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

## $show=home 1. a) Find all natural number, each of which can be written as the sum of two relatively prime relatively prime integers greater than$1$. b) Find all natural numbers, each of which can be written as the sum of three pairwise 2. Let$A B C$be a triangle with acute angle$\widehat{A B C}$. integers greater than$1 .$Let$K$be a point on the side$A B$, and$H$be its orthogonal projection on the line$B C .$A ray$B x$cuts the segment$K H$at$E$and cuts the line passing through$K$parallel to$B C$at$F .$Prove that$\widehat{A B C}=3 \widehat{C B F}$when and only when$E F=2 B K$. 3. Find all natural numbers n such that the product of the digits of$n$is equal to $$(n-86)^{2}\left(n^{2}-85 n+40\right)$$ 4. Prove that$a b+b c+c a<\sqrt{3} d^{2}$where$a, b, c, d$are real numbers satisfying the following conditions •$0<a, b, c<d$•$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}-\sqrt{\dfrac{1}{a^{2}}+\dfrac{1}{b^{2}}+\dfrac{1}{c^{2}}}=\dfrac{2}{d}$. 5. Solve the equation $$x^{4}+2 x^{3}+2 x^{2}-2 x+1=\left(x^{3}+x\right) \sqrt{\frac{1-x^{2}}{x}}$$ 6. Let$A B C D$be a square with sides equal to a. On the side$A D,$take the point$M$such that$A M=3 M D .$Draw the ray$B x$cutting the side$C D$at$I$such that$\widehat{A B M}=\widehat{M B I}$. The angle bisector of$\widehat{C B I}$cuts the side$C D$at$N$. Calculate the area of triangle$B M N$. 7. Let$B C$be a fixed chord (which is not a diameter of a circle. On the major arc$B C$of the circle, take a point$A$not coinciding with$B, C .$Let$H$be the orthocenter of triangle$A B C .$The second points of intersection of the line$B C$with the circumcircles of triangles$A B H$and$A C H$are$E$and$F$respectively. The line$E H$cuts the side$A C$at$M$and the line$F H$cuts the side$A B$at$N$. Determine the position of$A$so that the measure of the segment$M N$attains its least value. 8. How many are there natural 9-digit numbers with 3 distinct odd digits, 3 distinct even digits and every even digit in each number appears exactly two times (in this number 9. For every positive integer$n$, consider the function$f_{n}$defined on$\mathbb{R}$by $$f_{n}(x)=x^{2 n}+x^{2 n-1}+\ldots+x^{2}+x+1.$$ a) Prove that the function$f_{n}$attains its least value at a unique value$x_{n}$of$x$. b) Let$S_{n}$be the least value of$f_{n}$. Prove that •$S_{n}>\dfrac{1}{2}$for all$n$and there does not exist a real number$a>\dfrac{1}{2}$such that$S_{n}>a$for all$n$•$\left(S_{n}\right)(n=1,2, \ldots)$is a decreasing sequence and$\lim S_{n}=\dfrac{1}{2}$•$\lim x_{n}=-1$10. Let $$A=\sqrt{x^{2}+\sqrt{4 x^{2}+\sqrt{16 x^{2}+\sqrt{100 x^{2}+39 x+\sqrt{3}}}}}.$$ Find the greatest integer not exceeding$A$when$x=20062007$11. Let$A B C$be a triangle with$B C=a$,$C A=b$,$A B=c,$with inradius$r$and with incenter$I .$Let$A_{1}, B_{1}, C_{1}$be respectively the touching points of the sides$B C$,$C A$,$A B$with the incircle. The rays$I A$,$I B$,$I C$cut the incircle respectively at$A_{2}$,$B_{2}$,$C_{2}$. Let$B_{i}C_{i}=a_{i}$,$C_{i} A_{i}=b_{i}A_{i} B_{i}=c_{i}(i=1,2)$. Prove that $$\frac{a_{2}^{3} b_{2}^{3} c_{2}^{3}}{a_{1}^{2} b_{1}^{2} c_{1}^{2}} \geq \frac{216 r^{6}}{a b c}.$$ When does equality occur? 12. Let$O A B C$be a tetrahedron with $$\widehat{A O B}+\widehat{B O C}+\widehat{C O A}=180^{\circ},$$$O A_{1}$,$O B_{1}$,$O C_{1}$are internal angle bisectors respectively of the triangles$O B C$,$O C A$,$O A B$,$O A_{2}$,$O B_{2}$,$O C_{2}$are internal angle bisectors respectively of the triangles$O A A_{1}$,$O B B_{1}$,$O C C_{1}$. Prove that $$\left(\frac{A A_{1}}{A_{2} A_{1}}\right)^{2}+\left(\frac{B B_{1}}{B_{2} B_{1}}\right)^{2}+\left(\frac{C C_{1}}{C_{2} C_{1}}\right)^{2} \geq(2+\sqrt{3})^{2}.$$ 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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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2006 Issue 354
2006 Issue 354
Mathematics & Youth