Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions: (a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$; (b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$. Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]

Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

Two cyclists leave simultaneously a point $P$ in a circular runway with constant velocities $v_1, v_2 (v_1 > v_2)$ and in the same sense. A pedestrian leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1+v_2}{12}$ . If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist $91$ seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him $187$ seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.

Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly one of the trinomials $x^2-ax+b$ or $x^2-bx+a$ from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

[u]Set 1[/u] [b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs? [b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$? [b]G3.[/b] Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had. [u]Set 2[/u] [b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? [b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$. [b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$. [u]Set 3[/u] [b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have? [b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$. [b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$: $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real

Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$ \left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Find all functions $f:\mathbb R^{+} \longrightarrow \mathbb R^{+}$ so that $f(xy + f(x^y)) = x^y + xf(y)$ for all positive reals $x,y$.

A book contains $30$ stories. Each story has a different number of pages under $31$. The first story starts on page $1$ and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?

Let $f(x) = x^2 + 2018x + 1$. Let $f_1(x)=f(x)$ and $f_k(x)=f(f_{k-1}(x))$ for all $k\geqslant 2$. Prove that for any positive integer $n{}$, the equation $f_n(x)=0$ has at least two distinct real roots.

Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties: $\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$, $\bullet$ $f(30) = 1$, and $\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$. Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?

Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that $$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$ When does equality hold? [i](Walther Janous)[/i]

Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.

Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$). Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.

Find all non-negative real numbers $x, y, z$ satisfying $x^2y^2 + 1 = x^2 + xy$, $y^2z^2 + 1 = y^2 + yz$ and $z^2x^2 + 1 = z^2 + xz$.

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

There are relatively prime positive integers $s$ and $t$ such that $$\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}$$ Find $s + t$.

[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite. [b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime. [b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$. [b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$. [b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$. [b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$. [b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power. [b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

One considers the set $$A = \left\{ n \in N^* \big | 1 < \sqrt{1 + \sqrt{n}} < 2 \right\}$$ a) Find the set $A$. b) Find the set of numbers $n \in A$ such that $$\sqrt{n} \cdot \left| 1-\sqrt{1 + \sqrt{n}}\right| <1 ?$$