Question bank SMO 2020

If $S$ is the sum of all the real roots of the equation $x^2+\dfrac1{x^2}=2020^2+\dfrac1{2020^2}$, find $\lfloor S\rfloor$.

Find the largest positive integer $x$ that satisfies the equation
$$(\lfloor x\rfloor-2020)^2+(\lceil x\rceil-2030)^2=(\lfloor x\rfloor-\lceil x\rceil+10)^2.$$
(Note: If you think that the above equation has no solution in positive integers, enter your answer as “0”.)

Let $S_n=\dfrac1{1\times3}+\dfrac1{3\times5}+\dfrac1{5\times7}+\cdots+\dfrac1{(2n-1)\times(2n+1)}$. Find the value of $n$ such that $S_n$ takes the value of $0.48$.

Given that the three planes in the Cartesian space with equations $2x+4y+6z=5$, $3x+5y+2z=6$ and $8x+14y+az=b$ have a common line of intersection, find the value of $a+b$.

Let $i$ be the complex number $\sqrt{-1}$, and $n$ be the smallest positive integer such that $(\sqrt3+i)^n=a$, where $a$ is a real number. Find the value of $\lfloor n-a\rfloor$.

In the three-dimensional Cartesian space, let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote unit vectors along three mutually perpendicular directions $x$, $y$ and $z$-axes respectively. Three straight lines $l_1$, $l_2$ and $l_3$ have equations defined by
$$\begin{aligned}l_1:\ \mathbf r&=(4+\lambda)\mathbf i+(5+\lambda)\mathbf j+(6+\lambda)\mathbf k,\\l_2:\ \mathbf r&=(4+3\mu)\mathbf i+(5-\mu)\mathbf j+(6-2\mu)\mathbf k,\\l_3:\ \mathbf r&=(1+6\nu)\mathbf i+(2+2\nu)\mathbf j+(3+\nu)\mathbf k,\end{aligned}$$
where $\mu$, $\lambda$ and $\nu$ are real numbers. If the area of the triangle enclosed by the three lines $l_1$, $l_2$ and $l_3$ is denoted by $S$, find the value of $10S^2$.

Given that $f:\mathbb R\to\mathbb R$ such that
$$f(a^2-b^2)=(a-b)(f(a)+f(b))$$
for all real numbers $a$ and $b$, and that $f(1)=\dfrac1{101}$, find the value of $\displaystyle\sum_{k=1}^{100}f(k)$.

Find the sum of all the positive integers $n$ such that $n^4-4n^3+22n^2-36n+18$ is a perfect square.

(Note: If you think that there are infinitely many positive integer values of $n$ that satisfy the above conditions, enter your answer as “9999”.)

Assume that
$$(x+2+m)^{2019}=a_0+a_1(x+1)+a_2(x+1)^2+\cdots+a_{2019}(x+1)^{2019}.$$
Find the largest possible integer $m$ such that
$$(a_0+a_2+a_4+\cdots+a_{2018})^2-(a_1+a_3+a_5+\cdots+a_{2019})^2\le2020^{2019}.$$

Given that $S=\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\dfrac1{\sqrt{n(n+k)}}$, find the value of $\left\lfloor(S+2)^2\right\rfloor$.

Let $A=\{1,2,\ldots,10\}$. Count the number of ordered pairs $(S_1,S_2)$, where $S_1$ and $S_2$ are non-intersecting and non-empty subsets of $A$ such that the largest number in $S_1$ is smaller than the smallest number in $S_2$. For example, if $S_1=\{1,4\}$ and $S_2=\{5,6,7\}$, then $(S_1,S_2)$ is such an ordered pair.

Each cell of a $2020\times2020$ table is filled with a number which is either $1$ or $-1$. For $i=1,\ldots,2020$, let $R_i$ be the product of all the numbers in the $i$th row and let $C_i$ be the product of all the numbers in the $i$th column. Suppose $R_i+C_i=0$ for all $i=1,\ldots,2020$. What is the least number of $-1$'s in the table?

Assume that the sequence $\{a_k\}_{k=1}^{\infty}$ follows an arithmetic progression with $a_2+a_4+a_9=24$. Find the maximum value of $S_8\times S_{10}$, where $S_k$ denotes the sum $a_1+a_2+\cdots+a_k$.

Consider all functions $g:\mathbb R\to\mathbb R$ satisfying the conditions that

(i) $|g(a)-g(b)|\le|a-b|$ for any $a,b\in\mathbb R$;

(ii) $g(g(g(0)))=0$.

Find the largest possible value of $g(0)$.

A sequence $\{a_i\}_{i=1}^{\infty}$ is called a good sequence if $\dfrac{S_{2n}}{S_n}$ is a constant for all $n\ge1$, where $S_k$ denotes the sum $a_1+a_2+\cdots+a_k$. Suppose it is known that the sequence $\{a_i\}_{i=1}^{\infty}$ is a good sequence that follows an arithmetic progression. Determine $a_{2020}$ if $a_1=1\ne a_2$.

Determine the smallest positive integer $p$ such that the system
$$\begin{cases}6x+4y+3z=0,\\4xy+2yz+pxz=0\end{cases}$$
has more than one set of real solutions in $x,y,z$.

Let $ABC$ be a triangle with $a=BC$, $b=AC$ and $c=AB$. It is given that $c=100$ and
$$\frac{\cos A}{\cos B}=\frac ba=\frac43.$$
Let $P$ be a point on the inscribed circle of $\triangle ABC$. Find the maximum value of $PA^2+PB^2+PC^2$.

Find the largest positive integer $n$ less than $2020$ such that $\binom{n-1}{k}-(-1)^k$ is divisible by $n$ for $k=0,1,\ldots,n-1$.

Assume that $\{a_k\}_{k=1}^{\infty}$ is a sequence with the property that for any distinct positive integers $m,n,p,q$ with $m+n=p+q$, the following equality always holds:
$$\frac{a_m+a_n}{(a_m+1)(a_n+1)}=\frac{a_p+a_q}{(a_p+1)(a_q+1)}.$$
Given $a_1=0$ and $a_2=\dfrac12$, determine $\dfrac1{1-a_5}$.

(Hint: Consider $c_k=\dfrac1{a_k+1}-\dfrac12$ for all positive integer $k$.)

In the triangle $ABC$, the incircle touches the sides $BC,CA,AB$ at $D,E,F$ respectively. The line segments $ED$ and $AB$ are extended to intersect at $P$ such that $AB=BP=PD$. Suppose $CA=9$. Find the value of $(ABC)^2$, where $(ABC)$ is the area of the triangle $ABC$.