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

## $show=home 1. Find all integers$x,y,z$and$t$such that $$38\left(xyzt+xy+xt+zt+1\right)=49\left(yzt+y+t\right).$$ 2. Given an isosceles triangle$ABC$with the vertex angle$A$. Let$M$be a point inside the triangle such that$\widehat{AMB}>\widehat{AMC}$, and$N$is a point between$A$and$M$. Prove that$\widehat{ANB}>\widehat{ANC}$. 3. Find the maximal$k$such that there exist$k$positive integers which do not exceed 100 and have the property that any number among them cannot be a power of any remain one. 4. Assume that$ABC$be a triangle inscribed in the semicircle with the center$O$and the diameter$BC$. Two tangent lines to the semicircle$\left(O\right)$at$A$and$B$intersect at$D$. Prove that$DC$goes through the midpoint$I$of the altitude$AH$of$ABC$. 5. Find positive integer solutions of the equation $$xyz=x+2y+3z-5.$$ 6. Let$a,b$and$c$are real numbers such that$\left|a+b\right|+\left|b+c\right|+\left|c+a\right|=8$. Find the maximum and minimum values of the expression$P=a^{2}+b^{2}+c^{2}$. 7. Solve the system of equations $$\begin{cases} x & =2^{1-y},\\ y & =2^{1-x}. \end{cases}$$ 8. Let$ABC$be a triangle without any obtuse angle. Prove that $$\cos A\cos B+\cos B\cos C+\cos C\cos A\geq2\sqrt{\cos A\cos B\cos C}.$$ 9. Given$n$non-negative real numbers$x_{i}$,$i=1,2,\ldots,n\left(n\geq2\right)$satisfying$x_{1}+x_{2}+\ldots+x_{n}=1$. Show that $$\frac{1}{n}\left(\frac{x_{1}}{1+x_{1}}+\frac{x_{2}}{1+x_{2}}+\ldots\frac{x_{n}}{1+x_{n}}\right)<\frac{x_{1}^{2}}{1+x_{1}^{2}}+\frac{x_{2}^{2}}{1+x_{2}^{2}}+\ldots\frac{x_{n}^{2}}{1+x_{n}^{2}}.$$ 10. A natural number is called$k$-\emph{success} if the sum of its digits equals to$k$. Let$a_{ik}$be the number of all$k$-success numbers which have$i$digits. Find the sum$\sum_{k=1}^{m}\sum_{i=1}^{n}a_{ik}$where$k,i,m,n$are positive integers. 11. Find all triples$\left(x,y,p\right)$of two non-negative integers$x,y$and a prime number$p$such that$p^{x}-y^{p}=1$. 12. Given a triangle ABC and$H$is a point inside the triangle such that$\widehat{ABH}=\widehat{ACH}$. A circle which goes through$B$and$C$intersects the circle with the diameter$AH$at two different points$X$and$Y$,$AH$intersects$BC$at$D$. Let$K$be the perpendicular projection of$D$on$XY$. Prove that$\widehat{BKD}=\widehat{CKD}$. ##$type=three$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: 2016 Issue 470
2016 Issue 470
Mathematics & Youth