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

## $show=home 1. Find all natural numbers$N$such that$N$decreases by a factor of$1997$after truncating the last several digits. 2. Let$A B C$be a right triangle with right angle at$A$and$\widehat{A C B}=15^{\circ}$, Point$D$on edge$A C$such that the line passing through$D$and perpendicular to$B D$cuts$B C$at$E$and$D E=2 D A$. Find the measure of angle$A D B$. 3. Find all positive integers$n$such that$[A]=4951$where$A$is the sum of$n$terms $$A=\left(1+\frac{1}{2}\right)+\left(2+\frac{2}{2^{2}}\right)+\left(3+\frac{3}{2^{3}}\right)+\ldots+\left(n+\frac{n}{2^{n}}\right).$$ Here$[x]$denotes the largest integer not exceeding$x$4. Find the minimum value of the expression $$P=\frac{1+\sqrt{x}+\sqrt{y}+\sqrt{z}}{x y+y z+z x},$$ where$x, y, z$are positive numbers satisfying$x+y+z=3$5. Solve the equation $$x^{2}-2 x+7+\sqrt{x+3}=2 \sqrt{1+8 x}+\sqrt{1+\sqrt{1+8 x}}.$$ 6. Let$A B C$be a non-isosceles triangle with medians$A A^{\prime}$,$B B^{\prime}$and$C C^{\prime}$; and altitudes$A H$,$B F$and CK. Given that$C K=B B^{\prime}$,$B F=A A^{\prime}$. Determine the ratio$\dfrac{C C^{\prime}}{A H}$. 7.$a_{1}, a_{2}, \ldots, a_{n}(n \geq 3)$are positive numbers that $$\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{2}>\frac{3 n-1}{3}\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right).$$ Prove that for any triple$a_{i}, a_{j}, a_{k}$are three edge lengths of some triangle, where natural numbers$i, j,k$satisfying$0<i<j<k \leq n$. 8. The volume of a given parallelogrambased pyramid$S.ABCD$is$V$. Assume that plane$(P)$cuts$S A$,$S B$,$S C$,$S D$at$A^{\prime}$,$B^{\prime}$,$C^{\prime}$,$D^{\prime}$respectively such that $$\frac{S A}{S A^{\prime}}+\frac{S B}{S B^{\prime}}+\frac{S C}{S C^{\prime}}+\frac{S D}{S D^{\prime}}=8.$$ Denote the volume of the pyramid$S . A^{\prime} B^{\prime} C^{\prime}$by$V_{1}$and that of$S . A^{\prime} C^{\prime} D^{\prime}$by$V_{2}$. Prove the inequality $$\frac{1}{\sqrt{V_{1}}}+\frac{1}{\sqrt{V_{2}}} \leq \frac{4 \sqrt{2}}{\sqrt{V}}.$$ 9. Write$2012^{2013}$as a sum of$2013$positive interger$a_{1}, a_{2}, a_{3}, \ldots, a_{2013} ;$and let $$T=a_{1}^{13}+a_{2}^{13}+a_{3}^{13}+\ldots+a_{2013}^{13}.$$ Prove that$T+2012^{2013}$is not a perfect square. 10. The incircle$(I)$of a triangle$A B C$touches the edges$B C$,$C A$,$A B$at$D$,$E$,$F$, respectively.$M$is the intersection of$B C$and the internal angle bisector of angle$B I C$,$N$is the intersection of$E F$and the internal angle bisector of angle$E D F$. Prove that$A$,$M$,$N$are collinear. 11. If$p(x)$and$q(x)$are polynomials with integer coefficients, write$p(x) \equiv q(x) \pmod 2$if the coefficients of$p(x)-q(x)$are all even. A sequence of polynomials$p_{n}(x)$is such that$p_{1}(x)=p_{2}(x)=1$and $$p_{n+2}(x)=p_{n+1}(x)+x p_{n}(x),\,\forall n \geq 1.$$ Prove that$p_{2^{n}}(x) \equiv 1\pmod 2, \forall n \in \mathbb{N}$. 12. Let$A B C$be an acute triangle. Prove the inequality $$\frac{\cos B \cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos \frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq \frac{3}{4}$$ ##$type=three$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: 2012 Issue 425
2012 Issue 425
Mathematics & Youth