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

## $show=home 1. Let$a, b, c$be positive numbers such that$a^{3}+b^{3}=c^{3}$. Compare$a^{2007}+b^{2007}$and$c^{2007}$2. Let$A B C$be an isosceles triangle at$A$. Let$E$be an arbitrary point on the side$B C$. The line through$E$perpendicular to$A B$meets with the line through$C$perpendicular to$A C$at a point denoted by$D$. Let$K$be the midpoint of$B E$Find the measure angle$\widehat{A K D}$. 3. Find the least positive integer$n(n>1)$such that$\dfrac{1^{2}+2^{2}+\ldots+n^{2}}{n}$is a perfect square number. 4. Prove the following inequality $$\left(1+\frac{2 a}{b}\right)^{2}+\left(1+\frac{2 b}{c}\right)^{2}+\left(1+\frac{2 c}{a}\right)^{2} \geq \frac{9(a+b+c)^{2}}{a b+b c+c a}$$ where$a, b, c$are arbitrary positive real numbers. When does equality occur? 5. Solve the equation $$x^{3}-\sqrt{6+\sqrt{x+6}}=6.$$ 6. Let$A B C$be a right triangle at$A$. Draw the circle$(B)$with center at$B$and radius$B A$and draw a diameter$A D .$Choose two points$E$and$F$on the line$B C$such that$B$is their midpoint.$D E$and$D F$meets with$(B)$at$M$and$N$respectively. Prove that$M, N$and$C$are colinear. 7. Let$A B C$be an equilateral triangle with circumcircle$(O) .$Draw a circle$\left(O_{1}\right)$which is tangent to both side$B C$and arc$\widehat{B C}$. Construct the circles$\left(O_{2}\right)$,$\left(O_{3}\right)$similarly for the remaining sides$C A$and$A B .$Prove that$A O_{1}=B O_{2}=C O_{3}$iff (ie. if and only if) the circles$\left(O_{1}\right)$,$\left(O_{2}\right)$and$\left(O_{3}\right)$all have the same radius. (Notation:$(X)$is a circle with center at$X$.) 8. Let$\left(a_{n}\right)(n=0,1,2, \ldots)$be a sequence given by $$a_{0}=29,\, a_{1}=105,\,a_{2}=381,\, a_{n+3}=3 a_{n+2}+2 a_{n+1}+a_{n},\,\forall n=0,1,2, \ldots .$$ Prove that for each positive integer$m,$there exists a number$n$such that$a_{n}$,$a_{n+1}-1$,$a_{n+2}-2$are all divisible by$m$9. Let$f$be a continuous function on the interval [0,1] and satisfies the following properties $$f(0)=0,\,f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y \in [0,1].$$ Find$f\left(\dfrac{8}{23}\right)$. 10. Let$a, b, c$be non-negative real numbers with sum equal to$1 .$Prove that $$\sqrt{a+(b-c)^{2}}+\sqrt{b+(c-a)^{2}}+\sqrt{c+(a-b)^{2}} \geq \sqrt{3}$$ 11. Let$H$and$I$be, respectively, the orthorcenter and the incenter of an acute triangle$A B C$. The lines$A H$,$B H$and$C H$meet the circumcircles of triangles$HBC$,$HCA$and$HAB$at$A_{1}$,$B_{1}$,$C_{1}$respectively.$A I$,$B I$and$C I$meet the circumcircles of triangles$I B C$,$I C A$and$I A B$at$A_{2}$,$B_{2}$,$C_{2}$respectively. Prove that $$H A_{1} \cdot H B_{1} \cdot H C_{1}+64 R^{3} \geq 9 \cdot I A_{2} \cdot I B_{2} \cdot I C_{2}.$$ 12. Let$G$be the centroid of a tetrahedron$A B C D$. Let$A_{1}$,$B_{1}$,$C_{1}$,$D_{1}$be arbitrary points on$G A$,$G B$,$G C$and$G D$respectively. GA meets the plane$\left(B_{1} C_{1} D_{1}\right)$at$A_{2} .$Construct the points$B_{2}$,$C_{2}$,$D_{2}$similarly. Prove that $$3\left(\frac{G A}{G A_{1}}+\frac{G B}{G B_{1}}+\frac{G C}{G C_{1}}+\frac{G D}{G D_{1}}\right) = \frac{G A}{G A_{2}}+\frac{G B}{G B_{2}}+\frac{G C}{G C_{2}}+\frac{G D}{G D_{2}}$$ ##$type=three$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: 2007 Issue 358
2007 Issue 358
Mathematics & Youth