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

## $show=home 1. Do there exist two natural numbers$a, b$such that $$(3 a+2 b)(7 a+3 b)-4=\overline{22} * 12 * 2011 ?$$ 2. Equilateral triangles$A B E$and$B C F$are constructed outside triangle$A B C .$Let$G$be the centroid of triangle$A B E$and$I$be the midpoint of$A C .$Find the measure of angle$G I F$. 3. Find the smallest positive integer$n$such that$2^{n}-1$is divisible by$2011$. 4. Prove that for all integers$k$, the equation $$x^{4}-2010 x^{3}+(2009+k) x^{2}-2007 x+k=0$$ does not have two distinct integer roots. 5. From a point$M$outside the cycle$(O),$draw the tangents$M A, M B$and the secant$M C D$to$(O), C$lies between$M$and$D .A B$cuts$C D$at$N .$Prove that $$\frac{1}{M D}+\frac{1}{N D}=\frac{2}{C D}$$ 6. Let$A B C$be a right triangle, right angle at$A,$satisfying$A B+\sqrt{3} A C=2 B C$. Find the position of point$M$such that $$4 \sqrt{3} \cdot M A+3 \sqrt{7} \cdot M B+\sqrt{39} \cdot M C$$ is smallest possible. 7. Solve the equation $$\log _{3}\left(7^{x}+2\right)=\log _{5}\left(6^{x}+19\right)$$ 8. Let$A B C$be a triangle satisfying $$\tan \frac{A}{2} \tan \frac{B}{2}=\frac{1}{2}.$$ Prove that$A B C$is a right triangle iff $$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}=\frac{1}{10}.$$ 9. Let$A B C$be a triangle inscribed the circle$(O ; R), M$is a point not on the circle respectively. Let$r$,$r_{1}$be respectively the radii of the incircles of triangles$A B C$and$A_{1} B_{1} C_{1}$. Prove that $$\left|R^{2}-O M^{2}\right| \geq 4 r \cdot r_{1}$$ 10. Find the greatest positive constant$k$satisfying the inequality $$\frac{k}{a^{3}+b^{3}}+\frac{1}{a^{3}}+\frac{1}{b^{3}} \geq \frac{16+4 k}{(a+b)^{3}}.$$ 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f(x f(y)+y)+f(x y+x)=f(x+y)+2 x y.$$ 12. For each positive integer$n$consider a function$f_{n}$in$\mathbb{R}$defined by $$f_{n}(x)=\sum_{i=1}^{2 n} x^{i}+1.$$ Prove the following statements a)$f_{n}$obtains its minimum value at a unique point$x_{n},$for each positive integer$n .$Put$S_{n}=f_{n}\left(x_{n}\right)$. b)$S_{n}>\dfrac{1}{2}$for all$n \in \mathbb{N}^{*} .$Moreover,$\dfrac{1}{2}$is the best constant possible in the sense that there does not exist any real number$a>\dfrac{1}{2}$such that$S_{n}>a$for all$n \in \mathbb{N}^{*}$. c) The sequence$\left(S_{n}\right)(n=1,2, \ldots)$is decreasing and$\displaystyle\lim_{n\to\infty} S_{n}=\dfrac{1}{2}$. d)$\displaystyle\lim_{n\to\infty} x_{n}=-1$. ##$type=three$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: 2011 Issue 414
2011 Issue 414
Mathematics & Youth