SMO 2013 Open Round 1

The sum

$$\frac1{1\times2\times3}+\frac1{2\times3\times4}+\frac1{3\times4\times5}+\cdots+\frac1{100\times101\times102}$$

can be expressed as $a/b$, a fraction in its simplest form. Find $a+b$.

Determine the maximum value of $\dfrac{1+\cos x}{\sin x+\cos x+2}$, where $x$ ranges over all real numbers.

Let $\tan\alpha$ and $\tan\beta$ be two solutions of the equation $x^2-3x-3=0$. Find the value of

$$\left|\sin^2(\alpha+\beta)-3\sin(\alpha+\beta)\cos(\alpha+\beta)-3\cos^2(\alpha+\beta)\right|.$$

(Note: $|x|$ denotes the absolute value of $x$.)

Suppose that $a_1,a_2,a_3,a_4,\ldots$ is an arithmetic progression with $a_1>0$ and $3a_8=5a_{13}$. Let $S_n=a_1+a_2+\cdots+a_n$ for all integers $n\ge1$. Find the integer $n$ such that $S_n$ has the maximum value.

If $g(x)=\tan(x/2)$ for $0\lt x\lt \pi$ and $f(g(x))=\sin2x$, find the value of $k$ such that $k f(\sqrt2/2)=36\sqrt2$.

Let $g(x)$ be a strictly increasing function defined for all $x\ge0$. It is known that the range of $t$ satisfying

$$g(2t^2+t+5)\lt g(t^2-3t+2)$$

is $b\lt t\lt a$. Find $a-b$.

The figure below shows an $8\times9$ rectangular board.



How many squares are there in the above rectangular board?

Let $a,b,c$ be positive real numbers such that $a+b+c=2013$. Find the maximum value of $\sqrt{3a+12}+\sqrt{3b+12}+\sqrt{3c+12}$.

Let $A=\cos^2 10^\circ+\cos^2 50^\circ-\sin40^\circ\sin80^\circ$. Determine the value of $100A$.

Assume that $a_i\in\{1,-1\}$ for all $i=1,2,\ldots,2013$. Find the least positive number of the following expression:

$$\sum_{1\le i\lt j\le2013}a_i a_j.$$

Let $f$ be a function defined on nonzero real numbers such that

$$\frac{27f(-x)}x-x^2f\left(\frac1x\right)=-2x^2$$

for all $x\ne0$. Find $f(3)$.

In the figure below, $ABCD$ is a square with $AB=20$ cm (not drawn to scale). Assume that $M$ is a point such that the area of the shaded region is $40$ cm$^2$. Find $AM$ in centimetres.

In the triangle $ABC$, a circle passes through the point $A$, the midpoint $E$ of $AC$, the midpoint $F$ of $AB$ and is tangent to the side $BC$ at $D$. Suppose

$$\frac{AB}{AC}+\frac{AC}{AB}=4.$$

Determine the size of $\angle EDF$ in degrees.

Let $a_1,a_2,a_3,\ldots$ be a sequence of real numbers in a geometric progression. Let $S_n=a_1+a_2+\cdots+a_n$ for all integers $n\ge1$. Assume that $S_{10}=10$ and $S_{30}=70$. Find the value of $S_{40}$.

Find the number of three-digit numbers which are multiples of $3$ and are formed by the digits $0,1,2,3,4,5,6,7$ without repetition.

All the positive integers which are co-prime to $2012$ are grouped in an increasing order in such a way that the $n$th group has $2n-1$ numbers. So, the first three groups in this grouping are $(1)$, $(3,5,7)$, $(9,11,13,15,17)$. It is known that $2013$ belongs to the $k$th group. Find the value of $k$.

(Note: Two integers are said to be co-prime if their greatest common divisor is $1$.)

The numbers $1,2,3,\ldots,7$ are randomly divided into two non-empty subsets. The probability that the sum of the numbers in the two subsets being equal is $p/q$ expressed in the lowest term. Find $p+q$.

Find the number of real roots of the equation $\log_{10}^{2}x-\lfloor\log_{10}x\rfloor-2=0$.

(Note: $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.)

In the triangle $ABC$, $AB=AC$, $\angle A=90^\circ$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AC$ and $F$ is a point on $AB$ such that $BE$ intersects $CF$ at $P$ and $B,D,P,F$ lie on a circle. Let $AD$ intersect $CP$ at $H$. Given $AP=\sqrt5+2$, find the length of $PH$.

Find the total number of positive integers $n$ not more than $2013$ such that $n^4+5n^2+9$ is divisible by $5$.