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## $show=home 1. Let$a$,$b$,$c$be integers such that $$A=\frac{a^{2}+b^{2}+c^{2}-a b-b c-c a}{2}$$ is a perfect square. Prove that$a=b=c$. 2. Let$A B C$be a triangle in which$\widehat{B A C}=75^{\circ}$. Given that the altitude$A H$has length$A H=\dfrac{B C}{2}$. Prove that$A B C$is an isosceles triangle. 3. Let$x$,$y$be two nonnegative integers where$x>1$and$2 x^{2}-1=y^{15}$. Prove that$x$is divisible by$15$. 4. Let$A B C D$be a parallelogram. The ray$D x$from$D$is perpendicular to$D C$and lies on the half-plane divided by$C D$which does not contains$B$. Choose a point$E$on$D x$such that$D E=D C$. Draw an isosceles right triangle$B E F$(right angle at$F$) such that$F$and$D$are on the same half-plane divided by$B C \cdot E H$is the altitude onto$B C$. Prove that$FD$and$H$are collinear. 5. Let$a$,$b$,$c$be three sides of a right triangle,$a$is the hypotenuse. Prove that the equation$-a x^{2}+b x+c=0$has two distinct roots$x_{1}$,$x_{2}$such that$-\sqrt{2}<x_{1}<x_{2}<\sqrt{2}$. 6. Consider a convex polygon with$2009$vertices. A scissor is used to cut along all of its diagonals, thereby dividing the original polygon into smaller convex polygons. How many vertices are there of a resulting polygon with the greatest number of edges? 7. Determine the smallest value of the expression $$T=\frac{a b+b c}{a^{2}-b^{2}+c^{2}}+\frac{b c+c a}{b^{2}-c^{2}+a^{2}}+\frac{c a+a b}{c^{2}-a^{2}+b^{2}}$$ where$a$,$b$,$c$are three sides of a triangle$A B C$and$a b c=1$. 8. Let$A B C D . A_{1} B_{1} C_{1} D_{1}$be a cube. On its three skew edges, choose three points$M$,$N$,$P$. Determine the positions of the points$M$,$N$,$P$such that the triangle$M N P$has a) The smallest perimeter possible. b) The smallest area possible. ##$type=three$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: 2009 Issue 383
2009 Issue 383
Mathematics & Youth