In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(n+1)>\frac{f(n)+f(f(n))}{2}\] for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.

The rows $x_n$ and $y_n$ of positive real numbers are such that: $x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$ for each positive integer $n$. Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9

Evaluate \[\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\sqrt{2055+\ldots}}}}\]

A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism? [i]Proposed by Deyuan Li and Andrew Milas[/i]

$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear. [i]Proposed by chengbilly[/i]

Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$.

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$: (i) $f((xy)^p)=(f(x)f(y))^p$ (ii)$f((x+y)^q)=(f(x)+f(y))^q$ Find the function $f$.

For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

A possitive integer is called [i]special[/i] if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$-digit [i]special[/i] numbers (b) Are there $2000$-digit [i]special[/i] numbers?

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

Prove that in a group of $50$ people there are always two who have an even number (possibly zero) of common acquaintances within the group. (SI Tokarev)

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both? $\text{(A)}\ 18 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 25 \qquad \text{(E)}\ 27$

We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties: (i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$; (ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then $ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$; (iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$. A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties: If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$; If $ s_i\geq 0$, then $ f(S)\geq 0$; $ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$; $ f(ST)\equal{}f(S)f(T)$, $ f(S')\equal{}f(S)$ Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.

Let $P$ be a point inside isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $$\angle{PAD}=\angle{PDA}=90^\circ-\angle{BPC}.$$ If $PA=14, AB=18,$ and $CD=28,$ compute the area of $ABCD.$

Given a regular pentagon of area $1$, a pivot line is a line not passing through any of the pentagon's vertices such that there are $3$ vertices of the pentagon on one side of the line and $2$ on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points.

Let $ P(x)$ be a polynomial such that the following inequalities are satisfied: \[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\] and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\] Prove that for every positive natural number $ n,$ $ P(n)$ is positive.