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

## $show=home 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$M OD E$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$A R \perp B C$. ##$type=three$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 514
2020 Issue 514
Mathematics & Youth