SMO 2018 Senior Round 1

Question 1

The roots of the quadratic equation $x^2-3 x+1=0$ are $\alpha$ and $\beta$. Find a quadratic equation whose roots are $\alpha^2+\alpha$ and $\beta^2+\beta$.

$x^2+10 x-5=0$
$x^2-10 x+5=0$
$x^2-5 x-10=0$
$x^2+5 x-10=0$
$x^2-5 x-5=0$

Question 2

Simplify $\sqrt{2+\sqrt 3}+\sqrt{2-\sqrt 3}$.

$\sqrt 5$
$\sqrt 6$
$\sqrt 7$
$\sqrt 8$
$\sqrt{10}$

Question 3

Which of the following is true?

$10^{30}\gt 40^{15}\gt 30^{20}$
$30^{20}\gt 10^{30}\gt 40^{15}$
$10^{30}\gt 30^{20}\gt 40^{15}$
$30^{20}\gt 40^{15}\gt 10^{30}$
$40^{15}\gt 30^{20}\gt 10^{30}$

Question 4

Which of the following is the largest?

$\log_5 7-\log_5 6$
$\log_6 8-\log_6 7$
$\log_7 9-\log_7 8$
$\log_8 10-\log_8 9$
$\log_9 11-\log_9 10$

Question 5

In the expansion of $\left(x^2+\frac 1 x\right)^9$, find the coefficient of $x^{15}$.

$9$
$36$
$84$
$126$
None of the above

Question 6

The line $y=5 x-10$ meets the curve $x^2-4 x+y^2-32=0$ at two points P and Q. Find the length of the line segment PQ in meters, assuming that both the x and y axis are measured in meters.

Question 7

Suppose n is a positive integer, and x is measured in radians. If the equation $2\pi\sin nx=2\pi-x$, where $0\le x\lt 2\pi$, has exactly 2018 different solutions, find n.

Question 8

The polynomial $p(x)=x^3+Ax^2+Bx-3$ has a factor of $x^2+7 x+1$. Find the remainder when the polynomial p(x) is divided by $x-4$.

Question 9

In the figure below, the angles indicated at the vertex A,B,C,D,E,F,G are given by $a^\circ,b^\circ,c^\circ,d^\circ,e^\circ,f^\circ,g^\circ$ respectively. Find $a+b+c+d+e+f+g$.

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Question 10

Find x, where $0^\circ\le x^\circ\le 90^\circ$, such that $\cos x^\circ=\cos 49^\circ+\cos 71^\circ$.

Question 11

Let x and y be real numbers. Find the maximum value of $2 x^2-3 xy-2 y^2$ subject to the condition that $$25 x^2-20 xy+40 y^2=36.$$

Question 12

Suppose $\sin 2 x=\frac 79$. Find $108(\sin^6 x+\cos^6 x)$.

Question 13

Find the sum of all the positive integers x satisfying $$(4\log_2(\log_{16}x))(3\log_{16}(\log_2 x)-1)=1.$$

Question 14

Consider the function $f(x)=ax^2-c$, where a and c are some constants. Suppose that $-3\le f(1)\le-2$ and $1\le f(2)\le 6$. Find the largest possible value of $f(4)$.

Suppose a is the smallest number satisfying the inequality $$\left|\frac{|x+9|}{10}-\frac{|x-9|}{10}\right|\le x-1.$$ If $a=\frac mn$, where m and n are positive integers having no common factors larger than 1, find the value of $m+n$.

Question 16

Let P(x) be a polynomial of degree 4 such that $P(n)=\frac{120}{n}$ for n=1,2,3,4,5. Determine the value of P(6).

Question 17

Let $$L=\sum_{k=7}^{16}(1+\tan(15 k^\circ+15^\circ)\tan 15 k^\circ).$$ Find the largest integer which is smaller than or equal to L.

Question 18

Find the smallest three-digit positive integer whose square ends in the digits 129.

Question 19

In the figure below, the points A,B,C and D lie on the circle such that the lines AD and BC intersect at the point P. The line AF is tangent to the circle. The point E lies on the line AF such that the lines DE and PF intersect at C. If the line AP bisects the angle BAC, and $\angle CEA=(22+y)^\circ$ where $y^\circ=\angle BAP$, find the angle APC (in degrees).

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Question 20

Eleven identical boxes are arranged in a row. In how many ways can eight identical balls be put into the boxes if each box can hold at most one ball and no three empty boxes can appear consecutively next to each other?