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

## $show=home 1. Consider$n$consecutive points$A_{1}, A_{2}, A_{3}, \ldots, A_{n}$on the same line such that $$A_{1} A_{2}=A_{2} A_{3}=A_{3} A_{4}=\ldots=A_{n-1} A_{n}.$$ Find$n,$given that there are exactly$2025$segments on that line whose midpoints are one of these$n$points. 2. Let$A B C$be a right triangle with right angle at$A$. On the halfplane divided by$B C$which does not contain$A,$choose the points$D$,$E$such that$B D$is orthogonal with$B A$and$B D=B A$,$B E$is orthogonal with$B C$and$B E=B C$. Denote by$M$the midpoint of$C E .$Prove that$A$,$D$and$M$are colinear. 3. Find all positive integers$x$,$y$and$z$such that $$2 x y-1=z(x-1)(y-1).$$ 4. Solve for$x$$$4 x-x^{2}=3 \sqrt{4-3 \sqrt{10-3 x}}.$$ 5. Let$A B C$be a right triangle with right angle at$A$and let$A D$be the angle bisector at$A .$Denote by$M$and$N$the bases of the altitudes from$D$onto$A B$and$A C$respectively.$B N$meets$C M$at$K$and$A K$meets$D M$at$I$. Find the measure of angle$B I D$. 6. Let $$f(x)=2009 x^{5}-x^{4}-x^{3}-x^{2}-2006 x+1 .$$ Prove that$f(n)$,$f(f(n))$,$f(f(f(n)))$are pairwise coprime for any positive integer$n$. 7. Find$a$,$b$such that$\max _{0 \leq x \leq 16}|\sqrt{x}+a x+b|$is smallest possible. Find this minimum value. 8. The incircle of a triangle$A B C$touches$B C$,$C A$and$A B$at$A^{\prime}$,$B^{\prime},$and$C^{\prime}$respectively. Prove that $$A B^{\prime 2}+B C^{\prime 2}+C A^{\prime 2} \geq A B^{\prime} . B^{\prime} C^{\prime}+B C^{\prime} \cdot C^{\prime} A^{\prime}+C A^{\prime} . A^{\prime} B^{\prime} \geq B^{\prime} C^{\prime 2}+C^{\prime} A^{\prime 2}+A^{\prime} B^{\prime 2}.$$ When does equality occur? 9. Let$k$be a positive integer. Write$k$as a product of prime numbers (not necessarily distinct, for instance,$k=18=3.3 .2)$and let$T(k)$be the sum of all factors in the factorization above. Find the largest constant$C$such that$T(k) \geq C \ln k$for all positive integer$k$. 10. Let$a, b, c$be non-negative real numbers whose sum of squares equal$3 .$Find the maximum value of the following expression $$P=a b^{2}+b c^{2}+c a^{2}-a b c$$ 11. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a sequence, determined by the following recursive formula $$x_{1}=\frac{1}{2},\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2007}-x_{n}^{2008},\,\forall n \in \mathbb{N}^{*}.$$ Find the limit$\displaystyle \lim_{n \rightarrow+\infty} n x_{n}$. 12. Let$A B C D$be a tetrahedron whose altitudes are concurrent. Denote by$R$the radius of its circumcircle; by$h_{1}$,$h_{2}$,$h_{3}$,$h_{4}$the lengths of the altitudes corresponding to the vertices$A$,$B$,$C$,$D$respectively; and by$R_{1}$,$R_{2},R_{3}$,$R_{4}$the circumcircles's radius of the opposite faces of the vertices$A$,$B$,$C$,$D$respectively. Prove the following inequality $$\frac{1}{h_1+2 \sqrt{2} R_{1}}+\frac{1}{h_{2}+2 \sqrt{2} R_{2}}+\frac{1}{h_{3}+2 \sqrt{2} R_{3}}+\frac{1}{h_{4}+2 \sqrt{2} R_{4}} \geq \frac{1}{R}.$$ When does equality occur? ##$type=three$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: 2008 Issue 368
2008 Issue 368
Mathematics & Youth