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2018 Issue 494

  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 $\left(I_{2}\right)$.

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Mathematics & Youth: 2018 Issue 494
2018 Issue 494
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-494.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-494.html
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