Suppose that $(f_{n})_{n=1}^{\infty}$ is a sequence of continuous functions on the interval $[0,1]$ such that $$\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if}\;n=m\\ 0 & \text{if} \;n\ne m \end{cases}$$ and $\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty$. Show that there exists no subsequence $(f_{n_{k}})$ of $(f_{n})$ such that $\lim_{k\to \infty}f_{n_{k}}(x)$ exist for all $x\in [0,1]$.

For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions. 1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$. 2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$. Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following. For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]

Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\] [i]Proposed by Sutanay Bhattacharya[/i]

A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$, point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$?

The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_{n}$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n\geq 0$, $d$ divides $F_{n+2020}-F_n$. Compute the remainder when $d$ is divided by $1001$. [i]Proposed by Ankit Bisain[/i]

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

On an infinite square grid we place finitely many [i]cars[/i], which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a [i]move[/i], one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times. [i]Nikolai Beluhov[/i]

Let $\{a_k\}_{k\geq 0}$ be a sequence given by $a_0 = 0$, $a_{k+1}=3\cdot a_k+1$ for $k\in \mathbb{N}$. Prove that $11 \mid a_{155}$

The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Show that every natural number $n\leq2^{1\;000\;000}$ can be obtained first with 1 doing less than $1\;100\;000$ sums; more precisely, there is a finite sequence of natural numbers $x_0,\ x_1,\dots,\ x_k\mbox{ with }k\leq1\;100\;000,\ x_0=1,\ x_k=n$ such that for all $i=1,\ 2,\dots,\ k$ there exist $r,\ s$ with $0\leq{r}\leq{s}<i$ such that $x_i=x_r+x_s$.

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.