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SMO 2016 Senior Round 1

Question 1

If α,β are two distinct roots of the equation $2 x^2+4 x-45=0$, find the value of $\alpha^2\beta+\alpha\beta^2$.
  1. 15
  2. -15
  3. 45
  4. -45
  5. 90

Question 2

Simplify $$\frac{9^{8/3}-9^{5/3}}{4^{8/3}-4^{2/3}}.$$
  1. $\frac{3^{7/3}5}{2^{5/3}}$
  2. $3^{7/3}2^{5/3}$
  3. $\frac{3^{5/3}2^{5/3}}5$
  4. $\frac{3^{7/3}2^{5/3}}5$
  5. $\frac{3^{4/3}2^{1/3}}5$

Question 3

Given that $e^{2 x}e^y=e^2$ and $\ln(x+2 y)=\ln 5+\ln 2$, find the value of x+y.
  1. 2
  2. 4
  3. 8
  4. 10
  5. 12

Question 4

Solve for x in the following equation $$\log_9 x+\log_3(27 x)=30.$$
  1. $3^{16}$
  2. $3^{17}$
  3. $3^{18}$
  4. $3^{19}$
  5. $3^{20}$

Question 5

Given that $\cos\alpha=12/13$, where $0^\circ\le\alpha\le 90^\circ$, and that $\tan\beta=3/4$, where $180^\circ\le\beta\le 270^\circ$. Find $\sin(\beta-\alpha)$.
  1. $-16/65$
  2. $16/65$
  3. $-56/65$
  4. $56/65$
  5. $33/65$

Question 6

Which of the following is the greatest?
  1. $\sqrt{\frac 16}$
  2. $\left(\frac 1{218}\right)^{1/6}$
  3. $\left(\frac 1{15}\right)^{1/3}$
  4. $\left(\frac 17\right)^{1/3}\left(\frac 15\right)^{1/6}$
  5. $\left(\frac 18\right)^{1/4}\left(\frac 13\right)^{1/3}$

Question 7

Find all the positive values of x for which $$2|x^2-3 x|\le x.$$
  1. $5/2\le x\le 3$
  2. $1/2\le x\le 5/2$
  3. $5/2\le x\le 7/2$
  4. $0\le x\le 7/2$
  5. $1/2\le x\le 1$

Question 8

Suppose $0^\circ\le x\le 180^\circ$. Find all the possible values of x such that $$3\cos^2(2 x)+4\sin(2 x)-\sin^2(2 x)=0.$$
  1. $x=55^\circ,135^\circ$
  2. $x=75^\circ,125^\circ$
  3. $x=105^\circ,165^\circ$
  4. $x=95^\circ,145^\circ$
  5. $x=115^\circ,175^\circ$

Question 9

Which of the following is equal to $$\frac 1{\sqrt{2000}+\sqrt{2004}}+\frac 1{\sqrt{2004}+\sqrt{2008}}+\frac 1{\sqrt{2008}+\sqrt{2012}}+\frac 1{\sqrt{2012}+\sqrt{2016}}?$$
  1. $3\sqrt{14}-5\sqrt 5$
  2. $4\sqrt{14}-5\sqrt 5$
  3. $3\sqrt{14}-5\sqrt 3$
  4. $3\sqrt{14}-3\sqrt 5$
  5. $3\sqrt 7-\sqrt 5$

Question 10

Let a=cos 282°, b=cos 349°, c=sin 102° and d=sin 169°. Which of the following is true?
  1. $b\gt a\gt c\gt d$
  2. $b\gt c\gt a\gt d$
  3. $c\gt a\gt d\gt b$
  4. $a\gt d\gt b\gt c$
  5. $a\gt c\gt d\gt b$

Question 11

The expression $3 x^3+Ax^2+Bx-10$, where A and B are integers, is divisible by 3 x -1 but leaves a remainder of -14 when divided by x+2. Find the value of A+B.

Question 12

Find the largest positive integer p such that $-x^2+2(1+2 p)x-2 p-31$ is always negative.

Question 13

Find the largest integer smaller than $(2+\sqrt 2)^4$.

Question 14

Given that x>0, and $5^x+\frac 1{5^x}=\frac 52$, find the value of $2^{2/x}$.

Question 15

Find the largest value of x satisfying the following equation: $$\sqrt{3 x-47}+\sqrt{x-1}=6.$$

Question 16

Find the number of solutions for the following equation: $$|2\sin x+1|=2|\cos x|,$$ where $0^\circ\le x\le 360^\circ$.

Question 17

Find the maximum value of $6\cos^2 x-24\sin x\cos x-4\sin^2 x$, where $0^\circ\le x\le 360^\circ$.

Question 18

Given that $\tan\alpha=1/3$ and $\tan\beta=3$. Find the value of $$\frac{3\sin(\alpha+\beta)-6\sin\alpha\cos\beta}{2\sin\alpha\sin\beta+\cos(\alpha+\beta)}.$$

Question 19

Consider the function $$f(x)=3|x+2|+3|x|-|x-1|-5|x-2|,$$ where x is any real number. Suppose A and B are the maximum and minimum values of f(x) respectively. Find the value of A-B.

Question 20

Find the coefficient of $x^4$ in the expansion of $$\left(x^2+\frac 2{x^2}\right)\left(x+\frac 1{x^3}\right)^{10}.$$