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

## $show=home 1. Show that for every$n \in \mathbb{N}$,$4^{2^{n}}+2^{2^{n}}+1$is divisible by$7$. 2. Given an acute triangle$A B C$with$A B < A C$. Let$A D$be the altitude from$A$of the triangle. Let$M$and$N$so that$A B$is the perpendicular bisector of$M D$and$A C$is the perpendicular bisector of$N D$. The line$M N$intersects$A B$at$E$and intersects$A C$at$F .$The lines$B F$and$C E$meet at$H$. Let$I$,$K$,$O$respectively be the midpoints of$B H$,$C H$and$B C$. Prove that $$\widehat{E I F}=\widehat{E K F}=\widehat{E D F}=\widehat{E O F}.$$ 3. Consider the following sets $$\begin{array}{l}A=\{x \in \mathbb{N}: x=3 k+2 \text {with } k \in \mathbb{N} \text { and } k \leq 668\} \\ B=\{x \in \mathbb{N}: x=5 k+1 \text { with } k \in \mathbb{N} \text { and } k \leq 668\} \\ C=\{x \in \mathbb{N}: x \in A \text { and } x \in B\}\end{array}.$$ How many elements are there in the set$C$? 4. Given a right triangle$A B C$with the right angle$A$and$\hat{B}=60^{\circ}$. Let$A H$be the altitude from$A$of the triangle. Let$I$be the midpoint of$A B$. On the ray$I H$choose$K$so that$B K=B A$. Find the value of the angle$\widehat{B K C}$. 5. Given real numbers$a, b, c$satisfying$a b \neq 0$,$2 a\left(a^{2}+b^{2}\right)=b c$, and$b\left(a^{2}+15 b^{2}\right)=6 a c$. Compute the value of the expression $$P=\frac{a^{4}+6 a^{2} b^{2}+15 b^{4}}{15 a^{4}+b^{4}}.$$ 6. Solve the system of equations $$\begin{cases}x^{2}+y^{2} &=9 \\ \sqrt{5-x}+\sqrt{23+x-6 y} &=2 \sqrt{5}\end{cases}$$ 7. Given non-negative numbers$a,b,c$satisfying$25 a+45 b+52 c \leq 95$. Find the maximum value of the expression $$\frac{a+b}{a+b+1}+\frac{b+c}{b+c+1}+\frac{c+a}{c+a+1}.$$ 8. Given a triangle$A B C$. Its incircle$(I)$is tangent to$B C$,$C A$,$A B$respectively at$D$,$E$,$F$Let$M$be the reflection of$F$over$B$and$N$the reflection of$E$over$C$. The altitude$D H$of$D E F$intersects$M N$at$G$. Show that$D H=D G$. 9. Given non-negative numbers$x$,$y$,$z$satisfying$x+y+z=3$. Prove that $$\left(x^{3}+y^{3}+z^{3}\right)\left(x^{3} y^{3}+y^{3} z^{3}+z^{3} x^{3}\right) \leq 36(x y+y z+z x).$$ 10. The sequence$\left(u_{n}\right)$is determined as follows $$u_{1}=16,\, u_{2}=288,\quad u_{n+2}=18 u_{n+1}-17 u_{n},\, \forall n \geq 1.$$ Find the smallest possible value for$n$so that$u_{n}$is divisible by$2^{2020}$. 11. Find all functions$f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$satisfying $$f\left(\frac{f(n)}{n^{2020}}\right)=n^{2021},\,\forall n \in \mathbb{Z}.$$ 12. Given a triangle$A B C$. Let$A^{\prime}$be the reflection of$A$over the midpoint$M$of$B C$. The line$A A^{\prime}$intersects$\left(A^{\prime} B C\right)$at the second point$K$. Let$I$,$J$respectively be the centers of$(K A B)$and$(K A C)$.$I J$intersects$B C$at$S$. Show that$S A$is a tangent line of the circumcircle$(O)$of$A B C$. ##$type=three$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: 2020 Issue 522
2020 Issue 522
Mathematics & Youth