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$show=home 1. The number$s=\overline{3\ldots3}^{2}+\overline{5\ldots54\ldots4}^{2}$, written in decimal system, consist of$n+1$digits$3$,$n-1$digits$5$and$n$digits$4$. Given that$s=r^{2}$, find the value of$r$. 2. Let$AM$be the median of triangle$ABC$. On the half-plane containing$C$created by the side$AB$, draw line segment$AE$perpendicular to$AB$such that$AB=AE$. On the half-plane containing$B$created by$AC$, draw$AF$perpendiclar to$AC$such that$AF=AC$. Prove that$EF=2AM$and$EF\perp AM$. 3. Consider$n$positive integers$a_{1},a_{2},\ldots a_{n}$($n>1$) satisfying $a_{1}+a_{2}+\ldots+a_{n}=a_{1}a_{2}\ldots a_{n}.$ a) Prove that for any given value of$n$, the above equation always has solution. b) Determine all values of$n$such that the equation$a_{1}<a_{2}<\ldots<a_{n}$. 4. The numbers$a,b,c$satisfy $ab+bc+ca=2013abc,\quad2013(a+b+c)=1.$ Find the sum$A=a^{2013}+b^{2013}+c^{2013}$. 5. Prove that if in a trapezium$ABCD$($AB||CD$),$AC+CB=AD+DB$, then$ABCD$is an isosceles trapezium. 6. Solve the inequality on$\mathbb{R}$$(\sqrt{13}-\sqrt{2x^{2}-2x+5}-\sqrt{2x^{2}-4x+4})(x^{6}-x^{3}+x^{2}-x+1)\geq0.$ 7. The three angles of an acute triangle$ABC$are such that$A>\dfrac{\pi}{4}$,$B>\dfrac{\pi}{4}$,$C>\dfrac{\pi}{4}$. Determine the smallese value of the expression $\frac{\tan A-2}{\tan^{2}C}+\frac{\tan B-2}{\tan^{2}A}+\frac{\tan C-2}{\tan^{2}B}.$ 8. Let$S.ABCD$be a pyramid inscribed in a sphere centred at$O$, and$AB=a$,$CD=b$. Draw parallellograms$ADKB$and$SDHC$. Determine the ratio$\dfrac{HK}{EF}$in terms of$a$and$b$, where$E$is the point of intersection of$AD$and$BC$, and$F$is the point of intersecion of$AC$and$BD$. 9. How many positive integers$n$are there such that$n$has$2013$digits in decimal number system and$\dfrac{n}{7}$is a positive integer with$2013$odd digits?. 10. Find all funtions$f:\mathbb{R}\to\mathbb{R}$,$g:\mathbb{R}\to\mathbb{R}$such that the following two conditions are satisfied a)$f(x)-2g(x)=g(y)+4y$, for all$x,y\in\mathbb{R}$; b)$f(x)g(x)\leq33x^{2}$, for all$x\in\mathbb{R}$. 11. Find all polynomials$T(x,y)$such that $T(x,y)T(z,t)=T(xz+yt,xt+yz)$ for all$x,y,z,t\in\mathbb{R}$. 12. Let$ABCD$be a cyclic quadrilateral. The circle whose diameter is$AB$meets$CA$,$CB$,$DA$and$DB$at$E,F,I$and$J$respectively (all differ from$A$and$B$). Prove that the angle-bisector of an angle between$EF$and$IJ$is perpendicular to the line$CD.$$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 435
2013 Issue 435
Mathematics & Youth