Two men, $A$ and $B$, set out from town $M$ to town $N$, which is $15$ km away. Their walking speed is $6$ km/hr. They also have a bicycle which they can ride at $15$ km/hr. Both $A$ and $B$ start simultaneously, $A$ walking and $B$ riding a bicycle until $B$ meets a pedestrian girl, $C$, going from $N$ to $M$. Then $B$ lends his bicycle to $C$ and proceeds on foot; $C$ rides the bicycle until she meets $A$ and gives $A$ the bicycle which $A$ rides until he reaches $N$. The speed of $C$ is the same as that of $A$ and $B$. The time spent by $A$ and $B$ on their trip is measured from the moment they started from $M$ until the arrival of the last of them at $N$. a) When should the girl $C$ leave $N$ for $A$ and $B$ to arrive simultaneously in $N$? b) When should $C$ leave $N$ to minimize this time?

A function $ f$ from $ [0,1]$ to $ \mathbb{R}$ has the following properties: $ (i)$ $ f(1)\equal{}1;$ $ (ii)$ $ f(x) \ge 0$ for all $ x \in [0,1]$; $ (iii)$ If $ x,y,x\plus{}y \in [0,1]$, then $ f(x\plus{}y) \ge f(x)\plus{}f(y)$. Prove that $ f(x) \le 2x$ for all $ x \in [0,1]$.

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is: $ \textbf{(A)}\ 36.5 \qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 37.2\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 37.52 $

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

Find all polynomials $P(x)$ of degree $1$ such that $\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$

The numbers $12,14,37,65$ are one of the solutions of the equation $xy-xz+yt=182$. What number corresponds to which letter?

Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.

Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$. [i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]

Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied: (a) $xf(x,y,z) = zf(z,y,x)$, (b) $f(x, ky, k^2z) = kf(x,y,z)$, (c) $f(1, k, k+1) = k+1$. ([i]United Kingdom[/i])

Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$

Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$. Prove that $x + y + z$ is an integer.

Find the real numbers $ m $ which have the property that the equation $$ x^2-2mx+2m^2=25 $$ has two integer solutions.

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$. Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$. Determine the largest possible value of $T$.

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.

[u]Round 1[/u] [b]p1.[/b] A positive integer is said to be transcendent if it leaves a remainder of $1$ when divided by $2$. Find the $1010$th smallest positive integer that is transcendent. [b]p2.[/b] The two diagonals of a square are drawn, forming four triangles. Determine, in degrees, the sum of the interior angle measures in all four triangles. [b]p3.[/b] Janabel multiplied $2$ two-digit numbers together and the result was a four digit number. If the thousands digit was nine and hundreds digit was seven, what was the tens digit? [u]Round 2[/u] [b]p4.[/b] Two friends, Arthur and Brandon, are comparing their ages. Arthur notes that $10$ years ago, his age was a third of Brandon’s current age. Brandon points out that in $12$ years, his age will be double of Arthur’s current age. How old is Arthur now? [b]p5.[/b] A farmer makes the observation that gathering his chickens into groups of $2$ leaves $1$ chicken left over, groups of $3$ leaves $2$ chickens left over, and groups of $5$ leaves $4$ chickens left over. Find the smallest possible number of chickens that the farmer could have. [b]p6.[/b] Charles has a bookshelf with $3$ layers and $10$ indistinguishable books to arrange. If each layer must hold less books than the layer below it and a layer cannot be empty, how many ways are there for Charles to arrange his $10$ books? [u]Round 3[/u] [b]p7.[/b] Determine the number of factors of $2^{2019}$. [b]p8.[/b] The points $A$, $B$, $C$, and $D$ lie along a line in that order. It is given that $\overline{AB} : \overline{CD} = 1 : 7$ and $\overline{AC} : \overline{BD} = 2 : 5$. If $BC = 3$, find $AD$. [b]p9.[/b] A positive integer $n$ is equal to one-third the sum of the first $n$ positive integers. Find $n$. [u]Round 4[/u] [b]p10.[/b] Let the numbers $a,b,c$, and $d$ be in arithmetic progression. If $a +2b +3c +4d = 5$ and $a =\frac12$ , find $a +b +c +d$. [b]p11.[/b] Ten people playing brawl stars are split into five duos of $2$. Determine the probability that Jeff and Ephramare paired up. [b]p12.[/b] Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and for all $n\ge 2$, $$F_n = \left \lceil \frac{F_{n-1}+F_{n-2}}{2} \right \rceil +1,$$ where $\lceil r \rceil$ denotes the least integer greater than or equal to $r$ . Find $F_{2019}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$

Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$, $$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that $$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$