Let $k , n$ and $r$ be positive integers.
(a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying
$$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$
(b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in \mathbb { R }$ and for all $t \geq 0$, $$f ( x ) = f \left( e ^ { t } x \right)$$ Show that $f$ is a constant function.

Let $a \geq b \geq c > 0$ be real numbers such that for all $n \in \mathbb { N }$, there exist triangles of side lengths $a ^ { n } , b ^ { n } , c ^ { n }$. Prove that the triangles are isosceles.

Let $a , b , c \in \mathbb { N }$ be such that $$a ^ { 2 } + b ^ { 2 } = c ^ { 2 } \text { and } c - b = 1$$ Prove that ( i ) $a$ is odd, ( ii ) $b$ is divisible by 4 ,
(iii) $a ^ { b } + b ^ { a }$ is divisible by $c$.

Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r > 1$, are of the form $n = 2^{l}$ for some $l \geq 0$.

Let $f$ be a polynomial with integer coefficients. Define $$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$ and $$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$ If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius 1 is regular (i.e., has equal sides).

Consider a right-angled triangle with integer-valued sides $a < b < c$ where $a, b, c$ are pairwise co-prime. Let $d = c - b$. Suppose $d$ divides $a$. Then
(a) Prove that $d \leq 2$.
(b) Find all such triangles (i.e. all possible triplets $a, b, c$) with perimeter less than 100.

There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $n + 1$ friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below.
$$\begin{aligned} f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\ f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y ) \end{aligned}$$
for all integers $x , y$.
(i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.

Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.

Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

Let $a _ { 0 } , a _ { 1 } , \cdots , a _ { 19 } \in \mathbb { R }$ and
$$P ( x ) = x ^ { 20 } + \sum _ { i = 0 } ^ { 19 } a _ { i } x ^ { i } , \quad x \in \mathbb { R }$$
If $P ( x ) = P ( - x )$ for all $x \in \mathbb { R }$, and
$$P ( k ) = k ^ { 2 } , \text{ for } k = 0,1,2 \cdots , 9$$
then find
$$\lim _ { x \rightarrow 0 } \frac { P ( x ) } { \sin ^ { 2 } x }$$

If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

Let $a , b , c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a + b + c = 6$ and $a b + b c + a c = 9$. Suppose $a < b < c$. Show that
$$0 < a < 1 < b < 3 < c < 4$$

Let $a , b , c$ and $d$ be four non-negative real numbers where $a + b + c + d = 1$. The number of different ways one can choose these numbers such that $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + d ^ { 2 } = \max \{ a , b , c , d \}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .

Amongst all polynomials $p ( x ) = c _ { 0 } + c _ { 1 } x + \cdots + c _ { 10 } x ^ { 10 }$ with real coefficients satisfying $| p ( x ) | \leq | x |$ for all $x \in [ - 1,1 ]$, what is the maximum possible value of $\left( 2 c _ { 0 } + c _ { 1 } \right) ^ { 10 }$ ?
(A) $4 ^ { 10 }$
(B) $3 ^ { 10 }$
(C) $2 ^ { 10 }$
(D) 1

Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show that the equation $P(P(x)) = 0$ has at least as many distinct real roots as the equation $P(x) = 0$.

Let $\mathbb { Z }$ denote the set of integers. Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be such that $f ( x ) f ( y ) = f ( x + y ) + f ( x - y )$ for all $x , y \in \mathbb { Z }$. If $f ( 1 ) = 3$, then $f ( 7 )$ equals
(A) 840
(B) 844
(C) 843
(D) 842

If $x _ { 1 } > x _ { 2 } > \cdots > x _ { 10 }$ are real numbers, what is the least possible value of $$\left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 1 } - x _ { 2 } } \right) \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 2 } - x _ { 3 } } \right) \cdots \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 9 } - x _ { 10 } } \right) ?$$ (A) $10 ^ { 10 }$
(B) $10 ^ { 9 }$
(C) $9 ^ { 9 }$
(D) $9 ^ { 10 }$

In a triangle $A B C$, consider points $D$ and $E$ on $A C$ and $A B$, respectively, and assume that they do not coincide with any of the vertices $A , B , C$. If the segments $B D$ and $C E$ intersect at $F$, consider the areas $w , x , y , z$ of the quadrilateral $A E F D$ and the triangles $B E F , B F C , C D F$, respectively.
(a) Prove that $y ^ { 2 } > x z$.
(b) Determine $w$ in terms of $x , y , z$.

If $[ x ]$ denotes the largest integer less than or equal to $x$, then $$\left[ ( 9 + \sqrt { 80 } ) ^ { 20 } \right]$$ equals
(A) $( 9 + \sqrt { 80 } ) ^ { 20 } - ( 9 - \sqrt { 80 } ) ^ { 20 }$.
(B) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 20$.
(C) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 1$.
(D) $( 9 - \sqrt { 80 } ) ^ { 20 }$.

Suppose $n \geq 2$. Consider the polynomial
$$Q _ { n } ( x ) = 1 - x ^ { n } - ( 1 - x ) ^ { n } .$$
Show that the equation $Q _ { n } ( x ) = 0$ has only two real roots, namely 0 and 1.

Let $ABCD$ be a quadrilateral with all internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta _ { 1 } , \Delta _ { 2 } , \Delta _ { 3 }$ and $\Delta _ { 4 }$ denote the areas of the shaded triangles shown. Prove that
$$\Delta _ { 1 } - \Delta _ { 2 } + \Delta _ { 3 } - \Delta _ { 4 } = 0 .$$

Let $P ( x )$ be a polynomial with real coefficients. Let $\alpha _ { 1 } , \ldots , \alpha _ { k }$ be the distinct real roots of $P ( x ) = 0$. If $P ^ { \prime }$ is the derivative of $P$, show that for each $i = 1,2 , \ldots , k$,
$$\lim _ { x \rightarrow \alpha _ { i } } \frac { \left( x - \alpha _ { i } \right) P ^ { \prime } ( x ) } { P ( x ) } = r _ { i } ,$$
for some positive integer $r _ { i }$.