SMO 2023 Open Round 1

The graph $C$ with equation $y=\dfrac{ax^2+bx+c}{x+2}$ has an oblique asymptote with equation $y=4x-6$ and one of the stationary points at $x=-4$. Find the value of $a+b+c$.

If $x=\dfrac11+\dfrac1{1+2}+\dfrac1{1+2+3}+\dfrac1{1+2+3+4}+\cdots+\dfrac1{1+2+3+\cdots+100}$, find the value of $\lfloor1010x\rfloor$.

The set of all possible values of $x$ for which the sum of the infinite series $$1+\frac16(x^2-5x)+\frac1{6^2}(x^2-5x)^2+\frac1{6^3}(x^2-5x)^3+\cdots$$ exists can be expressed as $(a,b)\cup(c,d)$, where $a\lt b\lt c\lt d$. Find $d-a$.

Find the value of $\lfloor y\rfloor$, where $y=\displaystyle\sum_{k=0}^{\infty}(2k+1)(0.5)^{2k}$.

(Hint: Consider the series expansion of $(1-x)^{-2}$.)

The solution of the inequality $|x-1|+|x+1|\lt ax+b$ is $-1\lt x\lt 2$. Find the value of $\lfloor a+b\rfloor$.

The equation $x^4-4x^2+qx-r=0$ has three equal roots. Find the value of $\left\lfloor\dfrac{3q^2}{r^2}\right\rfloor$.

The parabolas $y=x^2-16x+50$ and $x=y^2$ intersect at 4 distinct points which lie on a circle centred at $(a,b)$. Find $|a-b|$.

In the 3-dimensional Euclidean space with origin $O$ and three mutually perpendicular $x$-, $y$- and $z$-axes, two planes $x+y+3z=4$ and $2x-z=6$ intersect at the line $$\mathbf r\times\begin{pmatrix}-1\\a\\b\end{pmatrix}=\begin{pmatrix}-2\\c\\d\end{pmatrix}.$$ Find the value of $|a+b+c+d|$.

Let $x,y,z$ be real numbers with $3x+4y+5z=100$. Find the minimum value of $x^2+y^2+z^2$.

Find the area of the region represented by the equation $\lfloor x\rfloor+\lfloor y\rfloor=1$ in the region $0\le x\lt 2$.

(Note: If you think that there is no area defined by the graph, enter “0”; if you think that the area is infinite, enter “9999”.)

Let $ABC$ be a triangle satisfying the following conditions that $\angle A+\angle C=2\angle B$, and $\dfrac1{\cos A}+\dfrac1{\cos C}=\dfrac{-\sqrt2}{\cos B}$. Determine the value of $\dfrac{2022\cos\left(\frac{A-C}{2}\right)}{\sqrt2}$.

Find $x$ which satisfies the following equation $$\frac{x-2019}{1}+\frac{x-2018}{2}+\frac{x-2017}{3}+\cdots+\frac{x+2}{2022}+\frac{x+3}{2023}=2023.$$

Assume that $x$ is a positive number such that $x-\dfrac1x=3$ and $$\frac{x^{10}+x^8+x^2+1}{x^{10}+x^6+x^4+1}=\frac mn,$$ where $m$ and $n$ are positive integers without common factors larger than $1$. Determine the value of $m+n$.

Consider the set of all possible pairs $(x,y)$ of real numbers that satisfy $(x-4)^2+(y-3)^2=9$. If $S$ is the largest possible value of $\dfrac yx$, find the value of $\lfloor7S\rfloor$.

Let $x,y$ be positive integers with $16x^2+y^2+7xy\le2023$. Find the maximum value of $4x+y$.

Let $x$ be the largest real number such that $$\sqrt{x-\frac1x}+\sqrt{1-\frac1x}=x.$$ Determine the value of $(2x-1)^4$.

Two positive integers $m$ and $n$ differ by $10$ and the digits in the decimal representation of $mn$ are all equal to $9$. Determine $m+n$.

Let $\{a_n\}$ be a sequence of positive numbers, and $S_n=a_1+a_2+a_3+\cdots+a_n$. For any positive integer $n$, let $b_n=\dfrac12\left(\dfrac{a_{n+1}}{a_n}+\dfrac{a_n}{a_{n+1}}\right)$. Given that $\dfrac{a_n+2}{2}=\sqrt{2S_n}$ holds for all positive integers $n$, determine the limit $\lim_{n\to\infty}(b_1+b_2+\cdots+b_n-n)$.

Let $ABC$ be a triangle with $AB=c$, $AC=b$ and $BC=a$, and satisfies the conditions $\tan C=\dfrac{\sin A+\sin B}{\cos A+\cos B}$, $\sin(B-A)=\cos C$ and that the area of the triangle $ABC=3+\sqrt3$. Determine the value of $a^2+c^2$.

Let $g:\mathbb R\to\mathbb R$, $g(0)=4$ and that $$g(xy+1)=g(x)g(y)-g(y)-x+2023.$$ Find the value of $g(2023)$.