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

## $show=home 1. Write$2005^{2006}$as a sum of natural numbers, then calculate the sum of all the digits occurred in these summands. Can one obtain either$2006$or$2007$in this way? Why? 2. Let$A B C$be an isosceles right triangle, right angle at$A$. Choose a point$D$in the half-plane on the side of$A B$that does not contain$C$such that$D A B$is also an isosceles right triangle, with right angle at$D$. Let$E$be a point (differs from$A$) on$A D .$The perpendicular line with$B E$through$E$intersects with$A C$at$F .$Prove that$E F=E B$. 3. Find the maximum and minimum values of the following expression $$\frac{x-y}{x^{4}+y^{4}+6}.$$ 4. Solve the equation $$\sqrt{2}\left(x^{2}+8\right)=5 \sqrt{x^{3}+8}.$$ 5. The two diagonals$A C$and$B D$of an inscribed quadrilateral$A B C D$intersect at$O$. The circumcircles$\left(S_{1}\right)$of$A B O$and$\left(S_{2}\right)$of$\mathrm{CDO}$meet at$\mathrm{O}$and$K$. Through$\mathrm{O},$draw parallel lines with$A B$and$C D$; they meet with$\left(S_{1}\right)$and$\left(S_{2}\right)$at$N$and$M,$respectively. Let$P$and$Q$be two points on$O N$and$O M$respectively such that$\dfrac{O P}{P N}=\dfrac{M Q}{Q}$. Prove that$O$,$K$,$P$and$Q$lie on the same circle. 6. There are nine cards in a box, labelled from$1$to$9$. How many cards should be taken from the box so that the probability of getting a card whose label is a multiple of 4 will be greater than$\dfrac{5}{6} ?$7. Let$x_{1}, x_{2}, \ldots, x_{n}$be$n$arbitrary real numbers chosen in a given closed interval$[a ; b]$. Prove the inequality $$\sum_{i=1}^{n}\left(x_{i}-\frac{a+b}{2}\right)^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\frac{1}{n} \sum_{i=1}^{n} x_{i}\right)^{2}.$$ When does equality occur? 8. Let$H$be the orthocenter of an acute triangle$A B C$whose edges are$B C=a, C A=b$, and$A B=c .$Prove that $$\frac{H A^{2}+H B^{2}+H C^{2}}{a^{2}+b^{2}+c^{2}} \leq(\cot A \cdot \cot B)^{2}+(\cot B \cdot \cot C)^{2}+(\cot C \cdot \cot A)^{2}.$$ 9. Let$p$be a prime number, greater than$3$and$k=\left[\dfrac{2 p}{3}\right]$. (The notation$[x]$denote the largest integer which is not exceeding$x$.) Prove that$\displaystyle \sum_{i=1}^{k} C_{p}^{i}$is a multiple of$p^{2}$. 10. Find the measures of the angles of triangle$A B C$, such that $$\cos \frac{5 A}{2}+\cos \frac{5 B}{2}+\cos \frac{5 C}{2}=\frac{3 \sqrt{3}}{2}.$$ 11. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a boundedabove sequence such that $$x_{n+2} \geq \frac{1}{4} x_{n+1}+\frac{3}{4} x_{n},\,\forall n=1,2, \ldots$$ Prove that this sequence has limit. 12. Let$\alpha, \alpha^{\prime} ; \beta, \beta^{\prime}$and$\gamma, \gamma^{\prime}$be the dihedral angles, opposite to the edges$B C$,$DA$,$CA$,$DB$and$A B$,$D C$respectively, of a tetrahedron$A B C D$. Prove the inequality $$\sin \alpha+\sin \alpha^{\prime}+\sin \beta+\sin \beta^{\prime}+\sin \gamma+\sin \gamma^{\prime} \leq 4 \sqrt{2}.$$ 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: 2007 Issue 365
2007 Issue 365
Mathematics & Youth