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

## $show=home 1. Find$2018$numbers so that each of them is the square of the sum of all remaining numbers. 2. Find the following sum $$S=(1+2.3+3.5+\ldots+101.201) +\left(1^{2}+2^{2}+3^{2}+\ldots+100^{2}\right).$$ 3. Find all pairs of positive integers$(m, n)$such that $$n^{3}-5 n+10=2^{m}.$$ 4. Given a triangle$A B C$with$B C=a$,$A C=b$,$A B=c$and$3 \hat{B}+2 \hat{C}=180^{\circ}$. Prove that $$b+c \leq \dfrac{5}{4} a.$$ 5. Solve the system of equations $$\begin{cases}x^{2}-y^{2}+\sqrt{x}-y+2&=0 \\ x+8 y+4 \sqrt{x}-8 \sqrt{y}-4 \sqrt{x y} &=0\end{cases}$$ 6. Given three positive numbers$a, b, c$satisfying$a+b+c=3 .$Show that $$\frac{1}{(a+b)^{2}+c^{2}}+\frac{1}{(b+c)^{2}+a^{2}}+\frac{1}{(c+a)^{2}+b^{2}} \leq \frac{3}{5}$$ 7. Solve the system of equations $$\begin{cases}x-1&=\sqrt{9+12 y-6 y^{2}} \\ y-1&=\sqrt{9+12 x-6 x^{2}}\end{cases}.$$ 8. Given a right prism with equilateral bases$A B C . A^{\prime} B^{\prime} C$. Let$\alpha$be the angle between the line$B C$and the plane$\left(A^{\prime} B C\right)$. Prove that$\sin \alpha \leq 2 \sqrt{3}-3$9. Given positive numbers$a$,$b$. Show that $$\frac{1}{2}\left[1-\frac{\min (a, b)}{\max (a, b)}\right]^{2} \leq \frac{b-a}{a}-\ln b+\ln a \leq \frac{1}{2}\left[\frac{\max (a, b)}{\min (a, b)}-1\right]^{2}.$$ 10. Find the maximal positive number$k$so that the following inequality $$a^{2}+b^{2}+c^{2}+k(a+b+c) \geq 3+k(a b+b c+c a)$$ holds true for all positive numbers$a, b, c$. 11. Given the sequence$\left(x_{n}\right)$$$x_{2}=x_{3}=1,\quad (n+1)(n-2) x_{n+1}=\left(n^{3}-n^{2}-n\right) x_{n}-(n-1)^{3} x_{n-1},\, \forall n \geq 3.$$ Find all indices$n$so that$x_{n}$is an integer. 12. Given a cyclic quadrilateral$A B C D .$Let$K$be the intersection between$A C$and$B D .$Let$M$,$N$,$P$and$Q$respectively be the perpendicular projection of$K$on$A B$,$B C$,$C D$and$D A$. And then, let$X$,$Y$,$Z$and$T$respectively be the perpendicular projection of$K$on$M N$,$N P$,$P Q$,$Q M$. Prove that$A X C Z$and$B Y D T$have equal areas. ##$type=three$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: 2018 Issue 493
2018 Issue 493
Mathematics & Youth