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

## $show=home 1. Find all natural numbers$m, n$and primes$p$satisfying each of the following equalities a)$p^{m}+p^{n}=p^{m+n}$. b)$p^{m}+p^{n}=p^{m n}$2. Given a triangle$A B C$. Let$M$,$N$and$P$respectively be the midpoints of$A B$,$A C$and$B C$. Let$O$be the intersection between$C M$and$P N$,$I$be the intersection between$A O$and$B C$and$D$be the intersection between$M I$and$A C$. Show that$A I$,$B D$,$M P$are concurrent. 3. Solve the equation $$\frac{1-4 \sqrt{x}}{2 x+1}=\frac{2 x}{x^{2}+1}-2$$ 4. Given a right triangle$A B C$with the right angle$A$. Let$A H$be the altitude. On the opposite ray of the ray$H A$pick an arbitrary point$D(D \neq H)$. Through$D$draw the line perpendicular to$B D .$That line intersects$A C$at$E$. Let$K$be the perpendicular projection of$E$on$A H .$Show that$D K$has a fixed length when$D$varies. 5. Given real numbers$x$,$y$such that$x^{2}+y^{2}=1$. Find the minimum and maximum values of the expression $$T=\sqrt{4+5 x}+\sqrt{4+5 y}$$ 6. Find conditions on$a$,$b$besides$a>b \geq-1$so that the system $$\begin{cases} x^{2} &=(a-y)(a+y+2) \\ y^{2} &=(b-x)(b+x+2)\end{cases}$$ has unique solution. 7. Given positive real numbers$a, b, c$. Prove that $$\frac{b(2 a-b)}{a(b+c)}+\frac{c(2 b-c)}{b(c+a)}+\frac{a(2 c-a)}{c(a+b)} \leq \frac{3}{2}$$ 8. Let$A$,$B$and$C$denote the angles of a triangle$A B C\left(A, B, C \neq \frac{\pi}{2}\right)$. Show that $$\frac{\sin A}{\tan B}+\frac{\sin B}{\tan C}+\frac{\sin C}{\tan A} \geq \frac{3}{2}$$ 9. Find all pairs of positive integers$(x ; y)$satisfying $$x^{3}+y^{3}=x^{2}+72 x y+y^{2}$$ 10. For any integer$n$, show that $$a_{n}=\frac{3+\sqrt{5}}{10}\left(\frac{7+3 \sqrt{5}}{2}\right)^{n}+\frac{3-\sqrt{5}}{10}\left(\frac{7-3 \sqrt{5}}{2}\right)^{n}+\frac{2}{5}$$ is a perfect square. 11. A class has$n$students attending$n-1$clubs. Show that we can choose a group of at least two students so that, for each club, there are an even number of students in that group attend it. 12. Given a square$A B C D$and$P$is an arbitrary point on the side$A B$. Let$\left(I_{1}\right)$,$\left(I_{2}\right)$respectively be the incircles of$A D P$,$C B P$. Assume that$D I_{1}$,$C I_{2}$intersect$A B$respectively at$E$,$F$. The line through$E$which is parallel to$A C$intersects$B D$at$M$and the line through$F$which is parallel to$B D$intersects$A C$at$N$. Show that$M N$is a common tangent to$\left(I_{1}\right)$and$\left(I_{2}\right)$. ##$type=three$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 494
2018 Issue 494
Mathematics & Youth