SMO 2019 Open Round 1

Find the largest positive integer $k$ such that $\frac{x(k-x)-400}{x^2-x+k-38}$ is always negative for all real values of $x$.

Among all the nonzero real numbers $x,y,z$ satisfying $x+y+z=0$, find the largest possible value of $$\frac{y+z}{yz}(y^2+z^2-x^2)+\frac{z+x}{zx}(z^2+x^2-y^2)+\frac{x+y}{xy}(x^2+y^2-z^2).$$

Assume that $0\lt \alpha\lt \frac\pi2$ and $\cos(\alpha+\frac\pi6)=\frac45$. Determine the smallest possible integer $R$ such that $100\sin(2\alpha+\frac\pi{12})\lt R$.

Determine the number of integer solutions of the following inequality: $$x-2019\lt \log_{10}(x+2019).$$

Assume that $(-1,0)$ is a point on the curve $y=ax^2+bx+c$, where $a,b,c$ are three fixed numbers not exceeding $\frac12$. If the inequality $x\le ax^2+bx+c\le\frac12(x^2+1)$ holds for all real numbers $x$, what is the value of $2020(a^2+b^2-c^2)$?

Determine the maximum value of $x+y$ where $x$ and $y$ are numbers satisfying $$(x+\sqrt{1+x^2})(y+\sqrt{1+y^2})=1.$$

Find the maximum integer $m$ with $m=y+x$ such that $x$ and $y$ are real numbers satisfying $x^2+4y^2+x+y=0$.

Among all the real numbers $a,b,c$ satisfying $a+b+c=0$ and $abc=10$, find the largest possible value of $a^3+b^3+c^3$.

Suppose that the solution set of $|x-1|+|x-6|\le9$ is given by $\{x:a\le x\le b\}$ for some real numbers $a,b$. Find the value of $b-a$.

Let $\{a_n\}$ be an infinite sequence with $a_1=2$ and $a_{n+1}=a_n+4n+2$ for all $n\ge1$. Determine $$\lim_{n\to\infty}2000\left(\frac1{a_2-2}+\frac1{a_3-2}+\cdots+\frac1{a_n-2}\right).$$

Let $a>b>0$. Let $A$ be the minimum value of $$2019a+\frac1{2019(a-b)b}.$$ Determine $[A]$.

Let $x_1+x_2+\cdots+x_{99}=9702$ and $x_1^2+x_2^2+\cdots+x_{99}^2=960498$. Determine the largest possible value of $x_1$.

Find the largest positive integer $n$ satisfying the following condition: there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ such that $|a_i-a_j|$ is a product of two distinct prime integers for all $i\ne j$.

Let $a,b,c$ be integers satisfying $ab+bc+ca=1$ and $(1+a^2)(1+b^2)(1+c^2)=6923904100$. Determine the value of $|(a+b)(b+c)(c+a)|$.

In the triangle $ABC$, $AB=209$, $BC=171$, $CA=190$, $M$ is the midpoint of $BC$. Point $L$ is in the extension of $BA$ and point $N$ is in the extension of $MA$ such that $C,L,N$ are collinear and $AL=NL$. Find the length of $AN$.

Let $\frac ab$ be the fraction with the smallest positive denominator such that $$\frac{386}{2019}\lt \frac ab\lt \frac{35}{183}.$$ Determine the value of $a+b$.

Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Suppose $AB=59/\sqrt3$, $TA=32$, and $TB=27$. Find $TP$.

In triangle $ABC$, $BC=10$, $CA=9$, $\cos C=5/14$. The bisector of $\angle C$ meets the circumcircle at $N\ (\ne C)$, the altitude from $A$ meets the circumcircle at $L\ (\ne A)$ and $NL$ intersects $BC$ at $M$. Find the length $MC$.

Let $x_1$ be the only real root of $\log_2x=2020-x$ and $x_2$ be the only real root of $2^x=2020-x$. Determine the value of $x_1+x_2$.

Determine the minimum integer $x$ such that the following inequality holds: $$\sqrt[3]{10x-200}+\sqrt[3]{10x-300}+20x-500>0.$$