SMO 2026 Open Round 1

Given that $A=\displaystyle\lim_{n\to\infty}\frac{\sum_{r=n}^{2n}r^2}{\sum_{r=1}^{n}r^2}$, find the value of $\lfloor A\rfloor$.

(Note: If you think that the limit is infinity, enter your answer as 9999.)

The solution of the inequality $|2x^2-3x+2|\lt |x^2-4x+3|$ is $a\lt x\lt b$. Find the value of $\lfloor a^2+b^2\rfloor$.

Find $\lfloor B\rfloor$, where $B=\displaystyle\sum_{k=1}^{10}\frac{k(2^k)}{(k+1)(k+2)}$.

It is known that $\displaystyle\int_0^{1/3}\sqrt{1-x^2}\,dx=\frac12\sin^{-1}\frac13+b$. Find $\left\lfloor\frac1{b^2}\right\rfloor$.

For two vectors $\mathbf{a}$ and $\mathbf{b}$ in the Cartesian space, it is known that $$|\mathbf{a}+\lambda\mathbf{b}|\ge|\mathbf{a}|$$ for all real numbers $\lambda$. Find the largest possible value of the scalar product $\mathbf{a}\cdot\mathbf{b}$.

(Note: If you think that the set of all possible values is unbounded, enter 9999.)

Suppose that the two quadratic equations $y^2-my-36=0$ and $y^2-6y-m=0$ have a common root $w$, find the value of $7w^2-w^3$.

Let $a,b,c$ be non-negative real numbers such that $a+b+c=100$. Find the maximum value of $ab+2ac+3bc$.

A sequence of numbers $\{x_n\}$ is such that $x_1=1$ and $x_{n+1}=2x_n+1$ for $n=1,2,3,\ldots$. Find the unit digit of the number $2027^{x_{10}+1}$.

Determine $\lfloor M\rfloor$, where $M=\frac1{\sqrt1}+\frac1{\sqrt2}+\cdots+\frac1{\sqrt{100}}$.

Let $f(x)=\frac{9^x}{9^x+3}$. If $$m=f\left(\frac1{2026}\right)+f\left(\frac2{2026}\right)+\cdots+f\left(\frac{2025}{2026}\right),$$ find the value of $\lfloor2m\rfloor$.

Find the least positive integer $n$ (if it exists) such that $\displaystyle\sum_{k=1}^n a_k\cdot k^2=2026$, where each $a_k$ is either $1$ or $-1$ for all $k=1,2,3,\ldots,n$.

(Note: If you think that such an $n$ does not exist, enter your answer as 0.)

Suppose that $x,y,z$ are positive real numbers such that $$\frac{x^2}{x^2+1}+\frac{y^2}{y^2+1}+\frac{z^2}{z^2+1}=2.$$ If $m$ is the maximum value of $$\frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1},$$ find the value of $\lfloor m^2\rfloor$.

Let $x$ and $y$ be real numbers such that $x^2+y^2=1$, and $p$ be the minimum value of $\frac{y-2}{x-1}$. Find the value of $\lfloor4p\rfloor$.

There are $n$ pairs of students, with each pair consisting of a boy and a girl. At each turn, Ah Meng chooses $1$ boy and $1$ girl (not necessarily belonging to the same pair), and gives each of them a peanut. The same student may be chosen more than once. After several turns of choosing $1$ boy and $1$ girl, Ah Meng finds out that for each pair, either the boy has $21$ peanuts more than the girl, or the girl has $36$ peanuts more than the boy. Determine the minimum possible value of $n$.

Let $a$ and $b$ be integers satisfying $a^3+a^2b+4b^3=7087$. Find $a^2+b^2$.

A circle $\omega$ with centre $I$ is inside another circle $\Omega$ with centre $O$. $AB$ is a chord of $\Omega$ which is tangent to $\omega$. Let $C$ be the circumcentre of $\triangle IAB$. Suppose the radii of $\omega$ and $\Omega$ are $18$ and $54$ respectively and $IO=24$. Find $OC$.

Let $f:\mathbb R\to\mathbb R$ be a function satisfying $f(x+y)=f(x)f(y)-4\sin x\sin y$ for all $x,y\in\mathbb R$. Determine the maximum value of $f$.

Among all sequences $(x_n)_{n=1}^{2026}$ of $2026$ numbers satisfying $x_1\ge x_2\ge x_3\ge\cdots\ge x_{2026}\ge0$, $x_1+\cdots+x_{2026}\le400$ and $x_1^2+\cdots+x_{2026}^2\ge10000$, determine the minimum value of $\sqrt{x_1}+\sqrt{x_2}$.

Find the largest positive integer $N$ such that there exist real numbers $a,b$ satisfying $N|x^2+ax+b|\le2026+x^2$, for all $|x|\le1$.

In the triangle $ABC$ with circumcircle $\omega$, $CA=12$, $BC=9$, $\angle C=90^\circ$, the tangent to $\omega$ at $C$ intersects $AB$ produced at $D$. Let $E$ be the midpoint of $CD$. The point $F$ is on $EB$ produced such that $AF$ is parallel to $CD$. Find the area of the triangle $ABF$.