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## $show=home 1. Determine all triple of prime numbers$a,b,c$(not necessarily distinct) such that $abc<ab+bc+ca.$ 2. Let$ABC$be a right triangle, with right angle at$A$and$AH$is the altitude from$A$,$\widehat{ACB}=30^{0}$. Construct an equilateral triangle$ACD$($D$and$B$are in opposite side$AC$). K is the foot of the perpendicular line from$H$onto$AC$. The line through$H$and parallel to$AD$meets$AB$at$M$. Prove the points$D,K,M$are colinear. 3. Consider a$6\times6$board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black. 4. Let$a,b,c$be real numbers in the interval$[1,2]$. Prove the inequality $a^{2}+b^{2}+c^{2}+3\sqrt{(abc)^{2}}\geq2(ab+bc+ca).$ 5. Let$ABC$be a non-right triangle ($AB<AC$) with altitude$AH$.$E,F$are the orthogonal projection of point$H$onto$AB$and$AC$respectively.$EF$meets$BC$at$D$. Draw a semicircle with diameter$CD$on the half-plane containing$A$with edge$CD$. The line through$B$and perpendicular to$CD$meets the semicircle at$K$. Prove that$DK$is tangent to the circumcircle of triangle$KEF$. 6. Given that the equation $ax^{3}-x^{2}+ax-b=0\quad(a\ne0,\,b\ne0)$ has three positive real roots. Determine the greatest value of the following expression $P=\frac{11a^{2}-3\sqrt{3}ab-\frac{1}{3}}{9b-10(\sqrt{3}a-1)}.$ 7. Solve the following system of equations $\begin{cases} \sqrt{x-\frac{1}{4}}+\sqrt{y-\frac{1}{4}} & =\sqrt{3}\\ \sqrt{y-\frac{1}{16}}+\sqrt{z-\frac{1}{16}} & =\sqrt{3}\\ \sqrt{z-\frac{9}{16}}+\sqrt{x-\frac{9}{16}} & =\sqrt{3}\end{cases}.$ 8. Let$a,b$be real constants such that$ab>0$. Let$\{u_{n}\}$be a sequence where$n=1,2,3,\ldots$given by $u_{1}=a,\quad u_{n+1}=u_{n}+bu_{n}^{2},\,\forall n\in\mathbb{N}^{*}.$ Determine the limit $\lim_{n\to\infty}\left(\frac{u_{1}}{u_{2}}+\frac{u_{2}}{u_{3}}+\ldots+\frac{u_{n}}{u_{n+1}}\right).$ 9. Find all positive integers$k$with the property that there exists a polynomial$f(x)$with integer coefficients of degree greater than 1 such that for all prime numbers$p$and natural numbers$a,b$if$p$divides$(ab-k)$then it also divides$(f(a)f(b)-k)$. 10. Given$a_{i}\in[0,\alpha]$($i=\overline{1,n}$), ($\alpha>0$). Prove the inequality $\prod_{i=1}^{n}(\alpha-a_{i})\leq\alpha^{n}\left(1-\sum_{i=1}^{n}\frac{a_{i}}{S_{i}+\alpha}\right)$ where${\displaystyle S_{i}=\sum_{k=1}^{n}a_{k}-a_{i}}$for all$i=\overline{1,n}$. 11. Point$O$is in the interior of triangle$ABC$. The ray$Ox$parallel to$AB$meets$BC$at$D$, ray$Oy$parallel to$BC$meets$CA$at$E$, ray$Oz$parallel to$CA$meets$AB$at$F$. Prove that a)$3S_{DEF}\leq S_{ABC}$. b)$OD\cdot OE\cdot OF\leq27AB\cdot BC\cdot CA$. 12. The circle$(O)$and$(O')$meet at points$A,B$. Point$C$is fixed on$(O)$and point$D$is fixed on$(O')$. A moving point$P$is on the opposite ray of ray$BA$. The circumcircles of traingles$PBC$,$PDB$intersect$BD$,$BC$at seconde points$E,F$respectively. Prove that the midpoint of line segment$EF$is always on a fixed segment$EF$is always on a fixed straight line. ##$type=three$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: 2013 Issue 428
2013 Issue 428
Mathematics & Youth