SMO 2019 Senior Round 1

Question 1

The roots of the quadratic equation $x^2-7mx+5n=0$ are $m$ and $n$, where $m\ne0$ and $n\ne0$. Find a quadratic equation whose roots are $\frac mn$ and $\frac nm$.

$6x^2-37x+1=0$
$6x^2-50x-7=0$
$6x^2-50x+7=0$
$6x^2-37x+6=0$
$x^2-37x+1=0$

Question 2

Simplify $\left(\sqrt{10}-\sqrt2\right)^{1/3}\left(\sqrt{10}+\sqrt2\right)^{7/3}$.

$24+4\sqrt5$
$24+6\sqrt5$
$24+8\sqrt5$
$24+10\sqrt5$
$24+12\sqrt5$

Question 3

Let $a=4^{3000}$, $b=6^{2500}$ and $c=7^{2000}$. Which of the following statement is true?

$a<b<c$
$a<c<b$
$b<a<c$
$c<a<b$
$c<b<a$

Question 4

If $\log_{21}3=x$, express $\log_7 9$ in terms of $x$.

$\frac{2x}{2-x}$
$\frac{2x}{1-x}$
$\frac{2x}{x-2}$
$\frac{2x}{x-1}$
$\frac{x}{1-x}$

Question 5

Suppose that $\sin x=\frac{12}{13}$ and $\cos y=-\frac45$, where $0^\circ\le x\le90^\circ$ and $90^\circ\le y\le180^\circ$. Find the value of $\cos(x+y)$.

$-\frac{56}{65}$
$\frac{56}{65}$
$-\frac{16}{65}$
$\frac{16}{65}$
None of the above

Question 6

Find the largest positive integer $n$ such that $n+8$ is a factor of $n^3+13n^2+40n+40$.

Question 7

Suppose $\tan x=5$. Find the value of $\frac{6+\sin2x}{1+\cos2x}$.

Question 8

Suppose $x$ and $y$ are real numbers such that $$|x-y|+3x-y=70,\qquad |y-x|+3y+x=50.$$ Find the maximum possible value of $x+2y$.

The coordinates of the vertices of a triangle $\triangle ABC$ are $A(6,0)$, $B(0,8)$ and $C(x,y)$ such that $x^2-16x+y^2-12y+91=0$. Find the largest possible value of the area of the triangle $\triangle ABC$.

Question 10

In the figure below, $AD$ is perpendicular to $BC$, $PQ$ is parallel to $BC$, and the triangle $\triangle PQR$ is an equilateral triangle whose area (in meter$^2$) is equal to the length of $AD$ (in meter). Find the smallest possible value of the length of $BC$.

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Question 11

Find the value of $448\left(\frac{\sin12^\circ\sin39^\circ\sin51^\circ}{\sin24^\circ}\right)$.

Question 12

In the figure below, the chord $AF$ passes through the origin $O$ of the circle, and is perpendicular to the chord $BC$. It is given that $AB=17$ cm, $CD=5$ cm. Suppose $\frac{BE}{ED}=\frac mn$, where $m$ and $n$ are positive integers which are relatively prime. What is the value of $m+n$?

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Question 13

Let $P(x)$ be the polynomial that results from the expansion of the following expression: $$(2x^3+3x^2+x)^5\left(\frac x6+\frac12\right)^5.$$ Find the sum of the coefficients of $x^{2k+1}$, where $k=0,1,2,3,\ldots,9$.

Question 14

Find the value of the following expression: $$\frac{2(1^2+2^2+3^2+\cdots+49^2+50^2)+(1\times2)+(2\times3)+(3\times4)+\cdots+(48\times49)+(49\times50)}{100}.$$

Question 15

Let $M$ be the maximum possible value of $\frac{15x^2-x^3-39x-54}{x+1}$, where $x$ is a positive integer. Find the value of $9M$.

Question 16

Find the maximum possible value of $x+y+z$ where $x,y,z$ are integers satisfying the following system of equations: $$x^2z+y^2z+8xy=200,\qquad 2x^2+2y^2+xyz=50.$$

Question 17

Find the remainder when $10^{43}$ is divided by $126$.

Question 18

Suppose $(\log_2 x)^2+4(\log_2(\log_3 y))^2=4(\log_2 x)(\log_2(\log_3 y))$. If $x=49$ and $y$ is a positive integer, find $y$.

Question 19

The figure below shows a rectangle $ABCD$ with $AB=16$ cm and $BC=15$ cm. Let $P$ be a point on the side $BC$ such that $BP=7$ cm, and let $Q$ be a point on the side $CD$ such that $CQ=6$ cm. Find the length of $AR$ (in cm), where $R$ is the foot of the perpendicular from $A$ to $PQ$.

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Question 20

A sequence $x_0,x_1,x_2,x_3,\ldots$ of integers satisfies the following conditions: $x_0=1$, and for any positive integer $n\ge1$, $|x_n-1|=|x_{n-1}+2|$. Find the maximum possible value of $2019-(x_1+x_2+\cdots+x_{2018})$.