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SMO 2005 Senior Round 1

Question 1

What is the smallest positive prime factor of the integer $2005^{2007}+2007^{2005}$?
  1. $5$
  2. $7$
  3. $2$
  4. $11$
  5. $3$

Question 2

What is the value of
$$\frac{2005^2+2\times2005\times1995+1995^2}{800}?$$
  1. $20000$
  2. $2000$
  3. $200000$
  4. $2000000$
  5. None of the above

Question 3

Let $p$ be a real number such that the equation $2y^2-8y=p$ has only one solution. Then which statement is true?
  1. $p\lt 8$
  2. $p=8$
  3. $p\gt -8$
  4. $p=-8$
  5. $p\lt -8$

Question 4

What is the sum of the last two digits of the integer $1!+2!+3!+\cdots+2005!$?
  1. $3$
  2. $4$
  3. $5$
  4. $6$
  5. $7$

Question 5

$5002^{2005\log_{5002}2005}$ is equal to which of the following?
  1. $1$
  2. $2005^{2005}$
  3. $5002^{2005}$
  4. $2005^{5002}$
  5. $5002^{5002}$

Question 6

In the diagram, $P,Q,R$ are three points on the circle whose centre is $O$. The lines $PO$ and $QR$ are produced to meet at $S$. Suppose that $RS=OP$, and $\angle PSQ=12^\circ$ and $\angle POQ=x^\circ$. Find the value of $x$.

Question illustration
  1. $36$
  2. $42$
  3. $48$
  4. $54$
  5. $60$

Question 7

Let $x,y$ be positive real numbers. What is the smallest possible value of
$$\frac{16}{x}+\frac{108}{y}+xy?$$
  1. $16$
  2. $18$
  3. $24$
  4. $30$
  5. $36$

Question 8

In the Cartesian plane, the graph of the function $y=\frac{2x-1}{x-1}$ is reflected about the line $y=-x$. What is the equation for the graph of the image of the reflection?
  1. $y=\frac{x-1}{2x+1}$
  2. $y=\frac{x-1}{x+2}$
  3. $y=\frac{2x-1}{x+1}$
  4. $y=\frac{x+1}{2-x}$
  5. $y=\frac{-1-x}{2+x}$

Question 9

Simplify
$$\sqrt{2\left(1+\sqrt{1+\left(\frac{x^4-1}{2x^2}\right)^2}\right)},$$
where $x$ is any positive real number.
  1. $\frac{x^2+1}{\sqrt2x}$
  2. $\frac{x^2+1}{x}$
  3. $\sqrt{\frac{x^2+1}{2x^2}}$
  4. $x^2+1$
  5. $\frac{x^2-1}{\sqrt2x}$

Question 10

Let $x,y$ be real numbers such that
$$x^2+y^2=2x-2y+2.$$
What is the largest possible value of $x^2+y^2$?
  1. $10+8\sqrt2$
  2. $8+6\sqrt2$
  3. $6+4\sqrt2$
  4. $4+2\sqrt2$
  5. None of the above

Question 11

Find the greatest integer less than $(2+\sqrt3)^4$.

Question 12

$\triangle ABC$ is a triangle such that $\angle C=90^\circ$. Suppose $AC=156$ cm, $AB=169$ cm and the perpendicular distance from $C$ to $AB$ is $x$ cm. Find the value of $x$.

Question 13

Find the sum of all the real numbers $x$ that satisfy the equation
$$(3^x-27)^2+(5^x-625)^2=(3^x+5^x-652)^2.$$

Question 14

Three positive integers are such that they differ from each other by at most $6$. It is also known that the product of these three integers is $2808$. What is the smallest integer among them?

Question 15

Simplify
$$\frac{2005^2(2004^2-2003)}{(2004^2-1)(2004^3+1)}\times\frac{2003^2(2004^2+2005)}{2004^3-1}.$$

Question 16

Consider the function $f(x)=\frac1{3^x+\sqrt3}$. Find the value of
$$\sqrt3\,[f(-5)+f(-4)+f(-3)+f(-2)+f(-1)+f(0)+f(1)+f(2)+f(3)+f(4)+f(5)+f(6)].$$

Question 17

Let $A$ and $B$ be two positive four-digit integers such that $A\times B=16^5+2^{10}$. Find the value of $A+B$.

Question 18

A triangle $ABC$ is inscribed in a circle of radius $4$ cm. Suppose that $\angle A=60^\circ$, $AC-AB=4$ cm, and the area of $ABC$ is $x$ cm$^2$. Find the value of $x^2$.

Question 19

Let $a,b,c$ be real numbers such that
$$a=8-b,\qquad c^2=ab-16.$$
Find the value of $a+c$.

Question 20

Let $a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8$ be positive integers such that
$$a_1^2+(2a_2)^2+(3a_3)^2+(4a_4)^2+(5a_5)^2+(6a_6)^2+(7a_7)^2+(8a_8)^2=204.$$
Find $a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8$.