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

## $show=home 1. Find the maximum calue of positive integer$n$such that$2013$can be written as the sum of$n$compound numbers. How does the answer change if$2013$is replaced by$2014$. 2. Let$ABC$be a right triangle, right angle at$A$,$\widehat{B}=60^{0}$. Point$E$on side$AC$such that$\widehat{ABE}=20^{0}$. Point$K$on the half line$BE$such that$EK=BC$. Find the measure of the angle$\widehat{BCK}$. 3. Solve the inequality $\frac{x^{2}+8}{x+1}+\frac{x^{3}+8}{x^{2}+1}+\frac{x^{4}+8}{x^{3}+8}+\ldots+\frac{x^{101}+8}{x^{100}+1}\geq800.$ 4. The quadrilateral$ABCD$is inscribed in circle$(O)$where angle$\widehat{BAD}$is obtuse. The rays through$A$and perpendicular to$AD,AB$meet$CB,CD$at$P$and$Q$respectively.$PQ$intersects$BD$at$M$. Prove that$\widehat{MAC}=90^{0}$. 5. Solve the system of equations $\begin{cases} \sqrt{2x-3}-\sqrt{y} & =2x-6\\ x^{3}+y^{3}+7(x+y)xy & =8xy\sqrt{2(x^{2}+y^{2})} \end{cases}.$ 6. The positive real numbers$a,b,c$satisfy the equation$abc=1$. Prove the inequality $a^{3}+b^{3}+c^{3}+\frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{9}{2}.$ 7. Let$ABC$be a triangle.$D$is the midpoint of side$BC$and$M$is an arbitrary point on segment$BD$.$MEAF$is a parallellogram where vertex$E$lies on$AB$,$F$lies on$AC$,$MF$and$AD$intersect at$H$. The line through$B$and parallel to$EH$intersects$MF$at$K$;$AK$meets$BC$at$I$. Find the ratio$\dfrac{IB}{ID}$. 8. The sequence$\{v_{n}\}_{n}$satisfies $v_{1}=5,\quad v_{n+1}=v_{n}^{4}-4v_{n}^{2}+2.$ Find a closed formular for$v_{n}$. ##$type=three$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 444
2014 Issue 444
Mathematics & Youth