Show that $\frac { \ln ( 12 ) } { \ln ( 18 ) }$ is irrational.

Show that $$\lim _ { x \rightarrow \infty } \frac { x ^ { 100 } \ln ( x ) } { e ^ { x } \tan ^ { - 1 } \left( \frac { \pi } { 3 } + \sin x \right) } = 0$$

a) Let $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H respectively be the midpoints of the sides $\mathrm { AB } , \mathrm { BC } , \mathrm { CD }$ and DA of a convex quadrilateral ABCD. Show that EFGH is a parallelogram whose area is half that of ABCD. b) Let $\mathrm { E } = ( 0,0 ) , \mathrm { F } = ( 0 , - 1 ) , \mathrm { G } = ( 1 , - 1 ) , \mathrm { H } = ( 1,0 )$. Find all points $\mathrm { A } = ( p , q )$ in the first quadrant such that $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H respectively are the midpoints of the sides $\mathrm { AB } , \mathrm { BC } , \mathrm { CD }$ and DA of a convex quadrilateral ABCD.

a) We want to choose subsets $A _ { 1 } , A _ { 2 } , \ldots , A _ { k }$ of $\{ 1,2 , \ldots , n \}$ such that any two of the chosen subsets have nonempty intersection. Show that the size $k$ of any such collection of subsets is at most $2 ^ { n - 1 }$. b) For $n > 2$ show that we can always find a collection of $2 ^ { n - 1 }$ subsets $A _ { 1 } , A _ { 2 } , \ldots$ of $\{ 1,2 , \ldots , n \}$ such that any two of the $A _ { i }$ intersect, but the intersection of all $A _ { i }$ is empty.

Define $$x = \sum _ { i = 1 } ^ { 10 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } \quad \text { and } \quad y = \sum _ { i = 0 } ^ { 9 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } .$$ Show that a) $x < \frac { \pi } { 6 } < y$ and b) $\frac { x + y } { 2 } < \frac { \pi } { 6 }$. (Hint: Relate these sums to an integral.)

Using the steps below, find the value of $x ^ { 2012 } + x ^ { - 2012 }$, where $x + x ^ { - 1 } = \frac { \sqrt { 5 } + 1 } { 2 }$. a) For any real $r$, show that $\left| r + r ^ { - 1 } \right| \geq 2$. What does this tell you about the given $x$? b) Show that $\cos \left( \frac { \pi } { 5 } \right) = \frac { \sqrt { 5 } + 1 } { 4 }$, e.g. compare $\sin \left( \frac { 2 \pi } { 5 } \right)$ and $\sin \left( \frac { 3 \pi } { 5 } \right)$. c) Combine conclusions of parts a and b to express $x$ and therefore the desired quantity in a suitable form.

For $n > 1$, a configuration consists of $2n$ distinct points in a plane, $n$ of them red, the remaining $n$ blue, with no three points collinear. A pairing consists of $n$ line segments, each with one blue and one red endpoint, such that each of the given $2n$ points is an endpoint of exactly one segment. Prove the following. a) For any configuration, there is a pairing in which no two of the $n$ segments intersect. (Hint: consider total length of segments.) b) Given $n$ red points (no three collinear), we can place $n$ blue points such that any pairing in the resulting configuration will have two segments that do not intersect. (Hint: First consider the case $n = 2$.)

Let $f ( x )$ be a polynomial with integer coefficients such that for each nonnegative integer $n , f ( n ) = \mathrm { a }$ perfect power of a prime number, i.e., of the form $p ^ { k }$, where $p$ is prime and $k$ a positive integer. ($p$ and $k$ can vary with $n$.) Show that $f$ must be a constant polynomial using the following steps or otherwise. a) If such a polynomial $f ( x )$ exists, then there is a polynomial $g ( x )$ with integer coefficients such that for each nonnegative integer $n , g ( n ) =$ a perfect power of a fixed prime number. b) Show that a polynomial $g ( x )$ as in part a must be constant.

Let $N$ be the set of non-negative integers. Suppose $f : N \rightarrow N$ is a function such that $f ( f ( f ( n ) ) ) < f ( n + 1 )$ for every $n \in N$. Prove that $f ( n ) = n$ for all $n$ using the following steps or otherwise. a) If $f ( n ) = 0$, then $n = 0$. b) If $f ( x ) < n$, then $x < n$. (Start by considering $n = 1$.) c) $f ( n ) < f ( n + 1 )$ and $n < f ( n + 1 )$ for all $n$. d) $f ( n ) = n$ for all $n$.

In triangle $ABC$, the bisector of angle $A$ meets side $BC$ in point $D$ and the bisector of angle $B$ meets side $AC$ in point $E$. Given that $DE$ is parallel to $AB$, show that $\mathrm{AE} = \mathrm{BD}$ and that the triangle $ABC$ is isosceles.

Define $f _ { k } ( n )$ to be the sum of all possible products of $k$ distinct integers chosen from the set $\{ 1,2 , \ldots , n \}$, i.e., $$f _ { k } ( n ) = \sum _ { 1 \leq i _ { 1 } < i _ { 2 } < \ldots < i _ { k } \leq n } i _ { 1 } i _ { 2 } \ldots i _ { k }$$ a) For $k > 1$, write a recursive formula for the function $f _ { k }$, i.e., a formula for $f _ { k } ( n )$ in terms of $f _ { \ell } ( m )$, where $\ell < k$ or ($\ell = k$ and $m < n$). b) Show that $f _ { k } ( n )$, as a function of $n$, is a polynomial of degree $2k$. c) Express $f _ { 2 } ( n )$ as a polynomial in variable $n$.

Let $p$, $q$ and $r$ be real numbers with $p^2 + q^2 + r^2 = 1$.
(a) Prove the inequality $3p^2 q + 3p^2 r + 2q^3 + 2r^3 \leq 2$.
(b) Also find the smallest possible value of $3p^2 q + 3p^2 r + 2q^3 + 2r^3$. Specify exactly when the smallest and the largest possible value is achieved.

Let $A$ be a non-empty finite sequence of $n$ distinct integers $a_{1} < a_{2} < \cdots < a_{n}$. Define
$$A + A = \left\{ a_{i} + a_{j} \mid 1 \leq i, j \leq n \right\}$$
i.e, the set of all pairwise sums of numbers from $A$. E.g., for $A = \{1,4\}$, $A + A = \{2,5,8\}$.
(a) Show that $|A + A| \geq 2n - 1$. Here $|A + A|$ means the number of elements in $A + A$.
(b) Prove that $|A + A| = 2n - 1$ if and only if the sequence $A$ is an arithmetic progression.
(c) Find a sequence $A$ of the form $0 < 1 < a_{3} < \cdots < a_{10}$ such that $|A + A| = 20$.

We want to construct a nonempty and proper subset $S$ of the set of non-negative integers. This set must have the following properties. For any $m$ and any $n$,
if $m \in S$ and $n \in S$ then $m + n \in S$ and if $m \in S$ and $m + n \in S$ then $n \in S$.
For each statement below, state if it is true or false.
(i) 0 must be in $S$.
(ii) 1 cannot be in $S$.
(iii) There are only finitely many ways to construct such a subset $S$.
(iv) There is such a subset $S$ that contains both $2015^{2016}$ and $2016^{2015}$.

A function $g$ satisfies the property that $g(k) = 3k + 5$ for each of the 15 integer values of $k$ in $[1,15]$.
For each statement below, state if it is true or false.
(i) If $g(x)$ is a linear polynomial, then $g(x) = 3x + 5$.
(ii) $g$ cannot be a polynomial of degree 10.
(iii) $g$ cannot be a polynomial of degree 20.
(iv) If $g$ is differentiable, then $g$ must be a polynomial.

Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers.
$T =$ set of slopes of all possible tangents to the graph of $f$.
$S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$.
For each statement below, state if it is true or false.
(i) If $f$ is differentiable, then $S \subset T$.
(ii) If $f$ is differentiable, then $T \subset S$.
(iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere.
(iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.

The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $$f(n) = n + \lfloor \sqrt{n} \rfloor$$ where $\lfloor k \rfloor$ denotes the nearest integer smaller than or equal to $k$. For example, $\lfloor \pi \rfloor = 3$, $\lfloor 4 \rfloor = 4$. Prove that for every natural number $m$ the following sequence contains at least one perfect square $$m, f(m), f^{2}(m), f^{3}(m), \ldots$$ The notation $f^{k}$ denotes the function obtained by composing $f$ with itself $k$ times, e.g., $f^{2} = f \circ f$.

Each integer is colored with exactly one of three possible colors - black, red or white satisfying the following two rules: the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black.
(a) Show that the negative of a white number must be colored black and the sum of two black numbers must be colored white.
(b) Determine all possible colorings of the integers that satisfy these rules.

You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
(a) A line segment has its endpoints on opposite edges of the hexagon. Show that it passes through the center of the hexagon if and only if it divides the two edges in the same ratio.
(b) Suppose a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that center of the square is same as that of the hexagon.
(c) Suppose the side of the hexagon is of length 1. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.

List in increasing order all positive integers $n \leq 40$ such that $n$ cannot be written in the form $a^{2} - b^{2}$, where $a$ and $b$ are positive integers.

Answer the following questions
(a) A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_{1}, \ldots, a_{k}$, each $a_{i} > 1$, such that $$\frac{1}{a_{1}} + \cdots + \frac{1}{a_{k}} = 1.$$ Show that if $k$ is stable then $k+1$ is also stable. Using this or otherwise, find all stable numbers.
(b) Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^{*}(y) := \max_{x \in A}\{yx - f(x)\}$$ whenever the above maximum is finite.
For the function $f(x) = -\ln(x)$, determine the set of points for which $f^{*}$ is defined and find an expression for $f^{*}(y)$ involving only $y$ and constants.

Let $f$ be a function on nonnegative integers defined as follows $$f(2n) = f(f(n)) \quad \text{and} \quad f(2n+1) = f(2n)+1$$
(a) If $f(0) = 0$, find $f(n)$ for every $n$.
(b) Show that $f(0)$ cannot equal 1.
(c) For what nonnegative integers $k$ (if any) can $f(0)$ equal $2^{k}$?

For how many natural numbers $n$ is $n^{6} + n^{4} + 1$ a square of a natural number?

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.

[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful.
(i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$.
(ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$).
(iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$.
Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.