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SMO 2017 Senior Round 1

Question 1

Find all the values of $k$ for which the expression $x^2+k(k-x)+3k$ is positive for all $x$.
  1. $0\lt k\lt 4$
  2. $-4\lt k\lt 0$
  3. $k\lt 0,\ k>4$
  4. $k>0,\ k\lt -4$
  5. None of the above

Question 2

Simplify $$\log_3 12+\log_9 16-\log_{27}8.$$
  1. $\log_3 4$
  2. $\log_3 8$
  3. $\log_3 16$
  4. $\log_3 24$
  5. $\log_3 32$

Question 3

Solve for $x$ in the following equation $$7^{x-1}+\frac17-\frac6{7^x}=0.$$
  1. $\log_6 7$
  2. $\log_6 14$
  3. $\log_7 6$
  4. $\log_7 12$
  5. $\log_7 14$

Question 4

When a polynomial is divided by $(x-2)$ and $x^2-3x+2$, the remainders are 11 and $Ax+5$ respectively, where $A$ is some integer. Find the remainder when it is divided by $(x-1)$.
  1. $7$
  2. $8$
  3. $9$
  4. $10$
  5. $11$

Question 5

Simplify $$\frac{(\sqrt5)^3-(\sqrt2)^3}{\sqrt5-\sqrt2}-\frac{(\sqrt5)^3+(\sqrt2)^3}{\sqrt5+\sqrt2}.$$
  1. $\sqrt{10}$
  2. $2\sqrt{10}$
  3. $3\sqrt{10}$
  4. $\sqrt5-\sqrt2$
  5. $\sqrt5+\sqrt2$

Question 6

Which of the following is the largest?
  1. $3^{100}$
  2. $90^{10}$
  3. $82^{25}$
  4. $8^{49}+7^{49}$
  5. $2^{150}$

Question 7

Suppose $270^\circ\lt \alpha\lt 360^\circ$. Which of the following is equal to $$\sqrt{\cos^3\alpha}-\sqrt{\sin^2\alpha\cos\alpha}\ ?$$
  1. $(\sin\alpha+\cos\alpha)\sqrt{\cos\alpha}$
  2. $(\sin\alpha-\cos\alpha)\sqrt{\cos\alpha}$
  3. $(\cos\alpha-\sin\alpha)\sqrt{\cos\alpha}$
  4. $(-\sin\alpha-\cos\alpha)\sqrt{\cos\alpha}$
  5. $\sqrt{\cos\alpha}$

Question 8

Find the range of values of $x$ that satisfy both of the following inequalities: $$(x+3)^2\le(3x-1)^2,\qquad3x\ge x^2.$$
  1. $0\le x\le1$
  2. $1\le x\le2$
  3. $2\le x\le3$
  4. $3\le x\le4$
  5. None of the above

Question 9

If $f(x)=-x+6\sqrt{x+16}-5$, where $-16\le x\le0$, find the range of $f(x)$.
  1. $-29\le f(x)\le20$
  2. $-29\le f(x)\le19$
  3. $11\le f(x)\le19$
  4. $19\le f(x)\le20$
  5. $11\le f(x)\le20$

Question 10

A circle is given by the equation $x^2-4x+y^2+8y=5$. Find the equation of the tangent line to the circle at the point $(-2,-1)$.
  1. $3y-4x-5=0$
  2. $-7y+x-5=0$
  3. $3y+2x+7=0$
  4. $3y-2x-1=0$
  5. $4y-3x-2=0$

Question 11

Suppose $x^2+3x+15$ is a factor of $2x^4+mx^2+30n$ where $m,n$ are integers. Find the value of $mn$.

Question 12

How many different real numbers $x$, where $0^\circ\le x\le360^\circ$, satisfy the equation $$3\sin x-4\sin^3x=\frac12\ ?$$

Question 13

Suppose $\frac{1-\sin2A}{1+\cos2A}=\tan A-1\ne0$. Find $\tan^2A$.

Question 14

Given that $\sin(\frac\pi4-x)=\frac35$, $0\lt x\lt \frac\pi4$, find $\frac{15\cos2x}{\cos(\frac\pi4+x)}$.

Question 15

Find the coefficient of $x^3$ in the expansion of $(1+2x+3x^2+4x^3)^8$.

Question 16

Find the maximum value of $\frac{1+x-99x^2}{x(x^2+1)}$, where $\frac1{100}\le x\le\frac1{10}$.

Question 17

Four points $O,P,Q,R$ have coordinates $(0,0)$, $(0,30)$, $(x,y)$ and $(20,0)$ respectively, where the point $Q$ lies on the curve $y=50-x^2$ in the first quadrant. Let $\Pi$ be the polygon $OPQR$ with four sides $OP$, $PQ$, $QR$ and $RO$. What is the maximum possible area of $\Pi$ if this area must be an integer?

Question 18

There are 5 girls and 5 boys in a junior class, and 4 girls and 9 boys in a senior class. A committee of 7 members is to be formed by selecting students from these two classes. Find the number of ways this can be done if the committee must have exactly 4 seniors and exactly 5 boys.

Question 19

Suppose $x$ is a real number. Find the largest possible integer that can be attained by the expression $$\frac{7770-|x-10|}{|x-5|+|x-15|}.$$

Question 20

Evaluate $$\left(10-\frac{4\cos(\frac{4\pi}7)\cos(\frac{3\pi}7)\cos(\frac\pi7)-\cos(\frac{2\pi}7)-1}{2\cos(\frac\pi7)}\right)^2.$$