SMO 2021 Senior Round 1

Let $p$ be a real number such that the equation $x^2-10x=p$ has no real solution. Which of the following is true?

Which of the following is the largest?

Find the value of $2021^{(\log_{2021}2020)(\log_{2020}2019)(\log_{2019}2018)}$.

Suppose $\sin\theta=\frac{n-3}{n+5}$ and $\cos\theta=\frac{4-2n}{n+5}$ for some integer $n$. Find the maximum value of $160\tan^2\theta$.

Select all the inequalities which hold for all real values of $x$ and $y$.

(i) $x\le x^2+y^2$

(ii) $xy\le x^2+y^2$

(iii) $x-y\le x^2+y^2$

(iv) $y+xy\le x^2+y^2$

(v) $x+y-1\le x^2+y^2$

Let $x$ be the integer such that $x=5^{\sqrt{2+4\log_x5}}$. Determine the value of $x$.

If $\cos A-\cos B=\frac12$ and $\sin A-\sin B=-\frac14$, find the value of $100\sin(A+B)$.

Find the constant in the expansion of $\left(\sqrt[3]{x}+\frac1{\sqrt{x}}\right)^6\left(\sqrt{x}+\frac1x\right)^{10}$.

A quadratic polynomial $P(x)=ax^2+bx+c$, where $a\ne0$, has the following properties: $$P(n)=\frac1{n^2}\quad\text{for all }n=-1,2,3.$$ Determine the smallest positive value of $k$, where $k\ne2,3$, such that $P(k)=\frac1{k^2}$.

The figure below shows a triangle $ABC$ such that $AD$ and $BE$ are altitudes to the sides $BC$ and $CA$ respectively. The lines $AD$ and $BE$ intersect at $H$. Determine the area (in cm$^2$) of the triangle $ABC$ if $AH=50$ cm, $DH=18$ cm and $BH=EH$.

In the figure below, $\angle GCB=\angle ACE=\angle DFE=90^\circ$, and $\angle GBC=\angle EAC=\angle EDF=\theta^\circ$. Also, $GB=6$ cm, $AE=10$ cm and $DE=8$ cm. Let $S$ denote the sum of the areas of the triangles $ABC$ and $CDE$. Find the maximum possible value of $S$ (in cm$^2$).

Find the sum of all the solutions to the equation $\sqrt[3]{x-110}-\sqrt[3]{x-381}=1$.

If $f(x)=\left(2x+4+\frac{x-2}{x+3}\right)^2$, where $-2\le x\le2$, find the maximum value of $f(x)$.

Given that $$D=\sqrt{\sqrt{x^2+(y-1)^2}+\sqrt{(x-1)^2+y^2}}$$ for real values of $x$ and $y$, find the minimum value of $D^8$.

Find the minimum value of $\frac8{\sin2\theta}+12\tan\theta$, where $0\lt\theta\lt\frac\pi2$.

Determine the largest angle $\theta$ (in degree), where $0^\circ\le\theta\le360^\circ$, such that $$\sin(\theta+18^\circ)+\sin(\theta+162^\circ)+\sin(\theta+234^\circ)+\sin(\theta+306^\circ)$$ $$=1+\cos(\theta+60^\circ)+\cos(\theta+300^\circ).$$

Let $O$ be the circumcentre of the triangle $ABC$ and that $\angle ABC=30^\circ$. Let $D$ be a point on the side $BC$ such that the length of $AD$ is the same as the radius of the circle. Determine the value of $\angle ADO$ (in degree) if $\angle OAB=10^\circ$.

A function $f$ satisfies $f(x)f(x+1)=x^2+3x$ for all real numbers $x$. If $f(1)+f(2)=\frac{25}{6}$ and $0\lt f(1)\lt2$, determine the value of $11\times f(10)$.

Find the value of $$\frac1{\sin^2 0.5^\circ}-\tan^2 0.5^\circ+\frac1{\sin^2 1.5^\circ}-\tan^2 1.5^\circ+\frac1{\sin^2 2.5^\circ}-\tan^2 2.5^\circ+\cdots+\frac1{\sin^2 179.5^\circ}-\tan^2 179.5^\circ.$$

Let $a_1,a_2,a_3$ be three distinct integers where $1000>a_1>a_2>a_3>0$. Suppose there exist real numbers $x,y,z$ such that $$\begin{aligned}(a_1-a_2)y+(a_1-a_3)z&=a_1+a_2+a_3,\\(a_1-a_2)x+(a_2-a_3)z&=a_1+a_2+a_3,\\(a_1-a_3)x+(a_2-a_3)y&=a_1+a_2+a_3.\end{aligned}$$ Find the largest possible value of $x+y+z$.