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2020 Issue 514

  1. Draw on a board $2019$ plus signs $(+)$ and $2020$ minus signs $(-)$. We perform a procedure as follows. We delete two arbitrary signs. If they are both plus or both minus, we will add back a plus sign. If not, we add back a minus sign. We do it for $4038$ times. What is the remain sign on the board?
  2. Given a triangle $A B C$ with $\hat{A}=60^{\circ}$ and $A B+A C=2 B C$. Show that the median $A M,$ the altitude $B H$ and the angle bisector $C I$ of the triangle are concurrent.
  3. Suppose that $a$, $b$, $c$ are positive numbers and $a+b+c=a b c$. Prove that $$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1} \leq \frac{c}{\sqrt{c^{2}+1}}.$$
  4. Given an acute triangle $A B C$ inscribed in a circle $O$. Draw the altitude $A D$. Let goi $E$ and $F$ respectively be the perpendicular projections of $D$ on the sides $A B$ and $A C$. Suppose that two line segments $O A$ and $E F$ meet at $I$. Show that $$A B \cdot A C \cdot A I=A D^{3}.$$
  5. Consider the polynomial $f(x)=(x+a)(x+b)$ where $a$ and $b$ are integers. Show that there always exists at least an integer $m$ so that $f(m)=f(2020) \cdot f(2021)$.
  6. Let $a$, $b$, $c$, $d$ be non-negative numbers whose sum is equal to $1$. Find the maximum and minimum values of the expression $$P=\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{d+1}+\frac{d}{a+1}.$$
  7. Given the system of equation $$\begin{cases}a_{11} x_{1}+a_{12} x_{2}+\ldots+a_{1 n} x_{n} &=0 \\ a_{21} x_{1}+a_{22} x_{2}+\ldots+a_{2 n} x_{n} &=0 \\ \ldots & \ldots \\ a_{n 1} x_{1}+a_{n 2} x_{2}+\ldots+a_{n n} x_{n} &=0\end{cases}$$ with the coefficients satisfying
    • $a_{i i}>0, \forall i=\overline{1, n}$
    • $a_{i j}<0, \forall i \neq j, \forall i, j=\overline{1, n}$
    • $\displaystyle \sum_{k=1}^{n} a_{i k}>0, \forall i=\overline{1, n}$.
      Prove the system of equation has unique solution $x_{1}=x_{2}=\ldots=x_{n}=0$.
    1. Given a circle $O$ and a point $M$ outside the circle. Draw a secant $M A B$ ($A$ is in between $M$ and $B$). The tangents at $A$ and $B$ intersect at $C$. Draw $C D$ perpendicular to $M O$ $D E$ perpendicular to $C A$ and $D F$ perpendicular to CB. Show that the line $E F$ always passes through a fixed point when the secant $M A B$ varies. 
    2. Given positive numbers $x$, $y$ satisfying $x<\sqrt{2} y$, $x \sqrt{y^{2}-\dfrac{x^{2}}{2}}=2 \sqrt{y^{2}-\dfrac{x^{2}}{4}}+x$. Find the minimum value of the expression $$P=x^{2} \sqrt{y^{2}-\frac{x^{2}}{2}}.$$
    3. Let $S=1 ! 2 ! \ldots 100 !$. Show that there exists an interger $k$, $1 \leq k \leq 100$, so that $\dfrac{S}{k !}$ is a perfect square. Is such $k$ unique? (Notice that $n !=1.2 .3 \ldots n$ with $n \in \mathbb{N}$ and $0 !:=1$.)
    4. Given a non-constant function $f(x)$ which is determined on $\mathbb{R}$. Show that there always exist a real number $a$, a non-empty proper subset $A$ and two functions $g(x), h(x)$ satisfying $g(x) \geq a$, $\forall x \in A$ and $h(x)<a$, $\forall x \in \bar{A}$ $(\bar{A}=\mathbb{R} \backslash A)$ so that $f(x)=g(x)+h(x)$, $\forall x \in \mathbb{R}$.
    5. Given an acute triangle $A B C$. Two points $M$, $N$ are inside the side $B C$ such that $B M=C N$. The line which passes through $M$ and is perpendicular to $\mathrm{CA}$ intersects $A B$ at $F .$ The line which passes through $N$ and is perpendicular to $A B$ intersects $A C$ at $E$. Two lines $M F$ and $N E$ intersect at $P$. On the circumcircle of $P E F$ choose $Q$ so that $P Q \parallel B C$. Let $R$ be the reflection point of $Q$ in the midpoint of $B C .$ Show that $A R \perp B C$.

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    Mathematics & Youth: 2020 Issue 514
    2020 Issue 514
    Mathematics & Youth
    https://www.molympiad.org/2020/09/2020-issue-514.html
    https://www.molympiad.org/
    https://www.molympiad.org/
    https://www.molympiad.org/2020/09/2020-issue-514.html
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