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

## $show=home 1. Find a four digits perfect square, given that all four digits are distinct, and if these digits are written in reverse order, the result is also a perfect square, and is divisible by the original number. 2. Determine all possible choices of three integers$x$,$y$and$z$such that $$x^{2}+y^{2}+z^{2}+3<x y+3 y+2 z.$$ 3. Let$A B C$be a triangle where the length of the altitudes$A H$is$6 \mathrm{cm}, B H$is$3 \mathrm{cm}$and the measure of angle$C A H$is three times the measure of angle$B A H$. Find the area of this triangle. 4. Find the greatest value of the expression $$M=\frac{232 y^{3}-x^{3}}{2 x y+24 y^{2}}+\frac{783 z^{3}-8 y^{3}}{6 y z+54 z^{2}}+\frac{29 x^{3}-27 z^{3}}{3 x z+6 x^{2}}$$ where$x, y$and$z$are positive numbers satisfying the condition$x+2 y+3 z=\dfrac{1}{4}$5. Let$A B C$be a right triangle with right angle at$A$and$A B<A C$. Let$H$be the projection of$A$onto$B C$, let$M$be the point reflection of$H$across$A B$.$M C$meets the circumcircle of triangle$A B H$at$P(P \neq M)$,$H P$meets the circumcircle of triangle$A P C$at$N(N \neq P)$. Let$E$and$K$be respectively the intersections of$A B$and$B C$with the circumcircle of triangle$A P C(E \neq A, K \neq C)$. Prove that a)$E N$is parallel to$B C$. b)$H$is the midpoint of$B K$. 6. Find the interger part of$A$, where $$A=\sqrt{2}+\sqrt{\frac{3}{2}}+\sqrt{\frac{4}{3}}+\ldots+2010 \sqrt{\frac{2010}{2009}}$$ 7. Find the least value of the following expression $$A=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}+\frac{z}{1+t^{2}}+\frac{t}{1+z^{2}}$$ where$x, y, z, t$are nonnegative real numbers satisfying$x+y+z+t=k$($k$is a given positive number). 8. Let$A B C$be a given triangle and$M$is a point which is not on its sides. Prove that $$P_{A /(M C B)}=P_{B /(M C A)}=P_{A /(M A B)}$$ if and only if$M$is the centroid of triangle$A B C$. ($P_{T /(X Y Z))}$is the power of the point$T$with respect to the circle through$X$,$Y$,$Z$.) 9. Let$A B C$be a triangle. A circle intersects with the sides$B C$,$C A$and$A B$at pairs of two points$(M, N)$;$(P, Q)$and$(S, T)$respectively, where$M$lies between$B$and$N$;$P$lies between$C$and$Q,$and$S$lies between$A$and$T$. Let$K$,$H$,$L$be respectively the intersections of$S N$and$Q M$;$Q M$and$T P$;$T P$and$S N$. Prove that the lines$A K$,$B H$,$C L$are concurrent. 10. Let$\left(a_{n}\right)$be a sequence of numbers such that $$a_{0}=10,\quad \left(6-a_{n}\right)\left(16+a_{n-1}\right)=96,\, n=0,1,2, \ldots$$ Find the sum $$S=\frac{1}{a_{0}}+\frac{1}{a_{1}}+\ldots+\frac{1}{a_{2010}}$$ 11. Find all functions$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that the following equality holds $$f(x+y)+f(x y)=x+y+x y,\,\forall x, y \in \mathbb{R}^{+}.$$ 12. Let$A B C$be a triangle. Prove that $$\cos A \cos B \cos C+8 \sqrt{3} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \geq 24 \cos ^{2} \frac{A}{2} \cos ^{2} \frac{B}{2} \cos ^{2} \frac{C}{2}-1$$ ##$type=three$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: 2010 Issue 399
2010 Issue 399
Mathematics & Youth