z+y+1}+(1-x)(1-y)(1-z).$ ##$type=three$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

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Mathematics & Youth: 2014 Issue 443
2014 Issue 443
Mathematics & Youth