SMO 2010 Senior Round 1

Find the value of

$$\frac{(1\times2\times3)+(2\times4\times6)+(3\times6\times9)+\cdots+(335\times670\times1005)}{(1\times3\times6)+(2\times6\times12)+(3\times9\times18)+\cdots+(335\times1005\times2010)}.$$

If $a,b,c,d$ are real numbers such that

$$\frac{b+c+d}{a}=\frac{a+c+d}{b}=\frac{a+b+d}{c}=\frac{a+b+c}{d}=r,$$

find the value of $r$.

If $0\lt x\lt \pi/2$, $\sin x-\cos x=\pi/4$ and

$$\tan x+\frac1{\tan x}=\frac{a}{b-\pi^c},$$

where $a,b,c$ are positive integers, find the value of $a+b+c$.

Find the value of $\sqrt{14^3+15^3+16^3+\cdots+24^3+25^3}$.

In the figure below, $ABC$ is an isosceles triangle inscribed in a circle with centre $O$ and diameter $AD$, with $AB=AC$. $AD$ intersects $BC$ at $E$, and $F$ is the midpoint of $OE$. Given that $BD$ is parallel to $FC$ and $BC=2\sqrt5$ cm, find the length of $CD$ in cm.

Find the number of ordered pairs $(x,y)$, where $x$ is an integer and $y$ is a perfect square, such that $y=(x-90)^2-4907$.

Let $S=\{1,2,3,\ldots,9,10\}$. A non-empty subset of $S$ is considered “Good” if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets $\{4,8\}$, $\{3,4,7,8\}$ and $\{1,3,6,8,10\}$ are “Good”. How many subsets of $S$ are “Good”?

If the graph of a quadratic function $f(x)=ax^2+bx+c$ $(a\ne0)$ passes through two distinct points $(r,k)$ and $(s,k)$, what is $f(r+s)$?

Find the number of positive integers $k\lt 100$ such that $2(3^{6n})+k(2^{3n+1})-1$ is divisible by $7$ for any positive integer $n$.

Let $ABCD$ be a trapezium with $AD$ parallel to $BC$ and $\angle ADC=90^\circ$, as shown in the figure below. Given that $M$ is the midpoint of $AB$ with $CM=13/2$ cm and $BC+CD+DA=17$ cm, find the area of the trapezium $ABCD$ in cm$^2$.

The area of a rectangle remains unchanged when either its length is increased by $6$ units and width decreased by $2$ units, or its length decreased by $12$ units and its width increased by $6$ units. If the perimeter of the original rectangle is $x$ units, find the value of $x$.

For $r=1,2,3,\ldots$, let $u_r=1+2+3+\cdots+r$. Find the value of

$$\frac1{1/u_1}+\frac2{1/u_1+1/u_2}+\frac3{1/u_1+1/u_2+1/u_3}+\cdots+\frac{100}{1/u_1+1/u_2+\cdots+1/u_{100}}.$$

If $2010!=M\times10^k$, where $M$ is an integer not divisible by $10$, find the value of $k$.

If $a>b>1$ and

$$\frac1{\log_a b}+\frac1{\log_b a}=\sqrt{1229},$$

find the value of $\dfrac1{\log_{ab}b}-\dfrac1{\log_{ab}a}$.

For any real number $x$, let $\lceil x\rceil$ denote the smallest integer that is greater than or equal to $x$ and $\lfloor x\rfloor$ denote the largest integer that is less than or equal to $x$ (for example, $\lceil1.23\rceil=2$ and $\lfloor1.23\rfloor=1$). Find the value of

$$\sum_{k=1}^{2010}\left\lceil\frac{2010}{k}-\left\lfloor\frac{2010}{k}\right\rfloor\right\rceil.$$

Let $f(x)=\dfrac{x^{2010}}{x^{2010}+(1-x)^{2010}}$. Find the value of

$$f\left(\frac1{2011}\right)+f\left(\frac2{2011}\right)+f\left(\frac3{2011}\right)+\cdots+f\left(\frac{2010}{2011}\right).$$

If $a,b,c$ are positive real numbers such that

$$ab+a+b=bc+b+c=ca+c+a=35,$$

find the value of $(a+1)(b+1)(c+1)$.

In the figure below, $AB$ and $CD$ are parallel chords of a circle with centre $O$ and radius $r$ cm. It is given that $AB=46$ cm, $CD=18$ cm and $\angle AOB=3\times\angle COD$. Find the value of $r$.

Find the number of ways that $2010$ can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most $1$.

Find the largest possible value of $n$ such that there exist $n$ consecutive positive integers whose sum is equal to $2010$.