2020 Issue 522

  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 $A B C$.




Mathematics & Youth: 2020 Issue 522
2020 Issue 522
Mathematics & Youth
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