SMO 2020 Senior Round 1

Let $b$ be a positive integer. If the minimum possible value of the quadratic function $5x^2+bx+506$ is $6$, find the value of $b$.

Which of the following is equal to $\sqrt{5+\sqrt3}+\sqrt{5-\sqrt3}$?

Simplify $\log_8 5\cdot(\log_5 3+\log_{25}9+\log_{125}27)$.

Let $a=50^{1/505}$, $b=10^{1/303}$ and $c=6^{1/202}$. Which of the following is true?

Let $p=\log_{10}(\sin x)$, $q=(\sin x)^{10}$, $r=10^{\sin x}$, where $0\lt x\lt \dfrac\pi2$. Which of the following is true?

Find the minimum possible value of $|x-10|-|x-20|+|x-30|$, where $x$ is any real number.

Parallelogram $ABCD$ has sides $AB=39$ cm and $BC=25$ cm. Find the length of diagonal $AC$ (in cm) if diagonal $BD=34$ cm.

Suppose $\sin(45^\circ-x)=-\dfrac13$, where $45^\circ\lt x\lt 90^\circ$. Find $(6\sin x-\sqrt2)^2$.

If $8\cos x-8\sin x=3$, find the value of $55\tan x+\dfrac{55}{\tan x}$.

Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are integers, such that
$$x^2+y^2-20x-14y+140\lt 0.$$

The figure below shows a right-angled triangle $ABC$ such that $\angle BAC=90^\circ$, $\angle ABC=30^\circ$ and $AB=48$ cm. Let $P$ be a point on side $AB$ such that $CP$ is the angle bisector of $\angle ACB$ and $Q$ be a point on side $BC$ such that line $AQ$ is perpendicular to line $CP$. Determine the length of $PQ$.



Editorial note: the retained scan includes handwritten annotations. Only the printed givens transcribed above are used; a new clean checked diagram is supplied in the solution.

In the figure below, the point $O$ is the center of the circle, $AD$ and $BC$ intersect at $E$, and $\angle AEB=70^\circ$, $\angle AOB=62^\circ$. Find the angle $\angle OCD$ (in degree $^\circ$).



Editorial note: the retained scan includes handwritten annotations. Only the printed givens transcribed above are used; a new clean checked diagram is supplied in the solution.

Find the value of $\dfrac{4\cos43^\circ}{\sin73^\circ}-\dfrac{12\sin43^\circ}{\sqrt3\sin253^\circ}$.

If $\dfrac{x^2}{5}+\dfrac{y^2}{7}=1$, find the largest possible value of $(x+y)^2$.

Find the coefficient of $x^6$ in the expansion of $(1+x+2x^2)^7$.

Suppose $(3x-y)^2+\sqrt{x+38+14\sqrt{x-11}}+|z+x-y|=7$. Find the value of $|x+y+z|$.

Suppose there are real numbers $x,y,z$ satisfying the following equations:
$$x+y+z=60,\qquad xy-z^2=900.$$
Find the maximum possible value of $|z|$.

Find the sum $\displaystyle\sum_{k=1}^{16}\log_2\left(\sqrt{\sin^2\frac{k\pi}{8}+1}-\sin\frac{k\pi}{8}\right)$.

Let $a,b$ be positive real numbers, where $a>b$. Suppose there exists a real number $x$ such that $(\log_2 ax)(\log_2 bx)+25=0$. Find the minimum possible value of $\dfrac ab$.

The figure below shows a rectangle $ABCD$ such that the diagonal $AC=20$ cm. Let $P$ be a point on side $CD$ such that $BP$ is perpendicular to diagonal $AC$. Find the area of rectangle $ABCD$ (in $\mathrm{cm}^2$) if $BP=15$ cm.