SMO 2013 Senior Round 1

Question 1

A shop sells two kinds of products $A$ and $B$. One day, a salesman sold both $A$ and $B$ at the same price of 2100 dollars to a customer. Suppose product $A$ makes a profit of $20\%$ but product $B$ makes a loss of $20\%$. Then this deal

makes a profit of 70 dollars
makes a loss of 70 dollars
makes a profit of 175 dollars
makes a loss of 175 dollars
makes no profit or loss

Question 2

How many integer solutions does the equation $(x^3-x-1)^{x+2013}=1$ have?

$0$
$1$
$2$
$3$
More than 3

Question 3

In the $xy$-plane, which of the following is the reflection of the graph of

$$y=\frac{1+x}{1+x^2}$$

about the line $y=2x$?

$x=\frac{1+y}{1+y^2}$
$x=\frac{-1+y}{1+y^2}$
$x=-\frac{1+y}{1+y^2}$
$x=\frac{1-y}{1+y^2}$
None of the above

Question 4

Let $n$ be a positive integer. Find the number of possible remainders when

$$2013^n-1803^n-1781^n+1774^n$$

is divided by $203$.

$1$
$2$
$3$
$4$
More than 4

Question 5

Find the number of integers $n$ such that the equation

$$xy^2+y^2-x-y=n$$

has an infinite number of integer solutions $(x,y)$.

$0$
$1$
$2$
$3$
More than 3

Question 6

If $0\lt \theta\lt \pi/4$ is such that $\operatorname{cosec}\theta-\sec\theta=\sqrt{13}/6$, then $\cot\theta-\tan\theta$ equals

$\frac{\sqrt{13}}6$
$\frac{\sqrt{12}}6$
$\frac{\sqrt5}6$
$\frac{13}6$
$\frac56$

Question 7

$ABCD$ is a trapezium inscribed in a circle centred at $O$. It is given that $AB\parallel CD$, $\angle COD=3\angle AOB$, and $AB/CD=2/5$. Find the ratio

$$\frac{\text{area of }\triangle BOC}{\text{area of }\triangle AOB}.$$

Question illustration

$\frac32$
$\frac74$
$\frac{\sqrt3}{\sqrt2}$
$\frac{\sqrt5}2$
$\frac{\sqrt7}{\sqrt2}$

Question 8

A square $ABCD$ and an equilateral triangle $PQR$ are inscribed in a circle centred at $O$ in such a way that $AB\parallel QR$. The sides $PQ$ and $PR$ of the triangle meet the side $AB$ of the square at $X$ and $Y$ respectively. The value of $\tan\angle XOY$ is

Question illustration

$\frac1{\sqrt3}$
$1$
$\frac{\sqrt6-\sqrt3}{\sqrt2}$
$\frac{2\sqrt2-2}{\sqrt3}$
$\sqrt3$

Question 9

Two people go to the same swimming pool between 2:00 p.m. and 5:00 p.m. at random time and each swims for one hour. What is the chance that they meet?

$\frac19$
$\frac29$
$\frac13$
$\frac49$
$\frac59$

Question 10

Given a triangle $ABC$, let $B^{\prime}$ and $C^{\prime}$ be points on the sides $AB$ and $AC$ such that $BB^{\prime}=CC^{\prime}$. Let $O$ and $O^{\prime}$ be the circumcentres (the centres of the circumscribed circles) of $ABC$ and $AB^{\prime}C^{\prime}$, respectively. Suppose $OO^{\prime}$ intersects lines $AB^{\prime}$ and $AC^{\prime}$ at $B^{\prime\prime}$ and $C^{\prime\prime}$, respectively. If $AB=\frac12 AC$, then

Question illustration

$AB^{\prime\prime}<\frac12AC^{\prime\prime}$
$AB^{\prime\prime}=\frac12AC^{\prime\prime}$
$\frac12AC^{\prime\prime}<AB^{\prime\prime}<AC^{\prime\prime}$
$AB^{\prime\prime}=AC^{\prime\prime}$
$AB^{\prime\prime}>AC^{\prime\prime}$

Question 11

Suppose a right-angled triangle is inscribed in a circle of radius $100$. Let $\alpha$ and $\beta$ be its acute angles. If $\tan\alpha=4\tan\beta$, find the area of the triangle.

Question 12

Let $f(x)=\dfrac{1+10x}{10-100x}$. Set $f^n=\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ terms}}$. Find the value of

$$f\left(\frac12\right)+f^2\left(\frac12\right)+f^3\left(\frac12\right)+\cdots+f^{6000}\left(\frac12\right).$$

Question 13

Let $AB$ and $CD$ be perpendicular segments intersecting at point $P$. Suppose that $AP=2$, $BP=3$ and $CP=1$. If all the points $A,B,C,D$ lie on a circle, find the length of $DP$.

Question 14