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

## $show=home 1. Let $$A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2011}-\frac{1}{2012},\quad B=\frac{1}{1007}+\frac{1}{1008}+\ldots+\frac{1}{2012}.$$ Compute the value of$\left(\dfrac{A}{B}\right)^{2012}$. 2. Let$f(x)$be a polynomial with integer coefficients such that$f(3) \cdot f(4)=5 .$Prove that$f(x)-6$does not have any integer solution. 3. Find all triple of integers$a, b, c$such that $$2^{a}+8 b^{2}-3^{c}=283.$$ 4. Given a triangle$A B C$,$B C=a$,$C A=b$,$A B=c$,$\widehat{A B C}=45^{\circ}$and$\widehat{A C B}=120^{\circ}$. Point$I$is taken on the opposite ray of$C B$such that$\widehat{A I B}=75^{\circ} .$Find the length of$A I$in term of$a$,$b$and$c$5. Point$K$lies on side$B C$of a triangle$A B C$. Prove that $$A K^{2}=A B \cdot A C - K B \cdot K C$$ if and only if$A B=A C$or$\widehat{B A K}=\widehat{C A K}$. 6. A non-isosceles triangle$A B C$has$B C=a$,$C A=b$,$A B=c$. Let$\left(A A_{1}, A A_{2}\right)$,$\left(B B_{1}, B B_{2}\right)$,$\left(C C_{1}, C C_{2}\right)$be the median and the altitude from vertices$A$,$B$and respectively. Prove that $$\frac{a^{2}}{b^{2}-c^{2}} \overline{A_{1} A_{2}}+\frac{b^{2}}{c^{2}-a^{2}} \overrightarrow{B_{1} B_{2}}+\frac{c^{2}}{a^{2}-b^{2}} \overrightarrow{C_{1} C_{2}}=\overrightarrow{0}$$ 7. Let$a, b, c \in(0 ; 1)$and $$a b+b c+c a+a+b+c=1+a b c.$$ Prove that $$\frac{1+a}{1+a^{2}}+\frac{1+b}{1+b^{2}}+\frac{1+c}{1+c^{2}} \leq \frac{3}{4}(3+\sqrt{3})$$ 8. Let$A B C$be an acute triangle with all angles greater than$45^{\circ}$. Prove that $$\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \leq 3(\sqrt{3}-1).$$ When does equality occur? 9. Two sequences$\left(a_{n}\right)$and$\left(b_{n}\right)$are defined inductively as follows $$a_{0}=3, b_{0}=-3,\quad a_{n}=3 a_{n-1}+2 b_{n-1},\,b_{n}=4_{n-1}+3 b_{n-1},\,\forall n \geq 1.$$ Find all natural numbers$n$such that$\displaystyle\prod_{k=0}^{n}\left(b_{k}^{2}+9\right)$is a perfect square. 10. Let$n$be a positive integer. How many strings of length$n: a_{1} a_{2} \ldots a_{n}$where$a_{i}$is chosen from$\{0,1,2, \ldots, 9\}(i=1,2, \ldots, n)$are there such that the number of occurrences of 0 is even? 11. Let$\left(u_{n}\right)$be a sequence defined by$u_{0}=a \in[0 ; 2), u_{n}=\dfrac{u_{n-1}^{2}-1}{n}$for all$n=1,2,3, \ldots$Find$\displaystyle\lim _{n \rightarrow+\infty}\left(u_{n} \sqrt{n}\right)$. 12. Let$A B C$be a triangle, inscribed in the circle$(O)$with altitudes$A D$,$B E$and$C F$.$A A^{\prime}$is a diameter of$(O)$.$A^{\prime} B$,$A^{\prime} C$intersect$A C$,$A B$at$M$,$N$respectively. Points$P$,$Q$are in$E F$, such that$P B$,$Q C$are perpendicular to$B C$. The line passing through$A$and orthogonal to$Q N$,$P M$cuts$(O)$at$X$,$Y$respectively. The tangents to circle$(O)$at$X$and$Y$meet at$J$. Prove that$J A^{\prime}$is perpendicular to$B C$. ##$type=three$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 422
2012 Issue 422
Mathematics & Youth