SMO 2021 Open Round 1

It is given that $\frac\pi2\lt \beta\lt \alpha\lt \frac{3\pi}4$, $\cos(\alpha-\beta)=\frac{12}{13}$ and $\sin(\alpha+\beta)=-\frac35$. Find $\lfloor|2021\sin(2\alpha)|\rfloor$.

Find the number of solutions of the equation $|x-3|+|x-5|=2$.

(Note: If you think that there are infinitely many solutions, enter your answer as “99999”.)

Evaluate $1\times2\times3+2\times3\times4+3\times4\times5+\cdots+10\times11\times12$.

It is given that the solution of the inequality $\sqrt{81-x^4}\le kx+1$ is $a\le x\le b$ with $b-a=2$, where $k>0$. Determine $\lfloor k\rfloor$.

The figure below shows a cross that is cut out from a $10\times9$ rectangular board.



Find the total number of rectangles in the above figure.

(Note: A square is a rectangle.)

Consider all the polynomials $P(x,y)$ in two variables such that $P(0,0)=2020$ and for all $x$ and $y$, $P(x,y)=P(x+y,y-x)$. Find the largest possible value of $P(1,1)$.

In the three dimensional Cartesian space with $\mathbf i$, $\mathbf j$ and $\mathbf k$ denoting the unit vectors along three perpendicular directions in a clockwise manner, the line $\ell$ with equation given by $\mathbf r\times(\mathbf i+2\mathbf j+3\mathbf k)=5\mathbf i-13\mathbf j+7\mathbf k$ intersects the plane $\Pi$ with equation $x+y+z=16$ at the point $(a,b,c)$. Find the value of $a+b+c$.

Find the minimum value of $(x+7)^2+(y+2)^2$ subject to the constraint $(x-5)^2+(y-7)^2=4$.

Find the largest possible value of $\alpha^4+\beta^4+\gamma^4$ among all possible sets of numbers $(\alpha,\beta,\gamma)$ that satisfy the equations
$$\begin{aligned}\alpha+\beta+\gamma&=2,\\\alpha^2+\beta^2+\gamma^2&=14,\\\alpha^3+\beta^3+\gamma^3&=20.\end{aligned}$$

If $p$ is the product of all the non-zero real roots of the equation $$\sqrt[9]{x^7+30x^5}=\sqrt[7]{x^9-30x^5},$$ find $\lfloor|p|\rfloor$.

Let $S$ be the sum of a convergent geometric series with first term $1$. If the third term of the series is the arithmetic mean of the first two terms, find $\lfloor3S+4\rfloor$.

Given that $\sin\alpha+\sin\beta=\frac1{10}$, and $\cos\alpha+\cos\beta=\frac19$, find $\lfloor\tan^2(\alpha+\beta)\rfloor$.

Determine the number of positive integers that are divisible by $2021$ and has exactly $2021$ divisors (including $1$ and itself).

Let $S=\displaystyle\sum_{k=0}^{25}\binom{100}{4k}-2^{98}$. Find $\left\lfloor\left|\dfrac{S}{2^{48}}\right|\right\rfloor$.

Assume that $ABC$ is an acute triangle with $\sin(A+B)=\frac35$ and $\sin(A-B)=\frac15$. If $AB=2022(\sqrt6-2)$, determine $\lfloor h\rfloor$, where $h$ is the height of the triangle from $C$ on $AB$.

Let $a_1,a_2,\ldots$ be a sequence with $a_1=1$ and $a_{n+1}=\frac{n+2}{n}S_n$ for all $n=1,2,\ldots$, where $S_n=a_1+a_2+\cdots+a_n$. Determine the minimum integer $n$ such that $a_n\ge2021$.

Each card of a stack of $101$ cards has one side colored red and the other colored blue. Initially all cards have the red side facing up and stacked together in a deck. On each turn, Ah Meng takes $8$ cards on the top, flip them over, and place them to the bottom deck. Determine the minimum number of turns required so that all the cards have the red sides facing up again.

Let $ABC$ be a triangle with $AB=10$ and $\frac{\cos A}{\cos B}=\frac{AC}{BC}=\frac43$. Let $P$ be a point on the inscribed circle of triangle $ABC$. Find the largest possible value of $PA^2+PB^2+PC^2$.

A basket contains $19$ apples labeled by the numbers $2,3,\ldots,20$, and $19$ bananas labeled by the numbers $2,3,\ldots,20$. Ah Beng picks $m$ apples and $n$ bananas from the basket. However he needs to ensure that for any apple labeled $a$ and any banana labeled $b$ that he picks, $a$ and $b$ are relatively prime. Determine the largest possible value of $mn$.

Let $p(x)=ax^2-bx+c$ be a polynomial where $a,b,c$ are positive integers and $p(x)$ has two distinct roots in $(0,1)$. Determine the least possible value of $abc$.