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2020 Issue 517

  1. Find all the prime numbers $p$ so that $2^{p}+p^{2}$ is a prime.
  2. Given an isosceles right triangle $A B C$ with the vertex angle $A$. The point $D$ inside $A B C$ such that $\widehat{A B D}=\widehat{B C D}=30^{\circ} .$ Compute the vertex angle $\widehat{C A D}$.
  3. Given positive numbers $a, b, c$ such that $a+b+c=1 .$ Find the maximum value of the expression $$P=\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}-\frac{1}{3(a b+b c+c a)}.$$
  4. Suppose that $A B C D$ is a thombus with $\widehat{B A C}=60^{\circ}$. Let $M$ be a point on the line segment $B C$ ($M$ is different from $B$ and $C$). The line $A M$ meets the line $C D$ at $N$ and then let $E$ be the intersection between the lines $D M$ and $B N .$ Show that the line $B C$ is tangent to the circumcircle of the triangle $CEN$.
  5. Solve the cquation $$8 x^{2}-11 x+1=(1-x) \sqrt{4 x^{2}-6 x+5}.$$
  6. Given real numbers $x, y, z, t$ satisfying $x^{2}+y^{2}+z^{2}+t^{2} \leq 2 .$ Find the maximum value of the expression $$P(x, y, z, t)=(x+3 y)^{2}+(z+3 t)^{2} + (x+y+z)^{2}+(x+z+t)^{2}.$$
  7. Find all the real roots of the equation $$\left\{\frac{2 x^{2}-5 x+2}{x^{2}-x+1}\right\}=\frac{1}{2}$$ where $\{a\}$ denotes the fractional part of the number $a$. 
  8. Given a convex quadrilateral $A B C D .$ Two diagonals $A C$ and $B D$ intersect at $O .$ Let $M$, $N$, $P$, $Q$ respectively be the perpendicular projections of $O$ on the lines $A B$, $B C$, $C D$, $D A$. Show that $A C$ and $B D$ are perpendicular if and only if $$\frac{1}{O M^{2}}+\frac{1}{O P^{2}}=\frac{1}{O N^{2}}+\frac{1}{O Q^{2}}$$
  9. Given positive numbers $a$, $b$, $c$ such that $a+b+c=3$. Find the minimum value of the expression $$P=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{6 a b c}{a b+b c+c a}.$$
  10. Find all the pairs $(p, k)$, where $p$ is a prime number and $k$ is a positive integer. such that $$k !=\left(p^{3}-1\right)\left(p^{3}-p\right)\left(p^{3}-p^{2}\right).$$
  11. Find all the functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(2 f(a)+f(b))=2 a+b-4,\, \forall a, b \in \mathbb{Z}.$$
  12. Given a triangle $A B C$ inscribed in a circle $(O)$ where $B$ and $C$ are fixed and $A$ can vary. There altitudes $A D$, $B E$, $C F$ meet at $H$. Let $\left(C_{1}\right)$, $\left(C_{2}\right)$, $\left(C_{3}\right)$ be the circles with diameters $A B$, $B C$, $CA$ respectively. The line $A H$ intersects $\left(C_{2}\right)$ at $M$. ($M$ belongs to the line scgment $A H$, $B H$ intersects $\left(C_{3}\right)$ at $N$ belongs to the line segment $B H$), $CH$ intersects $\left(C_{1}\right)$ at $P$ ($P$ belongs to the line segment $C H$), $A N$ intersects $\left(C_{1}\right)$ at $S$ which is different from $A$, $A P$ intersects $\left(C_{3}\right)$ at $T$ which is different from $A .$ The lines $N P$ intersects $\left(C_{1}\right)$ at $X$ which is different from $P$, intersects $\left(C_{3}\right)$ at $Y$ which is different from $N$. $X S$ meets $Y T$ at $Z$. The line passing through $A$ and perpendicular to $N P$ intersects $\left(C_{1}\right)$ at $J$ and intersect $\left(C_{3}\right)$ at $L$.
    a) The circle with center $A$ and radius $A N$ meets $\left(C_{1}\right)$ at $V,$ meets $\left(C_{3}\right)$ at $U$. Show that $V$, $N$, $I$ are collinear and $U$, $P$, $L$ collinear.
    b) Show that the incenter of the triangle $X Y Z$ is a fixed point.

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Mathematics & Youth: 2020 Issue 517
2020 Issue 517
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-517.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-517.html
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